diff --git "a/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/load_file.txt" "b/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/5dAzT4oBgHgl3EQfu_2Z/content/tmp_files/load_file.txt" @@ -0,0 +1,970 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf,len=969 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='01700v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='GT] 4 Jan 2023 Non-Adaptive Matroid Prophet Inequalities Kanstantsin Pashkovich, Alice Sayutina University of Waterloo Department of Combinatorics & Optimization 200 University Avenue West Waterloo, ON, Canada N2L 3G1 Abstract We consider the matroid prophet inequality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This problem has been extensively studied in the case of adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In par- ticular, there is a tight 2-competitive mechanism for all matroids [KW12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, it is not known what classes of matroids admit non-adaptive mechanisms with constant guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recently, in [CGKM20] it was shown that there are constant-competitive non-adaptive mechanisms for graphic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this work, we show that various known classes of matroids admit constant-competitive non-adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 1 Introduction Let us consider the classical prophet inequality problem [KS77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A gambler observes a sequence of non-negative independent random variables X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn, which correspond to a sequence of values for n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The gambler knows the distributions of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The gambler is allowed to accept at most one item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and the gambler is interested in maximizing the value of the accepted item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, the gambler cannot simply select an item of the maximum value, because the values of the n items are revealed to the gambler one by one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and each time a value of the current item is revealed the gambler has to make an irrevocable choice whether to accept the current item or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' What stopping rule the gambler should use to maximize the expected value of the item they accept?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The gambler knows only the distributions of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn while a prophet knows the realization of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, in contrast to the gambler the prophet can always obtain the maximum item’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The seminal result of Krengel and Sucheston [KS77] showed that the gambler can obtain at least a half of the expected value obtained by the prophet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The classical prophet inequality problem led to a series of works on different variants of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A natural variant of the problem is the generalization 1 of the problem where a gambler can buy more than one item, but the set of bought items should satisfy a known feasibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Formally, let us be given a collection S ⊆ 2[n] of item sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then both gambler and prophet can select any item set S from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So S defines a feasibility constraint for selecting items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In most standard examples of feasibility constraints, S can be defined as a collection of all item sets with cardinality at most k for some natural number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' More generally S can be defined as a collection of all independent sets in some matroid, in this case we speak about the matroid prophet inequality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The result in [SC84] showed that in the single-item setting a gambler can obtain at least half of the prophet value by using the following threshold-rule: determine a constant T as a function of known distributions and accept the first item exceeding T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This rule results in a 2-competitive mechanism, similar to the adaptive approach of [KS77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that this approximation guarantee is known to be tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There is also another method to set a threshold presented in [KW12], which also results in a 2-competitive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This was extended by Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' in [CHMS10] and [CGKM20] to the setting of several items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The results presented in [KW12] further extend to the matroid prophet in- equalities, where accepted items need to form an independent set in a known matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' It leads to a 2-competitive mechanism for every matroid, matching the single-item setting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, unlike the mechanism in the single-item setting, the mechanism for matroids is adaptive: the thresholds for items are computed based on the previously accepted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By [KW12], there also exists a constant-competitive adaptive mechanism for feasibility constraints defined as an intersection of constant number of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The mechanism by Kleinberg and Weinberg was further extended to a 2-competitive mechanism for polyma- troids by Dütting and Kleinberg in [DK15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Gravin and Wang [GW19] studied the bipartite matching version of this problem: in their version, the arriving items are the edges of the (known) bi- partite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Gravin and Wang obtained a 3-competitive non-adaptive mech- anism, which assigns thresholds to each vertex in the graph and an edge is accepted only if its weight is at least the sum of the thresholds associated with its endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Feldman, Svensson and Zenklusen [FSZ16] studied online item selection mechanisms called “online contention resolution schemes" (OCRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' They showed that given special properties, OCRS translate directly into a constant-competitive prophet inequality for the same problem against almighty adversary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' an ad- versary which knows in advance realizations of all the items and the random bits generated by an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' As a result, they develop a constant-competitive mechanism for prophet inequalities of the intersection of a constant number of matroids, knapsack and matching constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Those mechanisms are “almost” non-adaptive in a sense that they fix thresholds for all items, however their mech- anisms also impose a subconstraint: an item cannot be accepted if together with previously accepted items it forms one of the fixed forbidden sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Finally, in a later version of their paper [FSZ21], they prove that pure non- adaptive mechanisms cannot achieve a constant-competitive approximation even against a “normal” adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' They construct a family of gammoid matroids 2 showing a lower bound of Ω(log n/ log log n) for a guarantee of non-adaptive mechanisms on gammoids with n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There have been works studying similar setups with other goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [CHMS10] studied a Bayesian item selection process in a fixed item ar- rival order or against an adversary in control of the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' They studied it from a perspective of the revenue maximization for the auctioneer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The per- formance is constant-competitive compared to the well-known Myerson mech- anism [Mye81], which achieves the largest possible expected revenue among truthful mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The mechanism by Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [CHMS10] has an ad- vantage that it determines static thresholds together with a subconstraint so that each agent can be offered take-it-or-leave-it prices in an online fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recently, Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [CGKM20] developed a 32-competitive non-adaptive mechanism for graphic matroids against adversary item ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Our results First, we list the known results for non-adaptive mechanism that were mentioned in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 1 (Uniform Rank 1 Matroid [SC84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 2-competitive non-adaptive mechanism for single-item setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 2 (Graphic Matroid [CGKM20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 32-competitive non-adaptive mechanism for graphic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us list our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In case of a simple graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' a graph with no parallel edges or loops, we can slightly improve the above theorem by considering essentially the same mechanism as [CGKM20] but considering a different scaling of a point from the matroid polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We provide this result for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 16-competitive non-adaptive mechanism for graphic matroids in the case of simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Furthermore, the mechanism [CGKM20] can be generalized to the setting of k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This result we need later to obtain Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 4 (k-Column Sparse Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a (2k+2k)-competitive non-adaptive mechanism for k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that Theorem 2 of [CGKM20] follows from Theorem 4, since a graphic matroid is also a 2-column sparse matroid over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Using analogous approach to the one in [Sot13], we also develop a mechanism for cographic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 5 (Cographic Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 6-competitive non-adaptive mechanism for cographic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The approach in [Sot13] immediately leads to the following result for γ-sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3 Theorem 6 (γ-Sparse Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a γ-competitive non-adaptive mechanism for γ-sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Combining the above results and using classic Seymour’s decomposition re- sults we obtain the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 7 (Regular Matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 256-competitive non-adaptive mechanism for regular matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Subject to the Structural Hypothesis 1 due to Geelen, Gerards and Whittle, which is stated later, we can also derive the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Subject to the Structural Hypothesis 1, for every prime number p there exists a constant-competitive mechanism for every proper minor-closed class of matroids representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We also would like to observe that some of the recent results on “single sample prophet inequalities” (SSPI) lead to non-adaptive constant-competitive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For this, the single sample required by the gambler in SSPI can be directly sampled by our gambler from the available distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In partic- ular, the results in [AKW19] and [CFPP21] on laminar matroids and truncated partition matroids inspired by the mechanism in [MTW16] lead to non-adaptive mechanisms for prophet inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain these results, it is crucial that the mechanism in [MTW16] does not involve subconstraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' each item is accepted as long as the item is not in the “observation phase”, the item passes its threshold based only on the “observation phase” and the item forms an in- dependent set with previously accepted items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In comparison, it is not clear how from the results on regular matroids in [AKW19] based on the mechanism in [DK14] one can obtain non-adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So the following results can be directly obtained from [AKW19] and [CFPP21], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 9 (Laminar Matroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='6-competitive non-adaptive mechanism for laminar matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 10 (Truncated Partition Matroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists an 8-competitive non-adaptive mechanism for truncated partition matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2 Comparison to known results Our results for cographic matroids and k-column sparse matroids are obtained through modifications of the arguments in [Sot13] and [CGKM20], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The results on regular matroids and minor-closed families of matroids follow the approach outlined in [HN20] for the secretary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' As necessary building blocks we use our results for cographic and 2-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that a biggest challenge for us is the compatibility of non-adaptive thresholds with contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Indeed, standard tools for deriving mechanisms for contraction 4 minors need subconstraints, while subconstraints are not permitted in non- adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain our results, we resolve this issue only in the context of matroids representable over finite fields, see arguments in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' It would be interesting to see whether analogous results for contraction minors hold with no assumption about representability over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3 Preliminaries In this paper, we consider the matroid prophet inequality problem, where items arrive online in adversarial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, the adversary knows the distributions of all X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn and knows the gambler’s mechanism, but the realization of X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn is not known to the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Based on the available information, the adversary can decide on the order in which items and their values are observed by the gambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Prophet inequality Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M be a matroid on the ground set [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , n}, where [n] corresponds to n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let ⃗X := (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) be non-negative independent random variables representing the values of these n items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every subset of items S ⊆ [n] we define its weight as follows w(S) := � i∈S Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let PROPHM be the random variable corresponding to the value obtained by the prophet PROPHM := max S∈I(M) w(S) , where I(M) is a collection of independent sets for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let EPROPHM be the expectation of the value obtained by prophet EPROPHM := E[PROPHM] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us be given a number α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We call a mechanism α-competitive (alternatively, we say that the mech- anism guarantees an α-approximation) on the matroid M if the expected value obtained by the gambler via this mechanism is at least 1 αEPROPHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We call a mechanism α-competitive (alternatively, we say that the mech- anism guarantees an α-approximation) on the matroid class M if this mechanism is α-competitive for every matroid M ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Non-adaptive mechanism We say that a mechanism is non-adaptive if it has the following structure: Given the distributions of ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn), the mechanism determines the values of thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn), where each Ti, i ∈ [n] is a real number or +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If the value of item i ∈ [n] is observed, the gambler accepts the item i if and only both conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' the observed value of Xi is at least Ti 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' the item i together with all previously selected items forms an inde- pendent set with respect to the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that a non-adaptive mechanism does not change thresholds during its course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So, none of the thresholds depends on the realization of ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Another crucial feature of a non-adaptive mechanism is that the mechanism works only with the original matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A non-adaptive mechanism does not allow us to define a new matroid M ′, such that a set of items is independent in M ′ only if it is independent in M, and modify the condition (2) based on M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this work, we focus on non-adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' From here and later we use the term mechanism to refer to non-adaptive mechanisms exclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this work, non-adaptive mechanisms are allowed to make non- deterministic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, we allow a non-adaptive mechanism to construct the thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) non-deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To measure the performance of such a mechanism we use the expected total value, where the expectation is taken not only with respect to ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) but also with respect to ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='3 Matroids We provide a review of matroids here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Experienced readers should consider skipping or skimming this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For further results about matroids, consider consulting [Oxl06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid M = (E, S) is a pair of a finite ground set E and a collection S ⊆ 2E of independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The collection S ⊆ 2E of subsets of E satisfies the following conditions: (i) Empty set is an independent set, so ∅ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' (ii) The collection S is closed with respect to taking subsets, so for all A ⊆ B ⊆ E if B is in S then A is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' (iii) The collection S satisfies so called augmenation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In other words, for all A, B ⊆ E such that A, B ∈ S and |A| > |B|, there exists c ∈ A \\ B such that B ∪ {c} ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6 A subset of E is called dependent if it is not in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The inclusion-maximal independent sets are called bases and the inclusion-minimal dependent sets are called circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every two bases, their cardinalities are equal: for every bases A and B of M we have |A| = |B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A rank function for the matroid M is a function rM : 2E → N such that for every A ⊆ E the value rM(A) equals the cardinality of an inclusion-maximal independent subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the cases when the choice of the matroid is clear from the context, we write r instead of rM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a matroid M, we can define the dual matroid M ∗ over the same ground set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A set A is independent for matroid M ∗ if and only if E \\ A contains a basis of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' An element c ∈ E is called a loop in M if rM(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' An element c ∈ E is called a free element in M if rM∗(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To put it another way, an element c is free, if and only if for every set A, which is independent in M, A∪{c} is also independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that elements c and d ∈ E are parallel in matroid M, denoted by c ∥ d, if rM(c) = rM(d) = rM({c, d}) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' One can show that “being parallel” defines an equivalence relation on the non-loop elements of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid is called simple if it has no loops and no parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M = (E, S) be a matroid and A ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The contraction of M by A, de- noted as M/A, is a matroid over ground set E\\A with the following independent sets {S ⊆ E \\ A : S ∪ A′ ∈ S} , where A′ is an inclusion-maximal independent subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The restriction of M to A, denoted as M |A or M \\ A, is a matroid over the ground set A where a set S ⊆ A is independent in M |A if and only if it is independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid M ′ is called a simple version of M if M ′ is obtained from M by deleting all loops and contracting every parallel class of elements into a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For matroids M, N, we say that N is a minor of M = (E, S) if N is isomorphic to M/A\\B for some disjoint sets A, B ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid class M is called minor-closed if for any M ∈ M every minor of M is also in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now list some of the classical examples of matroids, which were ex- tensively studied in the context of various mathematical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A uniform matroid M = (E, S) of rank k is matroid where S := {A ⊆ E : |A| ≤ k} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' When |E| = n, we denote the uniform matroid of rank k as Uk,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A graphic matroid over graph G = (V, E) is a matroid M = (E, S), where S := {A ⊆ E : A is acyclic} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The graphic matroid over graph G is denoted as M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 7 A cographic matroid over graph G = (V, E) is a dual matroid M = (E, S) to the graphic matroid over the same graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this case we have S := {A ⊆ E : (V, E\\A) has the same number of components as (V, E)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A vector matroid M = (E, S) is a matroid such that there is a vector space V and a map φ : E → V satisfying S := {A ⊆ E : multiset φ(A) is linearly independent} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a field F, we say that M is representable over field F if M is iso- morphic to the vector matroid where V is a vector space over field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid is called regular if it is representable over every field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A matroid is called binary if it is representable over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A k-column sparse matroid M = (E, S) is a matroid such that there is a field F and dimension m and a map φ : E → Fm such that S := {A ⊆ E : multiset φ(A) is linearly independent over F} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and moreover φ(c) ∈ Fm has at most k nonzero coordinates for every c ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A γ-sparse matroid M = (E, S) is a matroid such that the inequality |S| ⩽ γrM(S) holds for every S ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A laminar matroid M = (E, S) is a matroid such that there exists a laminar family F over the ground set E and there are numbers cF ∈ N, F ∈ F such that S := {A ⊆ E : |A ∩ F| ≤ cF for every F ∈ F} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, if F = {E, E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ek}, where E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ek form a partition of the ground set E, then M is called a truncated partition matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall, that a family F is called laminar if for every A, B ∈ F we have A ⊆ B or B ⊆ A or A ∩ B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a matroid M = (E, S) we can define the corresponding polytope PM ⊆ RE as the convex hull of points corresponding to the characteristic vectors of independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The polytope PM is known to admit the following outer description [Sch03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' PM = {x ∈ RE : x ≥ 0 and x(S) ≤ rM(S) for every S ⊆ E} , where x(S) stands for � c∈S xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For a matroid M = (E, S) and a set A ⊆ E we can define the closure of A as the following set clM(A) := {c ∈ E | rM(A ∪ {c}) = rM(A)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 8 For a matroid M = (E, S), we call the following function ⊓M : E × E → Z a local connectivity function ⊓M(X, Y ) = r(X) + r(Y ) − r(X ∪ Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The following function λM : E → Z⩾0 is called a connectivity function λM(X) := ⊓M(X, E \\ X) = r(X) + r(E \\ X) − r(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Informally, connectivity functions measure dependence with respect to the matroid between parts of the ground set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To illustrate it, let us consider the connectivity function for vector matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose M = (E, S) is a vector matroid defined by a vector space V and a map φ : E → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have λM(S) =r(S) + r(E \\ S) − r(E) = dim(span φ(S)) + dim(span φ(E \\ S)) − dim(φ(E)) = dim ((span φ(S)) ∩ (span φ(E \\ S))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4 Ex-ante relaxation to the matroid polytope The goal of ex-ante relaxation [FSZ16] or [CGKM20] is to reduce the origi- nal problem to the problem where item values are distributed as independent Bernoulli random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that both problems are using the same ma- troid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the original problem item values ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) are independent ran- dom variables with known distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For i ∈ [n] let Fi be the cumulative distribution function of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The reduction of the original problem to a new problem is done using a point p in the matroid polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us first show that there is a point p ∈ PM with properties that prove to be desirable later following the argumentation in [CGKM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given a matroid M over the ground set [n] and random variables ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn), there exists p ∈ PM such that EPROPHM ⩽ n � i=1 piti , where ti := E[Xi | Xi ⩾ F −1 i (1 − pi)] for every i ∈ [n]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Iopt be a random variable indicating an optimal independent set in M with respect to ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In case when for some realization of ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn) there are several optimal independent sets, Iopt can be selected as any of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For i ∈ [n], let pi be the probability that element 1Here, we assume that for every i ∈ [n] the event Xi = F −1 i (1 − pi) happens with the zero probability, which is true for all continuous distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In case of discrete distributions one needs to introduce appropriate tie-breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 9 i is in Iopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , pm) is a convex combination of independent sets of M, and so lies in PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Due to EPROPHM = E[� i∈Iopt Xi], it remains to show that E[ � i∈Iopt Xi] ⩽ n � i=1 piti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We have E[ � i∈Iopt Xi] = n � i=1 P[i ∈ Iopt]E[Xi | i ∈ Iopt] = n � i=1 piE[Xi | i ∈ Iopt] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every i ∈ [n] we have that ti and E[Xi | i ∈ Iopt] are expectations of the same random variable Xi but conditioned on the event Xi ⩾ F −1 i (1−pi) and on the event i ∈ Iopt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that the probability of both these events equals pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, the expectation of Xi conditioned on Xi ⩾ F −1 i (1 − pi) is the “largest” conditional expectation of Xi on an event of probability pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have piE[Xi | i ∈ Iopt] ⩽ piti for every i ∈ [n] and so we get the desired inequality n � i=1 piE[Xi | i ∈ Iopt] ⩽ n � i=1 piti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us show how one can use the point p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , pn) guaranteed by Lemma 1 to reduce the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us define independent Bernoulli random variables ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) as follows, for each i ∈ [n] X′ i = � ti with probability pi 0 with probability 1 − pi , where ti := E[Xi | Xi ⩾ F −1 i (1 − pi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that we have a non-adaptive mechanism for the original ma- troid M and item values ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n), which sets nonnegative thresholds ⃗T ′ = (T ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , T ′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By definition of ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n), for every i ∈ [n] the exact value of T ′ i is not relevant per se, but it is crucial whether ti ≥ T ′ i or ti < T ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If for some i ∈ [n] we have T ′ i > ti then this item i is “inactive” and so is never selected by the gambler working with M and ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The key is to construct a non-adaptive mechanism for the original matroid M and item values ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) with positive thresholds ⃗T ′ = (T ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , T ′ n) such that for each item i ∈ [n] the probability that i is selected by the gambler is at least αpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we can use such a non-adaptive mechanism for the original matroid M and item values ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) to construct a non-adaptive α-competitive mechanism for the same matroid M and random variables ⃗X = 10 (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us define the thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) as follows, for every i ∈ [n] Ti := � +∞ if ti < T ′ i F −1 i (1 − pi) otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To see that the thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) lead to an α-competitive mech- anism for M and ⃗X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Xn), let us couple random variables X′ i with random variables Xi as follows X′ i := � ti if Xi ≥ F −1 i (1 − pi) 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that ⃗X′ = (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , X′ n) are independent Bernoulli random variables, where for each i ∈ [n] the variable X′ i equals ti with probability pi and equals 0 with probability 1 − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' When ⃗X′ are coupled with ⃗X this way, Xi and X′ i have the same expected value when conditioned on X′ i being ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The mechanism with thresholds ⃗T selects an item i ∈ [n] when run for ⃗X only if the mechanism with thresholds ⃗T ′ selects the item i when run for ⃗X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, for both of these algorithms, conditionally on the event that the item i is selected the expected value of i equals ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now, α-competitiveness guarantee of the thresholds ⃗T for M and ⃗X follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4 Graphic and k-column sparse matroids First, we construct a 16-competitive non-adaptive mechanism for graphic ma- troids without parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Our construction is done through the ex-ante re- laxation to the matroid polytope, following the works in [FSZ16] or [CGKM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Later, we present a constant-competitive non-adaptive mechanism for k-column sparse matroids whenever k is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Graphic matroids Now we are ready to provide a 16-competitive non-adaptive mechanism for graphic matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The provided mechanism is essentially the one constructed in [CGKM20] but with saving a factor of 2 in the guarantee, which is achieved by rescaling the point from the matroid polytope by 2 and not by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us be given a simple graph G = (V, E) and let us consider the corre- sponding graphic matroid M over the ground set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall that a subset of E is independent with respect to M if and only if it is acyclic in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us also assume that the graph G has n edges and so E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , en}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , pn) be a point in the polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we assume that for every i ∈ [n] the coordinate pi of p corresponds to the edge ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists an orientation of edges E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , en} in the graph G = (V, E) such that for every vertex v ∈ V we have � i∈[n]:ei∈δ−(v) pi ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that the average degree of a vertex in a forest on |V | vertices is at most (2|V | − 2)/|V | = 2 − 1/|V | ⩽ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us use this fact to prove the desired statement by induction on the number of vertices in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If the graph G has at most two vertices then the orientation is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Other- wise, since p is a convex combination of points corresponding to forests in G, we have that the average of the value � i∈[n]:ei∈δ(v) pi over all vertices v ∈ V is at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus there exists a vertex v ∈ V such that we have � i∈[n]:ei∈δ(v) pi ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We orient all edges incident to v as edges in δ−(v), so these edges are incoming with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we remove the vertex v and all edges incident to it and orient the remaining edges according to the orientation guaranteed by the inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we present an algorithm for graphic matroids of simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Algorithm 1 A non-adaptive 16-competitive mechanisms for graphic matroids of a simple graph 1: Let p be a point in the polytope PM so that the statement of Lemma 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Let the edges of the original graph G = (V, E) be oriented so that the statement of Lemma 2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently at random with probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4: Select a cut S ⊆ V uniformly at random, mark all edges not in [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] as “discarded".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] stands for the set of edges which are oriented such that their tail is in S and their head is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5: Set thresholds ⃗T = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Tn) as follows, for each i ∈ [n] Ti := � +∞ if ei is “discarded” F −1 i (1 − pi) otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every i ∈ [n], we have P[ei is selected | Xi ≥ Ti and ei is not “discarded”] ≥ 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that the vertex v is the head of the oriented edge ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us also assume that ei is not marked as “discarded” and Xi ≥ Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since the edge ei is not “discarded”, the edge ei is in the selected set [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, every not “discarded” edge incident to v has the vertex v as its head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, as long as no other edge with the head at the vertex v is selected by the gambler, the gambler has to select ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We claim, that with probability at least 1/2 no other edge with the head at v was selected by the gambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let I be the event indicating that "the gambler selected an edge ej, j ̸= i such that v is the head of ej", in other words “there is j ∈ [n], j ̸= i such that 12 v is the head of ej and Xj ≥ Tj and ej is not “discarded”".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let J indicate the event that "ei is not marked as “discarded” after the selection of the cut", in other words, "the head of ei is in S and the tail of ei is in S".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us show P[I | J] ≤ 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the union bound, we have P[I | J] ⩽ � j∈[n]\\{i}:ej∈δ−(v) P[Xj ≥ Tj and ej is not “discarded” | J] Note that for each edge ej ∈ δ−(v) we have P[Xj ≥ Tj|J] = pj and we also have P[ej is not “discarded”|J] = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that any edge is not “discarded” in Step 3 of Algorithm 1 with probability 1/2, and not “discarded” in Step 4 of Algorithm 1 with probability 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, since the probabilities are with respect to the edge ej ∈ δ−(v) and are counted conditioned on the event J, the conditioned probability of not being “discarded” in Step 4 of Algorithm 1 is 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, even conditioned on J the events "Xj ≥ Tj" and "ej is not “discarded”" are independent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we have � j∈[n]\\{i}:ej∈δ−(v) P[Xj ≥ Tj and ej is not “discarded” | J] ≤ � j∈[n]\\{i}:ej∈δ−(v) pj/4 ⩽ 1/2 , where the last inequality follows from the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We are ready to prove Theorem 3 by showing that Algorithm 1 is a 16- competitive for graphic matroids without parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 3 for every i ∈ [n] the probability of edge ei being accepted conditional on Xi ≥ Ti and being not “discarded” is at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Overall, the probability of edge ei being accepted is at least 1 16pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus mechanism guarantees at least �n i=1 1 16piti of the expected total value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 1, we have � i∈[n] 1 16piti ⩾ 1 16EPROPHM, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 k-column sparse matroids There are known constant-competitive mechanisms for k-column sparse ma- troids in the context of the secretary problem [Sot13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However they do not immediately lead to a non-adaptive mechanism of constant competitiveness guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The reason for that are not the updated thresholds but implicit changes to the considered matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, we present a constant competitive mechanism for k-column sparse matroid class for each constant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, graphic matroids form a subclass of 2-column sparse matroids .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Because of their significance, 2-column sparse matroids are also known in literature as represented frame matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Later, we use 2-column sparse matroids to prove results in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 13 Suppose M is a k-column sparse matroid over field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this section, we prove that there exists a (2k+2k)-competitive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose a k-sparse representation of M = (E, S) is defined by a map φ : E → Fd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, if for some element t ∈ E the vector φ(t) is a zero vector then c is a loop and therefore can be removed from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we consider an undirected hyper-multigraph G with vertex set [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Each matroid element t ∈ E induces a hyperedge et in this graph between non-zero coordinates of φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Formally, the hyperedge et is defined as follows et := {i ∈ [d] : φ(t)i ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that a vertex i ∈ [d] of the hyper-multigraph G is incident to every edge e of G such that i ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For a vertex i ∈ [d] we denote the collection of incident hyperedges by δ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The degree of a vertex i in the hyper-multigraph G equals |δ(i)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose I is an independent set of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then the average degree of a vertex is at most k when one considers the hyper-multigraph with vertices [d] and hyperedges {et : t ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that |I| ⩽ d because having more than d vectors in d-dimensional vector space Fd leads to a a linear dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since M is k-column sparse, we have that every edge in {et : t ∈ I} is incident to at most k vertices in [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, the total degree is at most kd and thus the average degree of a vertex is at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we consider orientations of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' An orientation of the graph G is a function ϕ which maps every edge et into one vertex of G incident to et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We call ϕ(et) to be the head of the edge et, and all other vertices, if any, to be tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every vertex i ∈ [d] we denote the set of incoming edges by δ−(i), formally δ−(i) = {et : ϕ(et) = i, t ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p be a point in the polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We assume that for every t ∈ E, the coordinate pt of p corresponds to the element t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists an orientation ϕ of hyperedges in the hyper-mulrigraph G such that for every vertex i ∈ [d] we have � t∈E:et∈δ−(i) pt ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The proof of Lemma 4 is analogous to the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us describe an algorithm for k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every t ∈ E we have P[t is selected | Xt ≥ Tt and t is not “discarded”] ≥ 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that item t ∈ E is accepted whenever Xt ≥ Tt and no other item was selected from non-discarded edges in δ−(ϕ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the union bound, for every event J we can upper bound the probability that P[there j ∈ E \\ {t} such that j is selected and ej ∈ δ−(ϕ(t)) | J] ⩽ � j∈E\\{t}:ej∈δ−(ϕ(t)) P[ej is not “discarded” and Xj ≥ Tj | J] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 14 Algorithm 2 A non-adaptive 2k+2k-competitive mechanisms for k-column sparse matroids 1: Let p be a point in the polytope PM so that the statement of Lemma 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Let the edges of the hyper-multigraph G be oriented so that the statement of Lemma 4 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3: For every edge ei ∈ E, i ∈ [n], mark the edge ei as “discarded" independently at random with probability 1 − 1 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4: Select a cut S ⊆ [d] uniformly at random, mark all edges not in [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] as “discarded”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] stands for the set of edges which are oriented such that all their tails are in S and their head is in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In particular, for t ∈ E we say that et lies in a cut [S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' S] with respect to the orientation ϕ if ϕ(et) ∈ S and for every i ∈ et \\ {ϕ(et)} we have i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5: Set thresholds {Tt : t ∈ E} as follows, for each t ∈ E Tt := � +∞ if t is “discarded” F −1 t (1 − pt) otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let J indicate the event that "et is not marked as “discarded” after the selection of the cut".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then for each j ∈ E \\{t} we have P[ej is not “discarded” and Xj ≥ Tj | J] ≤ 1 2kpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 4, we have � j∈E:ej∈δ−(ϕ(t)) pj ⩽ k, leading to the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that the proof of Lemma 5 is analogous to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We are ready to prove Theorem by showing that the Algorithm 2 is a 2k+2k-competitive for k-column sparse matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every item t ∈ E we have P[Xt ≥ Tt] = pt and P[t is not “discarded”] ≥ 1 2k+1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 5, we have that with probability at least 1/2 the item t is selected when it is not “discarded” and Xt ≥ Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus the expected total value of Algorithm 2 is at least � j∈E 1 2k+2kpjtj which is at least 1 2k+2kEPROPHM by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5 Cographic and gamma-sparse matroids 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Cographic matroids Let us revisit a mechanism of Soto [Sot13] for the cographic matroid secretary problem which is based on the following corollary of Edmond’s matroid parti- tioning theorem [Edm65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This mechanism leads to a non-adaptive mechanism for cographic matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 15 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G = (V, E) be a three edge-connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exist spanning trees H1, H2, H3 in G such that the union of their complements contains all the edges E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' E = (E \\ H1) ∪ (E \\ H2) ∪ (E \\ H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Algorithm 3 A non-adaptive 3-competitive mechanisms for cographic matroids in the case of three edge-connectivity 1: Let H1, H2 and H3 be the spanning trees as in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Uniformly at random select a spanning tree H∗ from H1, H2 and H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Set thresholds {Te : e ∈ E} as follows, for each e ∈ E Te := � +∞ if e is not in H∗ 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G = (V, E) be a three edge-connected graph and let M be the cographic matroid over G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then Algorithm 3 is a 3-competitive non-adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The expected total value of the mechanism provided by Algorithm 3 equals E[� e∈E\\H∗ Xe] which can be estimated as follows E[ � e∈E\\H∗ Xe] = 1 3E[ � i∈[3] � e∈E\\Hi Xe] ≥ 1 3E[ � e∈E Xe] ≥ 1 3EPROPHM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The next theorem provides a proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G = (V, E) be a graph and let M be the cographic matroid over G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then Algorithm 4 is a 6-competitive non-adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We can assume that G does not have bridges, because every such bridge is a loop in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus these edges can be selected neither by the gambler nor by the prophet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So we can assume G = G′ and M = M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the case when each connected component of G is three edge-connected, then Algorithm 4 runs Algorithm 3 for each component to obtain a 3-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Otherwise, there is one or more pairs of edges e,e′ such that {e, e′} corre- sponds to a cut in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In this case, the edges e,e′ correspond to parallel elements of the cographic matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Algorithm 4 considers the partition of E into classes of parallel elements C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us construct the matroid M ′′ from M by contracting all but one edge in each class C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that the ground set of M ′′ has k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Abusing the notation we refer to these elements of the ground set as 16 Algorithm 4 A non-adaptive 6-competitive mechanisms for cographic matroids 1: Delete all loops of M to obtain a matroid M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Remove all bridges from G = (V, E) and obtain a graph G′ = (V ′, E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2: Let C1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck be equivalence classes of M ′ with respect to the relation of being parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Construct the matroid M ′′ from M ′ by contracting all but one edge in each class C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that the ground set of M ′′ has k elements and matroid M ′′ is the cographic matroid over a graph G′′, where each connected component of G′′ is three edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Abusing the notation we refer to the elements of the ground set of M ′′ as C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3: Let H1, H2 and H3 be forests in G′′ such that the restriction of H1, H2 and H3 to each connected component of G′′ satisfies Proposition 6 for the respective connected component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 4: Uniformly at random select a forest H∗ from H1, H2 and H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5: For each i ∈ [k] select thresholds T e, e ∈ Ci according to Theorem 1 when the gambler is allowed to accept only one item of Ci and the distributions of Xe, e ∈ Ci are the same as original distributions of values for e ∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6: Set thresholds {Te : e ∈ E} as follows, for each e ∈ E Te := � T e if e ∈ Ci and Ci ∈ H∗ for some i ∈ [k] +∞ otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The matroid M ′′ is isomorphic to the cographic matroid over a graph G′′, where each connected component of G′′ is three edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Following Lemma 6, Algorithm 4 constructs forests H1, H2, H3 for the graph G′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So Algorithm 4 leads us to a 6-competitive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Indeed, the prophet with M and with the original distributions of Xe, e ∈ E performs exactly as the prophet with M ′′ and with the corresponding distributions of X′′ i := maxe∈Ci Xe, i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By selecting forests in Algorithm 4 the gambler acheives in expectation E[� i∈[k] X′′ i ]/3 when all classes C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Ck are singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' However, for classes that are not singletons we need to take into account an- other 2 approximation factor with respect to the prophet, who can achieve the expected value E[X′′ i ] for each i ∈ [k], while the gambler is guaranteed in ex- pectation to achieve only E[X′′ i ]/2 for each i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Gamma-sparse matroids Let us also revisit a mechanism of Soto [Sot13] for γ-sparse matroids to verify that it directly leads to a non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M = (E, S) be a γ-sparse matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There exists a γ- competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' First observe that the point x := 1/γ lies in the matroid polytope PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 17 Indeed, it is non-negative and for every set S ⊆ E(M) we have x(S) = |S|/γ ⩽ rM(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then x can be expressed as a convex combination of indicator variables corresponding to the independent sets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In other words, we have x = � S∈S αS1S for some α ⩾ 0, � S∈S αS = 1, where 1S refers to the characteristic vector of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now sample an independent set S in matroid M randomly with probabil- ity αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let the gambler select all items in S and let the gambler leave all the items not in S unselected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If Xe is the random variable corresponding to the weight of element e ∈ E(M), then this mechanism results in a total expected value as follows � S∈S αS � e∈S E[Xe] = � e∈E (1/γ)E[Xe] = E[ � e∈E Xe]/γ ⩾ EPROPH/γ , finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that Proposition 1 implies that for a three edge-connected graph G, the cographic matroid of G is 3-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus Lemma 6 is a corollary of Theo- rem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Similarly, for a planar graph G the graphic matroid is 3-sparse, leading us to the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let G is a planar graph and let M be the corresponding graphic matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There is a 3-competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6 Representable matroids Many results in the theory of matroids make use of minors coming from re- strictions and contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To get access to the toolbox provided by matroid theory, we need to understand how prophet inequality guarantees change when we consider minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Preliminaries Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M be a matroid and let matroid N be a restriction of the ma- troid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there exists an α-competitive non-adaptive mechanism on M, then there is an α-competitive non-adaptive mechanism for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain a mechanism for the matroid N, we can impose thresholds +∞ for the items that were removed from the ground set to obtain the restriction N from the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The remaining items are assigned the same thresholds in both mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A similar result for contractions is harder to obtain in the case of non- adaptive mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Indeed, a straightforward approach would require us to impose the thresholds +∞ for the contracted items, while using the given 18 mechanism on the remaining items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Unfortunately, this would also require us to “change" the underlying matroid, in other words a gambler might be forced to reject an item even though its value is over the assigned threshold and its addition to the currently selected items keeps the selected set independent with respect to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Because of this difficulty, in this work we provide a matching result for contractions only for matroids representable over a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This result is sufficient for the purpose of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M = (E, S) be a matroid representable over the field Fp for some p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ⊆ E be a subset of the ground set such that λM(T ) ⩽ k for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists S ⊆ T so that every set that is independent in M |S is also independent in M/T and EPROPHM|S ⩾ 1 pk+1 EPROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall that T stands for the complement of T with respect to the ground set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Consider the representation of the matroid M over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let φ : E → Fm p be the map describing the representation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, for every S ⊆ E we have that the set φ(S) = {φ(e) ∈ Fm p : e ∈ S} is independent over the field Fp if and only if S is an independent set for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since λM(T ) ⩽ k holds, by definition of λM we have rM(T ) + rM(T ) − rM(E) ⩽ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We have rM(R) = dim span(φ(R)) for every R ⊆ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have dim span φ(E) = dim span φ(T )+dim span φ(T )−dim � (span φ(T )) ∩ (span φ(T)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and so dim � (span φ(T )) ∩ (span φ(T )) � ⩽ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since we are working over the field Fp, the linear space L := (span φ(T )) ∩ (span φ(T )) has at most pk vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let C be the orthogonal complement of the linear space L in the space span φ(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we can represent span φ(T ) as L ⊕ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For every vector v ∈ span φ(T ) we denote v orthogonal projection to L and C by v |L and v |C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For each vector a ∈ L, define the set Ta := {t ∈ T : φ(t) |L= a, φ(t) ̸= a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition for every a ∈ L we have Ta ∩ L = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us select a uniformly at random from L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Ea[EPROPHM|Ta ] ≥ 1 pk EPROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To prove the desired inequality, we prove the corresponding inequality for any realization of item values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' From now on we consider the realization of item values fixed and thus we prove the following inequality Ea[PROPHM|Ta] ≥ 1 pk PROPHM/T 19 Let us consider the set Iopt on which the prophet achieves PROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that the set Iopt does not contain any item e such that φ(e) is in L, because every such an item e is a loop in M/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, the set Iopt can be partitioned into sets Iopt,a, a ∈ L where Iopt,a is a subset of Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The set Iopt is independent in M/T and so Iopt is also independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence the sets Iopt,a, a ∈ L are also independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus for every a ∈ L, PROPHM|Ta ⩾ w(Iopt,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have Ea[PROPHM|Ta] ⩾ � a∈L w(Iopt,a) |L| = 1 |L|w(Iopt) ⩾ 1 pk PROPHM/T , finishing the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now select a∗ ∈ L such that EPROPHM|Ta is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the previous claim, we have PROPHM|Ta∗ ⩾ 1 pk PROPHM/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now for every c ∈ C define set Hc := {t ∈ Ta∗ : (φ(t) |C) · c = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us select c uniformly at random from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Ec[EPROPHM|Hc ] ≥ 1 pEPROPHM|T a∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To prove the desired inequality, we prove the corresponding inequality for any realization of item values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' From now on we consider the realization of item values fixed and thus we prove the following inequality Ec[PROPHM|Hc ] ≥ 1 pPROPHM|T a∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Iopt be the set corresponding to PROPHM|Ta∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have that for every e ∈ Iopt, φ(e) is not in L and hence φ(e) |C is not the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Due to Pc[c · t = 1] = 1/p, for every t ∈ Ta∗, we have Ec[w(Iopt∩Hc)] = � t∈Iopt Pc[c·t = 1]w(t) = 1 p � t∈Iopt w(t) = 1 pw(Iopt) = PROPHM|Ta∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Finally, since Iopt is independent in M so is Iopt ∩ Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, we have Ec[PROPHM|Hc ] ≥ 1 pPROPHM|T a∗ , finishing the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us select c∗ so that EPROPHM|Hc is maximized and let S∗ := Hc∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have EPROPH(M |S∗) ⩾ 1 pk+1 EPROPH(M/T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Finally, we need to show that every set independent in M |S∗ is an indepen- dent set in M/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose the contrary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' there exists a set that is independent 20 in M |S∗ but is not an independent set in M/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then span S∗ has a non-trivial intersection with span T, suppose x ∈ (span φ(S∗)) ∪ (span φ(T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us show that x is a zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since x ∈ span S∗, we have x = � s∈S∗ αsφ(s) for some αs ∈ Fp, s ∈ S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us consider the projections of x on C and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since x ∈ span φ(T ) we have that x lies in L and so x |C is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus x |C= � s∈S∗ αs(φ(s) |C) is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition, φ(s) |L= a∗ and c∗ · (φ(s) |C) = 1 hold for every s ∈ S∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus over the field Fp we have � s∈S∗ αs = � s∈S∗ αs(c∗ · (φ(s) |C)) = c∗ · � � s∈S∗ αs(φ(s) |C) � = c∗ · (x |C) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us consider x |L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We have x |L= � s∈S∗ αs(φ(s) |L) = � � s∈S∗ αs � a∗ , where the last expression equals the zero vector since � s∈S∗ αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we have a vector x ∈ L ⊕ C such that both projections x |L and x |C are the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, the vector x is the zero vector, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Tree Decompositions Similarly to the approach [HN20] for the matroid secretary problem, we exten- sively use the tree decomposition of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A tree decomposition of bounded thickness allows us to construct non-adaptive mechanisms with good approxi- mation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Before proceeding with these constructions, let us introduce tree decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A tree decomposition of a matroid M = (E, S) is a pair (T, X) where T is a tree and X = {Xv ⊆ E : v ∈ V (T )}, where sets in X form a partition of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, we refer to the vertex and edge sets of the tree T as V (T ) and E(T ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given an edge e = {v1, v2} ∈ E(T ) of the tree T , let T1 and T2 be two connected components of T − e, in other word the removal of the edge e from T leads to two connected components T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The thickness of the edge e = (v1, v2) is denoted as λ(e) and is defined as follows λ(e) := λM(∪v∈V (T1)Xv) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The thickness of the tree decomposition is the maximum thickness of the edge e in E(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M be a family of matroids, M be a matroid and (T, X) be a tree decom- position of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that tree decomposition (T, X) is M-tree decomposition 21 if M |clM(Xv)∈ M holds for every v ∈ V (T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let tk(M) be a set of matroids which have M-tree decomposition of thickness at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Mα,p be the family of matroids which admit α-competitive non-adaptive mechanisms and are representable over the finite field Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then for every natural number k and every matroid M in tk(Mα,p), the matroid M has an (αpk+1)-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For a natural number m, let tk,m(Mα,p) be the set of matroids which have an Mα,p-tree decomposition (T, X) of thickness at most k satisfying |V (T )| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us prove the statement of the lemma by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The base case follows from the definition of the family Mα,p and the fact that Mα,p = tk,1(Mα,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now show how to do the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume m ≥ 2 and consider a matroid M = (E, S) in tk,m(Mα,p) with its Mα,p-tree decomposition (T, X) of thickness at most k and with |V (T )| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let ℓ be a leaf of the tree T and let u be the neighbour of the vertex ℓ in the tree T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Observe that the tree (V (T ) \\ {ℓ}, E(T )\\ {ℓu}) together with the subfamily {Xw : w ∈ V (T ) \\ {ℓ}} defines an Mα,p-tree decomposition of the matroid M \\ Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus the matroid M \\ Xℓ is in M ∈ tk,m���1(Mα,p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, by the inductive hypothesis there are thresholds T ′ e, e ∈ E \\ Xℓ guaranteeing αpk+1- competitiveness of the gambler in comparison to the prophet on the matroid M \\ Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' There are thresholds T ′′ e , e ∈ Xℓ leading to an (α · pk+1)-competitive non-adaptive mechanism for matroid M |Xℓ, such that the gambler always selects a set that is independent in M/Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 8 there exists a set S ⊆ Xl such that every set independent in M |S is also independent in the matroid M/Xℓ and EPROPHM|S ⩾ 1 pk+1 EPROPHM/Xℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By definition of Mα,p and the appearance of Xℓ in the tree decomposition, we have that M |Xℓ is in the family Mα,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 7, since S is a subset of Xℓ the matroid M |S is also in the family Mα,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, there are thresholds T ′′ e , e ∈ S that lead to an α-competitive non-adaptive mechanism on M |S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The thresholds T ′′ e , e ∈ Xℓ \\ S can be defined as +∞, finishing the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we can define thresholds Te, e ∈ E for all elements of the matroid M as follows Te := � T ′ e if e ̸∈ Xℓ T ′′ e otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now demonstrate that such thresholds Te, e ∈ E lead to an (αpk+1)- competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 22 First, by the above claim the selected items from Xℓ always form an inde- pendent set in M/Xℓ when used with the thresholds Te, e ∈ Xℓ on the matroid M |Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus the definition of the thresholds guarantees that in expectation the value of selected items from Xℓ is at least EPROPHM|Xℓ/(αpk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and in ex- pectation the value of selected items from E\\Xℓ is at least EPROPHM\\Xℓ/(αpk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To finish the proof, note that we have PROPHM|Xℓ + PROPHM\\Xℓ ≥ PROPHM and so EPROPHM|Xℓ + EPROPHM\\Xℓ ≥ EPROPHM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='3 Regular matroids In this section, we prove Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Before we proceed to the proof, let us define key notions related to regular matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A subset of the matroid’s ground set is called a circuit, if it is an inclusion- minimal dependent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' A cycle is a subset of the ground set which can be partitioned into a disjoint union of circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M1 = (E1, S1), M2 = (E2, S2) be two binary matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then the matroid sum M1△M2 has the ground set E1△E2 and the cycles of M1△M2 are all sets of the form C1△C2, where C1 is a cycle for M1 and C2 is a cycle for M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Consider two binary matroids M1 = (E1, S1), M2 = (E2, S2) and M = M1△M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If |E1 ∩ E2| = 0, and E1 ̸= ∅, E2 ̸= ∅, M is called a 1-sum of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If |E1 ∩ E2| = 1, |E1| ≥ 3, |E2| ≥ 3 and E1 ∩ E2 is not a loop of M1 or M2 or their dual matroids, M is called a 2-sum of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If |E1 ∩ E2| = 3, |E1| ≥ 7, |E2| ≥ 7 and E1 ∩ E2 is a circuit in both M1 and M2, and E1 ∩ E2 does not contain a circuit in their dual matroids, then M is called a 3-sum of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Seymour’s regular matroid decomposition theorem [Sey80], every regular matroid M can be obtained from graphic, cographic or a special matroid R10 through a sequence of 1-sums, 2-sums or 3-sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' This gives a tree decomposition (T, X) of thickness at most 2 so that each M |Xv, v ∈ V (T ) is either a graphic, cographic or a special matroid R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By performing parallel extensions of the elements to be deleted before each 2-sum and 3-sum, we construct a matroid M ′, so that M is a restriction of M ′ and M ′ has a tree decomposition (T, X ′) so that each M ′ |clM′ (X′v), v ∈ V (T ) is either graphic, cographic or a parallel extension of R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 23 By Theorem 2, every graphic matroid has a 32-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Theorem 5, every cographic matroid has a 6-competitive non- adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since matroid R10 has ground set of size 10, by Theorem 1 every parallel extension of R10 has a 20-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition every regular matroid is representable over finite field F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, by Theorem 13 with p = 2, k = 2 and α = 32 there is a 256- competitive non-adaptive mechanism for matroid M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since M is a restriction of M ′, by Lemma 7, there is a 256-competitive non-adaptive mechanism for M, finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4 Minor-closed representable matroid families In this section we show that every minor-clossed subclass of matroids repre- sentable over Fp has a constant-competitive non-adaptive mechanism, where the constant is a function only of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The proof of this fact is analogous to the proof in [HN20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Theorem 14 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='3 in [Gee11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given natural numbers q ⩾ 2 and n ⩾ 1, let M = (E, S) be a matroid with no U2,q+2 or M(Kn) minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then we have |E| ≤ qq3nrM(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Given natural numbers q ⩾ 2 and n ⩾ 1, let M = (E, S) be a matroid with no U2,q+2 or M(Kn) minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists a qq3n-competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If M has no U2,q+2 or M(Kn) minors, then every restriction of M also has no U2,q+2 or M(Kn) minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus for every X ⊆ E we have |X| ⩽ qq3nrM(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So, M is a qq3n-sparse matroid and by Theorem 12 there exists a qq3n-competitive non-adaptive mechanism for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 Projections and lifts Let M be a matroid and x be an element of the ground set, which is a not a loop and not a free element of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then M/x is called a projection of M \\ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' M \\ x is called a lift of M/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that here and later we write M/x and M \\ x instead of M/{x} and M \\ {x}, repsectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M and N be two matroids with the same ground set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We say that the distance between M and N is t, denoted by dist(M, N) = t if t is the smallest integer such that there exists a sequence of matroids P0, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , Pt where P0 = M and Pt = N and for every i ∈ [t] the matroid Pi is either a lift or a projection of Pi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N be a lift of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there is an α-competitive non- adaptive mechanism for M then there exists a (2α+2)-competitive non-adaptive mechanism for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N is a lift of M, there exists a matroid L = (E, S) and an element x of its ground set, such that M = L/x, N = L\\x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Here, x is not a loop and not a free element of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let P be the set of elements in L that are parallel to x, in other words P := {x′ ∈ E : x′ ∥ x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that N |P \\{x} is a uniform matroid of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note also that elements in P \\{x} are loops in M and so EPROPHM = EPROPHM\\P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ′ e, e ∈ E \\ {x} be the thresholds imposed by an α-competitive non- adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ′′ e , e ∈ P be the thresholds guaranteeing 2-competitive non-adaptive mechanism as in Theorem 1 for the uniform matroid of rank 1 on the ground set P \\{x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' and let T ′′ e , e ∈ E\\(P ∪{x}) be +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We select one of these two sets of thresholds for the matroid N as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The constructed mechanism for the matroid N selects one of those two sets at random, where first set of thresholds T ′ e, e ∈ E \\ {x} is selected with probability γ := α/(α + 1) and the second set T ′′ e , e ∈ E \\ {x} with probability 1 − γ = 1/(α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Next part is dedicated to the analysis of how thresholds T ′ e, e ∈ E \\ {x} perform on the matroid N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note, that these thresholds are coming from a mechanism for the matroid M, while they are used for the matroid N with probability γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We show that the total expected value achieved by thresholds T ′ e, e ∈ E \\ {x} on N is at least the total expected value achieved by these thresholds on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For this we can assume that for every realization of item values, the orders of items in matroid N and M are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To see that this assumption is valid, we can assume that the order for N is chosen in an adversarial way and is used also as the items order for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that the items order for M and N is the same for a given realization of item values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us also assume that for every item e ∈ E \\ {x} the threshold T ′ e is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then the gambler with matroid N selects all items that the gambler with matroid M selects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' We fix the item values realization and items order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let e1, e2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ek be the items with their values being at least their threshold and with the corre- sponding order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we need to show that if the gambler with matroid N selects items greedily from e1, e2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ek starting from e1, then the set of selected items is a superset of the items greedily selected by the gambler with matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If both gamblers end up selecting exactly the same set of items, then proof of the claim is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Otherwise consider the first index i ∈ [k] such that the item ei is selected by exactly one of the two gamblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N = L\\x and M = L/x we have that it is only possible if ei is selected by the gambler with the matroid N and rejected by the gambler with the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now we claim that every subsequent item, in other words an item in ei+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ek, is either selected by both gamblers or rejected by both gamblers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Suppose the contrary and consider the first item ej, i + 1 ≤ j ≤ k that is selected by one gambler and rejected by another gambler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let S := {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' , ej−1} and let T be the set of items selected by the gambler with M from the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus 25 the gambler with N selected T ∪ {ei} from the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So T ∪ {ei} is a basis of (L\\x) |S and T is a basis of (L/x) |S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, both T ∪{ei} and T ∪{x} are bases of L |S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If only one of the two gamblers accepts the item sj then the matroid L |S∪{sj} has two bases of different cardinality, attaining a contradiction and finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus we have that the thresholds T ′ e, e ∈ E\\{x} guarantee at least EPROPHM as the expected total value of the gambler with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To prove that the constructed mechanism is 1/(2α + 2)-competitive it is enough to show the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that in our construction we used α-competitive non-adaptive mechanism for the matroid M and 2-competitive non-adaptive mechanism for the uniform matroid of rank 1 on P \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' γ 1 αEPROPHM + (1 − γ) 1 2EPROPHP \\{x} ⩾ 1 2α+2EPROPHN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us consider the inclusion-maximal set Iopt on which the prophet achieves PROPHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let Copt be a random variable corresponding to the unique circuit of Iopt ∪ {x} in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Recall that x is not a free element of L so such a circuit exists and is unique and contains x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' First consider the events when |Copt| ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that by definition of a circuit, for every y ∈ Copt \\ {x} the set (Iopt ∪ {x}) \\ {y} is independent in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, for every y ∈ Copt \\ {x} the set Iopt \\ {y} is independent in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' So we have that conditioned on |Copt| ⩾ 3 we have PROPHM ≥ w(Iopt \\ {y}) for every y ∈ Copt \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let yopt be the random variable representing the element in Copt \\{x} of smallest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then conditioned on |Copt| ⩾ 3, we have w(Copt \\ {yopt, x}) ≥ w(C \\ {x})/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, conditioned on |Copt| ⩾ 3 we have PROPHM ⩾ w(Iopt \\ {yopt}) = w(Iopt \\ Copt) + w(Copt \\ {yopt}) ≥ w(Iopt \\ Copt) + 1 2w(Copt \\ {x}) ≥ 1 2w(Iopt) = 1 2PROPHN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Second consider the event that |Copt| < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since x is not a loop of L by definition, we have |Copt| = 2 and so Copt = {x, xopt} for some random variable element xopt ∈ P \\{x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For the event |Copt| ≥ 3 let us define the random variable element xopt to be an arbitrary element in Copt \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus, if |Copt| < 3 we have PROPHP \\{x} ≥ w(xopt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us define Jopt := Iopt \\ {xopt} and note that Jopt is independent in the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Moreover, since Iopt is the set on which the prophet achieves PROPHN, we have that conditioned on |Copt| < 3 the prophet achieves PROPHM on the set Jopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='Combining everything together we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='γ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='αEPROPHM + (1 − γ)1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2EPROPHP \\{x} = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1EPROPHM + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2EPROPHP \\{x} ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�w(xopt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2 + PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| < 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| < 3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| ⩾ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| ⩾ 3] = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�w(xopt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2 + w(Iopt \\ {xopt}) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| < 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| < 3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| ⩾ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| ⩾ 3] ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| < 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| < 3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='�PROPHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='α + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='���� |Copt| ⩾ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='P [|Copt| ⩾ 3] ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2α + 2EPROPHN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N be a matroid obtained from a matroid M by a sequence of t projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let L be the set of loops in the matroid N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let there exist an α-competitive non-adaptive mechanism for the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exists a non-adaptive mechanism for N\\L such that the expected total value of this mechanism is at least 1 α·3t EPROPHM\\L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' In the context of Lemma 10, every set that is independent for the matroid N\\ L is also independent for the matroid M \\ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Hence, we have EPROPHM\\L ≥ EPROPHN\\L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus in case t = 1, Lemma 10 leads us to the following corol- lary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N be a projection of the matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there is an α- competitive non-adaptive mechanism for M then there exists a 3α-competitive non-adaptive mechanism for N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us prove the statement by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Of course, in case t = 0 we have M = N and the statement is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us now assume that t is at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let N ′ be a matroid such that N ′ is obtained from the matroid M by a sequence of t − 1 projections and N is a projection of N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N is a projection of N ′ there is a matroid P = (E, S) and x ∈ E such that P \\ x = N ′ and P/x = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let L′ be the set of loops in the matroid N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By induction hypothesis, there exist thresholds T ′ e, e ∈ E \\ (L′ ∪ {x}) such that the gambler with the matroid N ′\\L′ achieves at least 1 α·3t−1 EPROPHM\\L′ as the expected total value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let us assume that to compute thresholds T ′ e, e ∈ E \\(L′ ∪{x}) the values of items in L were set to be 0 while the distribution of values for other items remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since L′ ⊆ L, analogously to Lemma 7 we can define thresholds T ′′ e := � +∞ if e ∈ L T ′ e otherwise 27 such that the gambler with the matroid N ′\\L achieves at least 1 α·3t−1 EPROPHM\\L as the expected total value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let T ′′′ e , e ∈ E \\(L∪{x}) be the thresholds guaran- teeing 2-competitive non-adaptive mechanism as in Theorem 1 for the uniform matroid of rank 1 on the ground set E \\ (L ∪ {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The constructed mechanism for the matroid N\\L selects one of two threshohold sets at random, where first set of thresholds T ′′ e , e ∈ E \\ (L ∪ {x}) is selected with probability 1/3 and the thresholds T ′′′ e , e ∈ E \\ (L ∪ {x}) with probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that the thresholds T ′′ e , e ∈ E \\ (L ∪ {x}) were designed for the matroid N ′ \\ L but are used for the matroid N \\ L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' hence less items might be selected than when it is used for N ′ \\ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Also note, that the thresholds T ′′′ e , e ∈ E \\ (L ∪ {x}) are used for N \\ L but were designed for the uniform matroid of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' For the analysis, let Ialg be the random variable indicating the items set selected by the gambler with matroid N ′ \\ L when the thresholds T ′′ e , e ∈ E \\ (L ∪ {x}) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Analogously to a claim in the proof of Lemma 9, we can assume that when the thresholds T ′′ e , e ∈ E\\(L∪{x}) are used the gambler with N \\L select all items in Ialg with an exception for possibly one item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let xopt be the random variable indicating the element of maximum value in E \\ (L ∪ {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To finish the proof it is enough to show the following inequality 1 3E[w(Ialg) − w(xopt)] + 2 3 1 2E[w(xopt)] ≥ 1 α · 3t EPROPHM\\L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' To obtain this inequality we can do estimations as follows 1 3E[w(Ialg)−w(xopt)]+2 3 1 2E[w(xopt)] = 1 3E[w(Ialg)] ≥ 1 3 1 α · 3t−1 EPROPHM\\L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Now let us combine Corollary 3 and Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M and N be matroids such that dist(M, N) ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' If there exists an α-competitive non-adaptive mechanism for the matroid M with α ≥ 2 then there exists a 3tα-competitive non-adaptive mechanism for the matroid N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Note that for α ≥ 2 we have 3α ≥ 2α + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Since N can be obtained from M by a sequence of t projection and lift steps, we can use Corollary 3 or Lemma 9 for each of these steps to obtain the desired competitiveness ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content='2 Minor-closed families theorem Lemma 12 (Lemma 6 in [HN20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p and n be integers such that p ⩽ n − 2 and p is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The matroid U2,n is not representable over the field Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The following Structural Hypothesis is due to Geelen, Gerards and Whittle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' The proof of this Structural Hypothesis has not appeared in print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 28 Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let p be a prime number and M is a proper minor-closed class of matroids representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then there exist k, n, t such that every M ∈ M is a restriction of an Fp- representable matroid M ′ having a full tree-decomposition (T, X) of thickness at most k so that for every v ∈ V (T ) if M ′ |clM′(Xv) has a M(Kn) minor, then there exists a 2-column sparse matroid N with dist(M ′ |clM′ (Xv), N) ⩽ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let k, n, t are as stated in the Structural Hypothesis 1 on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M1 be the set of matroids on distance t or less from some 2-column sparse matroid and are representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Theorem 4 all 2-column sparse matroids have a 32-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 11 there exists a (3t · 32)-competitive mechanism for matroids in M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Let M2 be the set of matroids without M(Kn) minor and are representable over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 12 all matroids in M2 do not have U2,p+2 as a minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Then by Corollary 2, we have that there is a pp3n-competitive non-adaptive mechanism for every matroid in M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By the Structural Hypothesis 1 we have that every M ∈ M is a restriction of some M ′ with a full tree-decomposition (T, X) of thickness at most k so that for every v ∈ V (T ) M ′ |clM′ (Xv)∈ M1 ∪ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Thus by Theorem 13, matroid M ′ has a γ := (max(3t · 32, pp3n) · pk+1)- competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' By Lemma 7 the matroid M has also a γ-competitive non-adaptive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' 29 References [AKW19] Pablo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Azar, Robert Kleinberg, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Matthew Weinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Prior independent mechanisms via prophet inequalities with limited in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Games and Economic Behavior, 118:511–532, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' [CFPP21] Constantine Caramanis, Matthew Faw, Orestis Papadigenopoulos, and Emmanouil Pountourakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAzT4oBgHgl3EQfu_2Z/content/2301.01700v1.pdf'} +page_content=' Single-sample prophet inequalities revisited.' metadata={'source': 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