structured/0704.0468.structured.json

{"paper_meta":{"paper_id":"arxiv:0704.0468","title":"0704.0468","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0704.0468v2 [cs.CC] 23 Mar 2009\nInapproximability of Maximum Weighted Edge\nBiclique and Its Applications\nJinsong Tan\nDepartment of Computer and Information Science\nSchool of Engineering and Applied Science\nUniversity of Pennsylvania, Philadelphia, PA 19104, USA\njinsong@seas.upenn.edu\nAbstract. Given a bipartite graph G = (V1, V2, E) where edges take\non both positive and negative weights from set S, the maximum weighted\nedge biclique problem, or S-MWEB for short, asks to find a bipartite sub-\ngraph whose sum of edge weights is maximized. This problem has various\napplications in bioinformatics, machine learning and databases and its\n(in)approximability remains open. In this paper, we show that for a wide\nrange of choices of S, specifically when\n ̨ ̨ min S\nmax S\n ̨ ̨ ∈Ω(ηδ−1/2) ∩O(η1/2−δ)\n(where η = max{|V1|, |V2|}, and δ ∈(0, 1/2]), no polynomial time algo-\nrithm can approximate S-MWEB within a factor of nǫ for some ǫ > 0\nunless RP = NP. This hardness result gives justification of the heuristic\napproaches adopted for various applied problems in the aforementioned\nareas, and indicates that good approximation algorithms are unlikely to\nexist. Specifically, we give two applications by showing that: 1) finding\nstatistically significant biclusters in the SAMBA model, proposed in [18]\nfor the analysis of microarray data, is nǫ-inapproximable; and 2) no poly-\nnomial time algorithm exists for the Minimum Description Length with\nHoles problem [4] unless RP = NP.\n1\nIntroduction\nLet G = (V1, V2, E) be an undirected bipartite graph. A biclique subgraph in G\nis a complete bipartite subgraph of G and maximum edge biclique (MEB) is the\nproblem of finding a biclique subgraph with the most number of edges. MEB is\na well-known problem and received much attention in recent years because of\nits wide range of applications in areas including machine learning [14], manage-\nment science [16] and bioinformatics, where it is found particularly relevant in\nthe formulation of numerous biclustering problems for biological data analysis\n[5,2,18,19,17], and we refer readers to the survey by Madeira and Oliveira [13]\nfor a fairly extensive discussion on this. Maximum edge biclique is shown to be\nNP-hard by Peeters [15] via a reduction from 3SAT. Its approximability status,\non the other hand, remains an open question despite considerable efforts [7,8,12]\n1. In particular, Feige and Kogan [8] conjectured that maximum edge biclique\n1 Note it might be easy to confuse the MEB problem with the Bipartite Clique problem\ndiscussed by Khot in [12]. Bipartite Clique, which also known as Balanced Complete\n\n2\nJinsong Tan\nis hard to approximate within a factor of nǫ for some ǫ > 0. In this paper, we\nconsider a weighted formulation of this problem defined as follows\nDefinition 1. S-Maximum Weighted Edge Biclique (S-MWEB)\nInstance: A complete bipartite graph G = (V1, V2, E) (throughout the paper, let\nη = max{|V1|, |V2|} and n = |V1| + |V2|), a weight function wG : E →S, where\nS is a set consisting of both positive and negative integers.\nQuestion: Find a biclique subgraph of G where the sum of weights on edges is\nmaximized.\nA few comments are in order. First note it is not a lose of generality but a\ntechnical convenience to require the graph be complete, one can always think of\nan incomplete bipartite graph as complete where non-edges are assigned weight\n0. Also note we require that both positive and negative weights be in S at the\nsame time because otherwise S-MWEB becomes a trivial problem.\nOur study of S-MWEB is motivated by the problem of finding statistically\nsignificant biclusters in microarray data analysis in the SAMBA model [18]\nand the Minimum Description Length with Holes (MDLH) problem [3,4,10];\ndetailed discussion of the two problems can be found in Sect. 4. Our main\ntechnical contribution of this paper is to show that if S satisfies the condition\n| min S\nmax S | ∈Ω(ηδ−1/2) ∩O(η1/2−δ), where δ > 0 is any arbitrarily small constant,\nthen no polynomial time algorithm can approximate S-MWEB within a factor\nof nǫ for some ǫ > 0 unless RP = NP. This result enables us to answer open\nquestions regarding the hardness of the SAMBA model and the MDLH prob-\nlem. Since maximum edge biclique can be characterized as a special case of\nS-MWEB with S = {−η, 1}, the nǫ-inapproximability result also provides inter-\nesting insights into the conjectured nǫ-inapproximability [8] of maximum edge\nbiclique.\nThe rest of the paper is organized in three sections. In Sect. 2, we present\nthe main technical result by proving the aforementioned inapproximability of S-\nMWEB. We give applications of this by answering hardness questions regarding\ntwo applied problems in Sect. 3. We conclude this work by raising a few open\nproblems in the last section.\n2\nApproximating S-Maximum Edge Biclique is Hard\nWe start this section by giving two lemmas about CLIQUE, which will be used\nin establishing inapproximability for the biclique problems we consider later.\nLemma 1 is a recent result by Zuckerman [20], obtained by a derandomization\nof results of H ̊astad [11]; Lemma 2 follows immediately from Lemma 1.\nLemma 1. ([20]) It is NP-hard to approximate CLIQUE within a factor of\nn1−ǫ, for any ǫ > 0.\nBipartite Subgraph [8], aims to maximize the number of vertices of a balanced sub-\ngraph whereas MEB aims to maximize the total weights on edges in a (not necessarily\nbalanced) subgraph.\n\nInapproximability of Maximum Weighted Edge Biclique and Its Applications\n3\nLemma 2. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate CLIQUE within a factor of n1−ǫ with probability at least\n1\npoly(n) unless\nRP = NP.\n2.1\nA Technical Lemma\nWe first describe the construction of a structure called {γ, {α, β}}-Product,\nwhich will be used in the proof of our main technical lemma.\nDefinition 2. ({γ, {α, β}}-Product)\nInput: An instance of S-MWEB on complete bipartite graph G = V1×V2, where\nγ ∈S and α < γ < β; an integer N.\nOutput: Complete bipartite graph GN = V N\n1\n× V N\n2\nconstructed as follows: V N\n1\nand V N\n2\nare N duplicates of V1 and V2, respectively. For each edge (i, j) ∈GN,\nlet (φ(i), φ(j)) be the corresponding edge in G. If wG(φ(i), φ(j)) = γ, assign\nweight α or β to (i, j) independently and identically at random with expectation\nbeing γ, denote the weight by random variable X. If wG(φ(i), φ(j)) ̸= γ, then\nkeep the weight unchanged. Call the weight function constructed this way w(·).\nFor any subgraph H of GN, denote by wγ(H) (resp., w−γ(H)) the total\nweight of H contributed by former-γ-edges (resp., other edges). Clearly, w(H) =\nwγ(H) + w−γ(H).\nWith a graph product constructed in this randomized fashion, we have the fol-\nlowing lemma.\nLemma 3. Given an S-MWEB instance G = (V1, V2, E) where γ ∈S, and a\nnumber δ ∈(0, 1\n2]; let η = max (|V1|, |V2|), N = η\nδ(3−2δ)+3\nδ(1+2δ) , GN = (V N\n1 , V N\n2 , E)\nbe the {γ, {α, β}}-product of G and S′ = (S ∪{α, β}) −{γ}. If\n1. |β −α| = O((Nη)\n1\n2 −δ); and\n2. there is a polynomial time algorithm that approximates the S′-MWEB\ninstance within a factor of λ, where λ is some arbitrary function in the size of\nthe S′-MWEB instance\nthen there exists a polynomial time algorithm that approximates the S-MWEB\ninstance within a factor of λ, with probability at least\n1\npoly(n).\nProof. For notational convenience, we denote η\n1\n2 −δ by f(η) throughout the proof.\nDefine random variable Y = X −γ, clearly E[Y ] = 0. Suppose there is a poly-\nnomial time algorithm A that approximates S′-MWEB within a factor of λ, we\ncan then run A on GN, the output biclique G∗\nB corresponds to N 2 bicliques in\nG (not necessarily all distinct). Let G∗\nA be the most weighted among these N 2\nsubgraphs of G, in the rest of the proof we show that with high probability, G∗\nA\nis a λ-approximation of S-MWEB on G.\nDenote by E1 the event that G∗\nB does not imply a λ-approximation on G.\nLet H be the set of subgraphs of GN that do not imply a λ-approximation on G,\n\n4\nJinsong Tan\nclearly, |H| ≤4Nη. Let H′ be an arbitrary element in H, we have the following\ninequalities\nPr {E1} ≤Pr\n \nat least one element in H is a λ-approximation of GN \n≤4Nη · Pr\n \nH′ is a λ-approximation of GN \n= 4Nη · Pr{E2}\nwhere E2 is the event that H′ is a λ-approximation of GN.\nLet the weight of an optimal solution U1×U2 of G be K, denote by U N\n1 ×U N\n2\nthe corresponding N 2-duplication in GN. Let x1 and x2 be the number of former-\nγ-edges in H′ and U N\n1 × U N\n2 , respectively. Suppose E2 happens, then we must\nhave\nw−γ(H′) + x1γ ≤N 2( K\nλ −1)\nw−γ(H′) + wγ(H′) ≥1\nλ(w−γ(U N\n1 × U N\n2 ) + wγ(U N\n1 × U N\n2 ))\nwhere the first inequality follows from the fact that we only consider integer\nweights. Since w−γ(U N\n1 × U N\n2 ) = N 2K −x2γ, it implies\n(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2\nso we have the following statement on probability\nPr{E2} ≤Pr\n \n(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2 \nLet z1 (resp., z2 and z3) be the number of edges in E(H′) −E(U N\n1 × U N\n2 )\n( resp., E(U N\n1 × U N\n2 ) −E(H′) and E(U N\n1 × U N\n2 ) ∩E(H′) ) transformed from\nformer-γ-edges in G. We have\nPr\n \n(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2 \n= Pr\nnPz1\ni=1 Yi −1\nλ\nPz2\nj=1 Yj + λ−1\nλ\nPz3\nk=1 Yk ≥N 2o\n= Pr\nnPz1\ni=1 Yi + 1\nλ\nPz2\nj=1 (−Yj) + λ−1\nλ\nPz3\nk=1 Yk ≥N 2o\n≤Pr\nnPz1\ni=1 Yi ≥N 2\n3\no\n+ Pr\nn\n1\nλ\nPz2\nj=1 (−Yj) ≥N 2\n3\no\n+ Pr\nn\nλ−1\nλ\nPz3\nk=1 Yk ≥N 2\n3\no\n≤Pr\nnPz1\ni=1 Yi ≥N 2\n3\no\n+ Pr\nnPz2\nj=1 (−Yj) ≥N 2\n3\no\n+ Pr\nnPz3\nk=1 Yk ≥N 2\n3\no\n≤P\ni∈{1,2,3}\n \nexp\n \n−2zi\n \nN 2\n3zi(c1f(Nη))\n 2 \n(Hoeffding bound)\n≤3 · exp\n \n−c2 · N 1+2δ\nη3−2δ\n \n(zi ≤η2N 2)\nwhere c1, c2 are constants (c2 > 0). Now if we set N = η\n3−2δ\n1+2δ +θ for some θ, we\nhave\nPr {E1} ≤4Nη · Pr {E2} ≤3 · exp\n \nln 4 · η\n4\n(1+2δ) +θ −c2 · η(1+2δ)θ \nFor this probability to be bounded by 1\n2 as η is large enough, we need to have\n4\n1+2δ +θ < (1+2δ)θ. Solving this inequality gives θ >\n2\nδ(1+2δ). Therefore, for any\nδ ∈(0, 1\n2], by setting N = η\nδ(3−2δ)+3\nδ(1+2δ) , we have Pr{E1}, i.e. the probability that\n\nInapproximability of Maximum Weighted Edge Biclique and Its Applications\n5\nthe solution returned by A does not imply a λ-approximation of G, is bounded\nfrom above by 1\n2 once input size is large enough. This gives a polynomial time\nalgorithm that approximates S-MWEB within a factor of λ with probability at\nleast 1\n2.\n⊓⊔\nThis lemma immediately leads to the following corollary.\nCorollary 1. Following the construction in Lemma 3, if S′-MWEB can be ap-\nproximated within a factor of nǫ′, for some ǫ′ > 0, then there exists a polyno-\nmial time algorithm that approximates S-MWEB within a factor of nǫ, where\nǫ = (1 + δ(3−2δ)+3\nδ(1+2δ) )ǫ′, with probability at least\n1\npoly(n). 2\nProof. Let |G| and |GN| be the number of nodes in the S-MWEB and S′-MWEB\nproblem, respectively. Since λ = |GN|ǫ′ ≤|G|(1+ δ(3−2δ)+3\nδ(1+2δ)\n)ǫ′, our claim follows\nfrom Lemma 3.\n⊓⊔\n2.2\n{−1, 0, 1}-MWEB\nIn this section, we prove inapproximability of {−1, 0, 1}-MWEB by giving a\nreduction from CLIQUE; in subsequence sections, we prove inapproximability\nresults for more general S-MWEB by constructing randomized reduction from\n{−1, 0, 1}-MWEB.\nLemma 4. The decision version of the {−1, 0, 1}-MWEB problem is NP-complete.\nProof. We prove this by describing a reduction from CLIQUE. Given a CLIQUE\ninstance G = (V, E), construct G′ = (V ′, E′) such that V ′ = V1∪V2 where V1, V2\nare duplicates of V in that there exist bijections φ1 : V1 →V and φ2 : V2 →V .\nAnd\nE′ = E1 ∪E2 ∪E3\nE1 = {(u, v) | u ∈V1, v ∈V2 and (φ1(u), φ2(v)) ∈E}\nE2 = {(u, v) | u ∈V1, v ∈V2, φ1(u) ̸= φ2(v) and (φ1(u), φ2(v)) /∈E}\nE3 = {(u, v) | u ∈V1, v ∈V2, and φ1(u) = φ2(v)}\nClearly, G′ is a biclique. Now assign weight 0 to edges in E1, −1 to edges in\nE2 and 1 to edges in E3. We then claim that there is a clique of size k in G if\nand only if there is a biclique of total edge weight k in G′.\nFirst consider the case where there is a clique of size k in G, let U be the set\nof vertices of the clique, then taking the subgraph induced by φ−1\n1 (U) × φ−1\n2 (U)\nin G′ gives us a biclique of total weight k.\nNow suppose that there is a biclique U1 ×U2 of total weight k in G′. Without\nloss of generality, assume U1 and U2 correspond to the same subset of vertices in\n2 Note we are slightly abusing notation here by always representing the size of a given\nproblem under discussion by n. Here n refers to the size of S′-MWEB (resp. S-\nMWEB) when we are talking about approximation factor nǫ′ (resp. nǫ). We adopt\nthe same convention in the sequel.\n\n6\nJinsong Tan\nV because if (φ1(U1) −φ2(U2)) ∪(φ2(U2) −φ1(U1)) is not empty, then removing\n(U1 −U2) ∪(U2 −U1) will never decrease the total weight of the solution. Given\nφ1(U1) = φ2(U2), we argue that there is no edge of weight −1 in biclique U1×U2;\nsuppose otherwise there exists a weight −1 edge (i1, j2) (i1 ∈U1, and j2 ∈U2),\nthen the corresponding edge (j1, i2) (j1 ∈U1, and i2 ∈U2) must be of weight\n−1 too and removing i1, i2 from the solution biclique will increase total weight\nby at least 1 because among all edges incident to i1 and i2, (i1, i2) is of weight 1,\n(i1, j2) and (i2, j1) are of weight −1 and the rest are of weights either 0 or −1.\nTherefore, we have shown that if there is a solution U1 × U2 of weight k in\nG′, U1 and U2 correspond to the same set of vertices U ∈V and U is a clique of\nsize k. It is clear that the reduction can be performed in polynomial time and\nthe problem is NP, and thus NP-complete.\n⊓⊔\nGiven Lemma 1, the following corollary follows immediately from the above\nreduction.\nTheorem 1. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate problem {−1, 0, 1}-MWEB within a factor of n1−ǫ unless P = NP.\nProof. It is obvious that the reduction given in the proof of Lemma 4 preserves\ninapproximability exactly, and given that CLIQUE is hard to approximate within\na factor of n1−ǫ unless P = NP, the theorem follows.\n⊓⊔\nTheorem 2. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate {−1, 0, 1}-MWEB within a factor of n1−ǫ with probability at least\n1\npoly(n)\nunless RP = NP.\nProof. If there exists such a randomized algorithm for {−1, 0, 1}-MWEB, com-\nbining it with the reduction given in Lemma 4, we obtain an RP algorithm for\nCLIQUE. This is impossible unless RP = NP.\n⊓⊔\n2.3\n{−1, 1}-MWEB\nLemma 5. If there exists a polynomial time algorithm that approximates {−1, 1}-\nMWEB within a factor of nǫ, then there exists a polynomial time algorithm that\napproximates {−1, 0, 1}-MWEB within a factor of n5ǫ with probability at least\n1\npoly(n).\nProof. We prove this by constructing a {γ, {α, β}}-Product from {−1, 0, 1}-\nMWEB to {−1, 1}-MWEB by setting γ = 0, α = −1 and β = 1. Since δ = 1\n2,\naccording to Corollary 1, it is sufficient to set N = η4 so that the probability of\nobtaining a n5ǫ-approximation for {−1, 0, 1}-MWEB is at least\n1\npoly(n).\n⊓⊔\nTheorem 3. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate {−1, 1}-MWEB within a factor of n\n1\n5 −ǫ with probability at least\n1\npoly(n)\nunless RP = NP.\nProof. This follows directly from Theorem 2 and Lemma 5.\n⊓⊔\n\nInapproximability of Maximum Weighted Edge Biclique and Its Applications\n7\n2.4\n{−η\n1\n2 −δ, 1}-MWEB and {−ηδ−1\n2 , 1}-MWEB\nIn this section, we consider the generalized cases of the S-MWEB problem.\nTheorem 4. For any δ ∈(0, 1\n2], there exists some constant ǫ such that no poly-\nnomial time algorithm can approximate {−η\n1\n2 −δ, 1}-MWEB within a factor of\nnǫ with probability at least\n1\npoly(n) unless RP = NP. The same statement holds\nfor {−ηδ−1\n2 , 1}-MWEB.\nProof. We prove this by first construct a {γ, {α, β}}-Product from {−1, 1}-\nMWEB to {−η\n1\n2 −δ, 1}-MWEB by setting γ = −1, α = −(Nη)\n1\n2 −δ and β = 1. By\nCorollary 1, we know that for any δ ∈(0, 1\n2], if there exists a polynomial time al-\ngorithm that approximates {−η\n1\n2 −δ, 1}-MWEB within a factor of nǫ, then there\nexists a polynomial time algorithm that approximates {−1, 1}-MWEB within a\nfactor of n(1+ δ(3−2δ)+3\nδ(1+2δ)\n)ǫ with probability at least\n1\npoly(n). So invoking the hardness\nresult in Theorem 3 gives the desired hardness result for {−η\n1\n2 −δ, 1}-MWEB.\nThe same conclusion applies to {−1, η\n1\n2 −δ}-MWEB by setting γ = 1, α = −1\nand β = (Nη)\n1\n2 −δ. Since η is a constant for any given graph, we can simply divide\neach weight in {−1, η\n1\n2 −δ} by η\n1\n2 −δ.\n⊓⊔\nTheorem 4 leads to the following general statement.\nTheorem 5. For any small constant δ ∈(0, 1\n2], if\n min S\nmax S\n ∈Ω(ηδ−1/2)∩O(η1/2−δ),\nthen there exists some constant ǫ such that no polynomial time algorithm can ap-\nproximate S-MWEB within a factor of nǫ with probability at least\n1\npoly(n) unless\nRP = NP.\n3\nTwo Applications\nIn this section, we describe two applications of the results establish in Sect. 3 by\nproving hardness and inapproximability of problems found in practice.\n3.1\nSAMBA Model is Hard\nMicroarray technology has been the latest technological breakthrough in biolog-\nical and biomedical research; in many applications, a key step in analyzing gene\nexpression data obtained through microarray is the identification of a bicluster\nsatisfying certain properties and with largest area (see the survey [13] for a fairly\nextensive discussion on this).\nIn particular, Tanay et. al. [18] considered the Statistical-Algorithmic Method\nfor Bicluster Analysis (SAMBA) model. In their formulation, a complete bipar-\ntite graph is given where one side corresponds to genes and the other size cor-\nresponds to conditions. An edges (u, v) is assigned a real weight which could be\neither positive or negative, depending on the expression level of gene u in condi-\ntion v, in a way such that heavy subgraphs corresponds to statistically significant\n\n8\nJinsong Tan\nbiclusters. Two weight-assigning schemes are considered in their paper. In the\nfirst, or simple statistical model, a tight upper-bound on the probability of an\nobserved biclusters in computed; in the second, or refined statistical model, the\nweights are assigned in a way such that a maximum weight biclique subgraph\ncorresponds to a maximum likelihood bicluster.\nThe Simple SAMBA Statistical Model: Let H = (V ′\n1, V ′\n2, E′) be a subgraph\nof G = (V1, V2, E), E′ = {V ′\n1 × V ′\n2} −E′ and p =\n|E|\n|V1||V2|. The simple statistical\nmodel assumes that edges occur independently and identically at random with\nprobability p. Denote by BT (k, p, n) the probability of observing k or more\nsuccesses in n binomial trials, the probability of observing a graph at least as\ndense as H is thus p(H) = BT (|E′|, p, |V ′\n1||V ′\n2|). This model assumes p < 1\n2 and\n|V ′\n1||V ′\n2| ≪|V1||V2|, therefore p(H) is upper bounded by\np∗(H) = 2|V ′\n1 ||V ′\n2|p|E′|(1 −p)|V ′\n1 ||V ′\n2|−|E′|\nThe goal of this model is thus to find a subgraph H with the smallest p∗(H).\nThis is equivalent to maximizing\n−log p∗(H) = |E′|(−1 −log p) + (|V ′\n1||V ′\n2| −|E′|)(−1 −log (1 −p))\nwhich is essentially solving a S-MWEB problem that assigns either positive\nweight (−1 −log p) or negative weight (−1 −log (1 −p)) to an edge (u, v), de-\npending on whether gene u express or not in condition v, respectively. The\nsummation of edge weights over H is defined as the statistical significance of H.\nSince\n1\nη2 ≤p < 1\n2, asymptotically we have −1−log (1−p)\n−1−log p\n∈Ω(\n1\nlog η) ∩O(1).\nInvoking Theorem 5 gives the following.\nTheorem 6. For the Simple SAMBA Statistical model, there exists some ǫ > 0\nsuch that no polynomial time algorithm, possibly randomized, can find a bicluster\nwhose statistical significance is within a factor of nǫ of optimal unless RP = NP.\nThe Refined SAMBA Statistical Model: In the refined model, each edge\n(u, v) is assumed to take an independent Bernoulli trial with parameter pu,v,\ntherefore p(H) = (Q\n(u,v)∈E′ pu,v)(Q\n(u,v)∈E′(1 −pu,v)) is the probability of ob-\nserving a subgraph H. Since p(H) generally decreases as the size of H increases,\nTanay et al. aims to find a bicluster with the largest (normalized) likelihood ra-\ntio L(H) =\n(Q\n(u,v)∈E′ pc)(Q\n(u,v)∈E′(1 −pc))\np(H)\n, where pc > max(u,v)∈E pu,v is a\nconstant probability and chosen with biologically sound assumptions. Note this\nis equivalent to maximizing the log-likelihood ratio\nlog L(H) =\nX\n(u,v)∈E′\nlog pc\npu,v\n+\nX\n(u,v)∈E′\nlog 1 −pc\n1 −pu,v\nWith this formulation, each edge is assigned weight either log\npc\npu,v\n> 0 or\nlog\n1−pc\n1−pu,v < 0 and finding the most statistically significant bicluster is equiva-\nlent to solving S-MWEB with S = {log\n1−pc\n1−pu,v , log\npc\npu,v }. Since pc is a constant\n\nInapproximability of Maximum Weighted Edge Biclique and Its Applications\n9\nand\n1\nη2 ≤pu,v < pc, we have log (1−pc)−log (1−pu,v)\nlog pc−log pu,v\n∈Ω(\n1\nlog η) ∩O(1). Invoking\nTheorem 5 gives the following.\nTheorem 7. For the Refined SAMBA Statistical model, there exists some ǫ > 0\nsuch that no polynomial time algorithm, possibly randomized, can find a bicluster\nwhose log-likelihood is within a factor of nǫ of optimal unless RP = NP.\n3.2\nMinimum Description Length with Holes (MDLH) is Hard\nBu et. al [4] considered the Minimum Description Length with Holes problem\n(defined in the following); the 2-dimensional case is claimed NP-hard in this\npaper and the proof is referred to [3]. However, the proof given in [3] suffers\nfrom an error in its reduction3, thus whether MDLH is NP-complete remains\nunsettled. In this section, by employing the results established in the previous\nsections, we show that no polynomial time algorithm exists for MDLH, under\nthe slightly weaker (than P ̸= NP) but widely believed assumption RP ̸= NP.\nWe first briefly describe the Minimum Description Length summarization\nwith Holes problem; for a detailed discussion of the subject, we refer the readers\nto [3,4].\nSuppose one is given a k-dimensional binary matrix M, where each entry is\nof value either 1, which is of interest, or of value 0, which is not of interest. Be-\nsides, there are also k hierarchies (trees) associated with each dimension, namely\nT1, T2, ..., Tk, each of height l1, l2, ..., lk respectively. Define level l = maxi(li).\nFor each Ti, there is a bijection between its leafs and the ’hyperplanes’ in the\nith dimension (e.g. in a 2-dimensional matrix, these hyperplanes corresponds to\nrows and columns). A region is a tuple (x1, x2, ..., xk), where xi is a leaf node\nor an internal node in hierarchy Ti. Region (x1, x2, ..., xk) is said to cover cell\n(c1, c2, ..., ck) if ci is a descendant of xi, for all 1 ≤i ≤k. A k-dimensional l-level\nMDLH summary is defined as two sets S and H, where 1) S is a set of regions\ncovering all the 1-entries in M; and 2) H is the set of 0-entries covered (unde-\nsirably) by S and to be excluded from the summary. The length of a summary\nis defined as |S| + |H|, and the MDLH problem asks the question if there exists\na MDLH summary of length at most K, for a given K > 0.\nIn an effort to establish hardness of MDLH, we first define the following\nproblem, which serves as an intermediate problem bridging {−1, 1}-MWEB and\nMDLH.\nDefinition 3. (Problem P)\nInstance: A complete bipartite graph G = (V1, V2, E) where each edge takes on\na value in {−1, 1}, and a positive integer k.\nQuestion:\nDoes there exist an induced subgraph (a biclique U1 × U2) whose\ntotal weight of edges is ω, such that |U1| + |U2| + ω ≥k.\nLemma 6. No polynomial time algorithm exists for Problem P unless RP = NP.\n3 In Lemma 3.2.1 of [3], the reduction from CLIQUE to CEW is incorrect.\n\n10\nJinsong Tan\nProof. We prove this by constructing a reduction from {−1, 1}-MWEB to Prob-\nlem P as follows: for the given input biclique G = (V1, V2, E), make N duplicates\nof V1 and N duplicates of V2, where N = (|V1| + |V2|)2. Connect each copy of\nV1 to each copy of V2 in a way that is identical to the input biclique, we then\nclaim that there is a size k solution to {−1, 1}-MWEB if and only if there is a\nsize N 2k solution to Problem P.\nIf there is a size k solution to {−1, 1}-MWEB, then it is straightforward that\nthere is a solution to Problem P of size at least N 2k. For the reverse direction, we\nshow that if no solution to {−1, 1}-MWEB is of size at least k, then the maximum\nsolution to Problem P is strictly less than N 2k. Note a solution U N\n1 × U N\n2\nto\nProblem P consists of at most N 2 (not necessarily all distinct) solutions to\n{−1, 1}-MWEB, and each of them can contribute at most (k −1) in weight to\nU N\n1 × U N\n2 , so the total weight gained from edges is at most N 2(k −1). And note\nthe total weight gained from vertices is at most N(|V1|+|V2|) = N\n√\nN, therefore\nthe weight is upper bounded by N\n√\nN + N 2(k −1) < N 2k and this completes\nthe proof.\nAs a conclusion, we have a polynomial time reduction from {−1, 1}-MWEB\nto Problem P. Since no polynomial time algorithm exists for {−1, 1}-MWEB\nunless RP = NP, the same holds for Problem P.\n⊓⊔\nTheorem 8. No polynomial time algorithm exists for MDLH summarization,\neven in the 2-dimension 2-level case, unless RP = NP.\nProof. We prove this by showing that Problem P is a complementary problem\nof 2-dimensional 2-level MDLH.\nLet the input 2D matrix M be of size n1×n2, with a tree of height 2 associated\nwith each dimension. Without loss of generality, we only consider the ’sparse’\ncase where the number of 1-entries is less than the number of 0-entries by at\nleast 2 so that the optimal solution will never contain the whole matrix as one\nof its regions. Let S be the set of regions in a solution. Let R and C be the set\nof rows and columns not included in S. Let Z be the set of all zero entries in M.\nLet z be the total number of zero entries in the R × C ’leftover’ matrix and let\nw be the total number of 1-entries in it. MDLH tries to minimize the following:\n(n1 −|R|) + (n2 −|C|) + (|Z| −z) + w = (n1 + n2 + |Z|) −(|R| + |C| + z −w)\nSince (n1 + n2 + |Z|) is a fixed quantity for any given input matrix, the 2-\ndimensional 2-level MDLH problem is equivalent to maximizing (|R|+|C|+z−w),\nwhich is precisely the definition of Problem P.\nTherefore, 2-dimensional 2-level MDLH is a complementary problem to Prob-\nlem P and by Lemma 6 we conclude that no polynomial time algorithm exists\nfor 2-dimensional 2-level MDLH unless RP = NP.\n⊓⊔\n4\nConcluding Remarks\nMaximum weighted edge biclique and its variants have received much atten-\ntion in recently years because of it wide range of applications in various fields\n\nInapproximability of Maximum Weighted Edge Biclique and Its Applications\n11\nincluding machine learning, database, and particularly bioinformatics and com-\nputational biology, where many computational problems for the analysis of mi-\ncroarray data are closely related. To tackle these applied problems, various kinds\nof heuristics are proposed and experimented and it is not known whether these\nalgorithms give provable approximations. In this work, we answer this question\nby showing that it is highly unlikely (under the assumption RP ̸= NP) that good\npolynomial time approximation algorithm exists for maximum weighted edge\nbiclique for a wide range of choices of weight; and we further give specific appli-\ncations of this result to two applied problems. We conclude our work by listing\na few open questions.\n1. We have shown that {Θ(−ηδ), 1}-MWEB is nǫ-inapproximable for δ ∈\n(−1\n2, 1\n2); also it is easy to see that (i) the problem is in P when δ ≤−1, where\nthe entire input graph is the optimal solution; (ii) for any δ ≥1, the problem is\nequivalent to MEB, which is conjectured to be nǫ-inapproximable [8]. Therefore\nit is natural to ask what is the approximability of the {−nδ, 1}-MWEB problem\nwhen δ ∈(−1, −1\n2] and δ ∈[ 1\n2, 1]. In particular, can this be answered by a better\nanalysis of Lemma 3?\n2. We are especially interested in {−1, 1}-MWEB, which is closely related\nto the formulations of many natural problems [1,3,4,18]. We have shown that\nno polynomial time algorithm exists for this problem unless RP = NP, and we\nbelieve this problem is NP-complete, however a proof has eluded us so far.\nReferences\n1. N. Bansal, A. Blum, and S. Chawla. Correlation clustering, Machine Learning,\n56:89-113, 2004.\n2. A. Ben-Dor, B. Chor, R. Karp, and Z. Yakhini. Discovering local structure in\ngene expression data: The Order-Preserving Submatrix Problem. In Proceedings of\nRECOMB’02, 49-57, 2002.\n3. S. Bu. The summarization of hierarchical data with exceptions. Master The-\nsis, Department of Computer Science, University of British Columbia, 2004.\nhttp://www.cs.ubc.ca/grads/resources/thesis/Nov04/Shaofeng Bu.pdf\n4. S. Bu, L. V. S. Lakshmanan, R. T. Ng. MDL Summarization with Holes. In Pro-\nceedings of VLDB’05, 433-444, 2005.\n5. Y. Cheng, and G. Church. Biclustering of expression data. In Proceedings of\nISMB’00, 93-103. AAAI Press, 2000.\n6. M. Dawande, P. Keskinocak, J. M. Swaminathan, and S. Tayur. On Bipartite and\nmultipartite clique problems. Journal of Algorithms, 41(2):388-403, 2001.\n7. U. Feige. Relations between average case complexity and approximation complex-\nity. In Proceedings of STOC’02, 534-543, 2002.\n8. U. Feige and S. Kogan. Hardness of approximation of the Balanced Complete\nBipartite Subgraph problem. Technical Report MCS04-04, The Weizmann Institute\nof Science, 2004.\n9. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the\nTheory of NP-completeness. Freeman, San Francisco, 1979.\n10. P. Fontana, S. Guha and J. Tan. Recursive MDL Summarization and Approxima-\ntion Algorithms. Preprint, 2007.\n\n12\nJinsong Tan\n11. J. H ̊astad. Clique is hard to approximate within n1−ǫ. Acta Mathematica, 182:105-\n142, 1999.\n12. S. Khot. Ruling out PTAS for Graph Min-Bisection, Densest Subgraph and Bipar-\ntite Clique. In Proceedings of FOCS’04, 136-145, 2004.\n13. S. C. Madeira, and A. L. Oliveira. Biclustering algorithms for biological data anal-\nysis: a survey. IEEE/ACM Transactions on Computational Biology and Bioinfor-\nmatics, 1:24-45, 2004.\n14. N. Mishra, D. Ron, and R. Swaminathan. On finding large conjunctive clusters. In\nProceedings of COLT’03, 448-462, 2003.\n15. R. Peeters. The maximum edge biclique problem is NP-complete. Discrete Applied\nMathematics, 131:651-654, 2003.\n16. J. M. Swaminathan and S. Tayur. Managing Broader Product Lines Through De-\nlayed Differentiation Using Vanilla Boxes. Management Science, 44:161-172, 1998.\n17. J. Tan, K. Chua, L. Zhang, and S. Zhu. Algorithmic and Complexity Issues of\nThree Clustering Methods in Microarray Data Analysis Algorithmica, 48(2): 203-\n219, 2007.\n18. A. Tanay, R. Sharan, and R. Shamir. Discovering statistically significant biclusters\nin gene expression data. Bioinformatics, 18, Supplement 1:136-144, 2002.\n19. L. Zhang, and S. Zhu. A New Clustering Method for Microarray Data Analysis.\nIn Proceedings of CSB’02, 268-275, 2002.\n20. D. Zuckerman. Linear Degree Extractors and the Inapproximability of Max Clique\nand Chromatic Number. In Proceedings of STOC’06, 681-690, 2006.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0704.0468v2 [cs.CC] 23 Mar 2009\nInapproximability of Maximum Weighted Edge\nBiclique and Its Applications\nJinsong Tan\nDepartment of Computer and Information Science\nSchool of Engineering and Applied Science\nUniversity of Pennsylvania, Philadelphia, PA 19104, USA\njinsong@seas.upenn.edu\nAbstract. Given a bipartite graph G = (V1, V2, E) where edges take\non both positive and negative weights from set S, the maximum weighted\nedge biclique problem, or S-MWEB for short, asks to find a bipartite sub-\ngraph whose sum of edge weights is maximized. This problem has various\napplications in bioinformatics, machine learning and databases and its\n(in)approximability remains open. In this paper, we show that for a wide\nrange of choices of S, specifically when\n ̨ ̨ min S\nmax S\n ̨ ̨ ∈Ω(ηδ−1/2) ∩O(η1/2−δ)\n(where η = max{|V1|, |V2|}, and δ ∈(0, 1/2]), no polynomial time algo-\nrithm can approximate S-MWEB within a factor of nǫ for some ǫ > 0\nunless RP = NP. This hardness result gives justification of the heuristic\napproaches adopted for various applied problems in the aforementioned\nareas, and indicates that good approximation algorithms are unlikely to\nexist. Specifically, we give two applications by showing that: 1) finding\nstatistically significant biclusters in the SAMBA model, proposed in [18]\nfor the analysis of microarray data, is nǫ-inapproximable; and 2) no poly-\nnomial time algorithm exists for the Minimum Description Length with\nHoles problem [4] unless RP = NP.\n1\nIntroduction\nLet G = (V1, V2, E) be an undirected bipartite graph. A biclique subgraph in G\nis a complete bipartite subgraph of G and maximum edge biclique (MEB) is the\nproblem of finding a biclique subgraph with the most number of edges. MEB is\na well-known problem and received much attention in recent years because of\nits wide range of applications in areas including machine learning [14], manage-\nment science [16] and bioinformatics, where it is found particularly relevant in\nthe formulation of numerous biclustering problems for biological data analysis\n[5,2,18,19,17], and we refer readers to the survey by Madeira and Oliveira [13]\nfor a fairly extensive discussion on this. Maximum edge biclique is shown to be\nNP-hard by Peeters [15] via a reduction from 3SAT. Its approximability status,\non the other hand, remains an open question despite considerable efforts [7,8,12]\n1. In particular, Feige and Kogan [8] conjectured that maximum edge biclique\n1 Note it might be easy to confuse the MEB problem with the Bipartite Clique problem\ndiscussed by Khot in [12]. Bipartite Clique, which also known as Balanced Complete"},{"paragraph_id":"p2","order":2,"text":"2\nJinsong Tan\nis hard to approximate within a factor of nǫ for some ǫ > 0. In this paper, we\nconsider a weighted formulation of this problem defined as follows\nDefinition 1. S-Maximum Weighted Edge Biclique (S-MWEB)\nInstance: A complete bipartite graph G = (V1, V2, E) (throughout the paper, let\nη = max{|V1|, |V2|} and n = |V1| + |V2|), a weight function wG : E →S, where\nS is a set consisting of both positive and negative integers.\nQuestion: Find a biclique subgraph of G where the sum of weights on edges is\nmaximized.\nA few comments are in order. First note it is not a lose of generality but a\ntechnical convenience to require the graph be complete, one can always think of\nan incomplete bipartite graph as complete where non-edges are assigned weight\n0. Also note we require that both positive and negative weights be in S at the\nsame time because otherwise S-MWEB becomes a trivial problem.\nOur study of S-MWEB is motivated by the problem of finding statistically\nsignificant biclusters in microarray data analysis in the SAMBA model [18]\nand the Minimum Description Length with Holes (MDLH) problem [3,4,10];\ndetailed discussion of the two problems can be found in Sect. 4. Our main\ntechnical contribution of this paper is to show that if S satisfies the condition\n| min S\nmax S | ∈Ω(ηδ−1/2) ∩O(η1/2−δ), where δ > 0 is any arbitrarily small constant,\nthen no polynomial time algorithm can approximate S-MWEB within a factor\nof nǫ for some ǫ > 0 unless RP = NP. This result enables us to answer open\nquestions regarding the hardness of the SAMBA model and the MDLH prob-\nlem. Since maximum edge biclique can be characterized as a special case of\nS-MWEB with S = {−η, 1}, the nǫ-inapproximability result also provides inter-\nesting insights into the conjectured nǫ-inapproximability [8] of maximum edge\nbiclique.\nThe rest of the paper is organized in three sections. In Sect. 2, we present\nthe main technical result by proving the aforementioned inapproximability of S-\nMWEB. We give applications of this by answering hardness questions regarding\ntwo applied problems in Sect. 3. We conclude this work by raising a few open\nproblems in the last section.\n2\nApproximating S-Maximum Edge Biclique is Hard\nWe start this section by giving two lemmas about CLIQUE, which will be used\nin establishing inapproximability for the biclique problems we consider later.\nLemma 1 is a recent result by Zuckerman [20], obtained by a derandomization\nof results of H ̊astad [11]; Lemma 2 follows immediately from Lemma 1.\nLemma 1. ([20]) It is NP-hard to approximate CLIQUE within a factor of\nn1−ǫ, for any ǫ > 0.\nBipartite Subgraph [8], aims to maximize the number of vertices of a balanced sub-\ngraph whereas MEB aims to maximize the total weights on edges in a (not necessarily\nbalanced) subgraph."},{"paragraph_id":"p3","order":3,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n3\nLemma 2. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate CLIQUE within a factor of n1−ǫ with probability at least\n1\npoly(n) unless\nRP = NP.\n2.1\nA Technical Lemma\nWe first describe the construction of a structure called {γ, {α, β}}-Product,\nwhich will be used in the proof of our main technical lemma.\nDefinition 2. ({γ, {α, β}}-Product)\nInput: An instance of S-MWEB on complete bipartite graph G = V1×V2, where\nγ ∈S and α < γ < β; an integer N.\nOutput: Complete bipartite graph GN = V N\n1\n× V N\n2\nconstructed as follows: V N\n1\nand V N\n2\nare N duplicates of V1 and V2, respectively. For each edge (i, j) ∈GN,\nlet (φ(i), φ(j)) be the corresponding edge in G. If wG(φ(i), φ(j)) = γ, assign\nweight α or β to (i, j) independently and identically at random with expectation\nbeing γ, denote the weight by random variable X. If wG(φ(i), φ(j)) ̸= γ, then\nkeep the weight unchanged. Call the weight function constructed this way w(·).\nFor any subgraph H of GN, denote by wγ(H) (resp., w−γ(H)) the total\nweight of H contributed by former-γ-edges (resp., other edges). Clearly, w(H) =\nwγ(H) + w−γ(H).\nWith a graph product constructed in this randomized fashion, we have the fol-\nlowing lemma.\nLemma 3. Given an S-MWEB instance G = (V1, V2, E) where γ ∈S, and a\nnumber δ ∈(0, 1\n2]; let η = max (|V1|, |V2|), N = η\nδ(3−2δ)+3\nδ(1+2δ) , GN = (V N\n1 , V N\n2 , E)\nbe the {γ, {α, β}}-product of G and S′ = (S ∪{α, β}) −{γ}. If\n1. |β −α| = O((Nη)\n1\n2 −δ); and\n2. there is a polynomial time algorithm that approximates the S′-MWEB\ninstance within a factor of λ, where λ is some arbitrary function in the size of\nthe S′-MWEB instance\nthen there exists a polynomial time algorithm that approximates the S-MWEB\ninstance within a factor of λ, with probability at least\n1\npoly(n).\nProof. For notational convenience, we denote η\n1\n2 −δ by f(η) throughout the proof.\nDefine random variable Y = X −γ, clearly E[Y ] = 0. Suppose there is a poly-\nnomial time algorithm A that approximates S′-MWEB within a factor of λ, we\ncan then run A on GN, the output biclique G∗\nB corresponds to N 2 bicliques in\nG (not necessarily all distinct). Let G∗\nA be the most weighted among these N 2\nsubgraphs of G, in the rest of the proof we show that with high probability, G∗\nA\nis a λ-approximation of S-MWEB on G.\nDenote by E1 the event that G∗\nB does not imply a λ-approximation on G.\nLet H be the set of subgraphs of GN that do not imply a λ-approximation on G,"},{"paragraph_id":"p4","order":4,"text":"4\nJinsong Tan\nclearly, |H| ≤4Nη. Let H′ be an arbitrary element in H, we have the following\ninequalities\nPr {E1} ≤Pr"},{"paragraph_id":"p5","order":5,"text":"at least one element in H is a λ-approximation of GN \n≤4Nη · Pr"},{"paragraph_id":"p6","order":6,"text":"H′ is a λ-approximation of GN \n= 4Nη · Pr{E2}\nwhere E2 is the event that H′ is a λ-approximation of GN.\nLet the weight of an optimal solution U1×U2 of G be K, denote by U N\n1 ×U N\n2\nthe corresponding N 2-duplication in GN. Let x1 and x2 be the number of former-\nγ-edges in H′ and U N\n1 × U N\n2 , respectively. Suppose E2 happens, then we must\nhave\nw−γ(H′) + x1γ ≤N 2( K\nλ −1)\nw−γ(H′) + wγ(H′) ≥1\nλ(w−γ(U N\n1 × U N\n2 ) + wγ(U N\n1 × U N\n2 ))\nwhere the first inequality follows from the fact that we only consider integer\nweights. Since w−γ(U N\n1 × U N\n2 ) = N 2K −x2γ, it implies\n(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2\nso we have the following statement on probability\nPr{E2} ≤Pr"},{"paragraph_id":"p7","order":7,"text":"(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2 \nLet z1 (resp., z2 and z3) be the number of edges in E(H′) −E(U N\n1 × U N\n2 )\n( resp., E(U N\n1 × U N\n2 ) −E(H′) and E(U N\n1 × U N\n2 ) ∩E(H′) ) transformed from\nformer-γ-edges in G. We have\nPr"},{"paragraph_id":"p8","order":8,"text":"(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2 \n= Pr\nnPz1\ni=1 Yi −1\nλ\nPz2\nj=1 Yj + λ−1\nλ\nPz3\nk=1 Yk ≥N 2o\n= Pr\nnPz1\ni=1 Yi + 1\nλ\nPz2\nj=1 (−Yj) + λ−1\nλ\nPz3\nk=1 Yk ≥N 2o\n≤Pr\nnPz1\ni=1 Yi ≥N 2\n3\no\n+ Pr\nn\n1\nλ\nPz2\nj=1 (−Yj) ≥N 2\n3\no\n+ Pr\nn\nλ−1\nλ\nPz3\nk=1 Yk ≥N 2\n3\no\n≤Pr\nnPz1\ni=1 Yi ≥N 2\n3\no\n+ Pr\nnPz2\nj=1 (−Yj) ≥N 2\n3\no\n+ Pr\nnPz3\nk=1 Yk ≥N 2\n3\no\n≤P\ni∈{1,2,3}"},{"paragraph_id":"p9","order":9,"text":"exp"},{"paragraph_id":"p10","order":10,"text":"−2zi"},{"paragraph_id":"p11","order":11,"text":"N 2\n3zi(c1f(Nη))\n 2 \n(Hoeffding bound)\n≤3 · exp"},{"paragraph_id":"p12","order":12,"text":"−c2 · N 1+2δ\nη3−2δ"},{"paragraph_id":"p13","order":13,"text":"(zi ≤η2N 2)\nwhere c1, c2 are constants (c2 > 0). Now if we set N = η\n3−2δ\n1+2δ +θ for some θ, we\nhave\nPr {E1} ≤4Nη · Pr {E2} ≤3 · exp"},{"paragraph_id":"p14","order":14,"text":"ln 4 · η\n4\n(1+2δ) +θ −c2 · η(1+2δ)θ \nFor this probability to be bounded by 1\n2 as η is large enough, we need to have\n4\n1+2δ +θ < (1+2δ)θ. Solving this inequality gives θ >\n2\nδ(1+2δ). Therefore, for any\nδ ∈(0, 1\n2], by setting N = η\nδ(3−2δ)+3\nδ(1+2δ) , we have Pr{E1}, i.e. the probability that"},{"paragraph_id":"p15","order":15,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n5\nthe solution returned by A does not imply a λ-approximation of G, is bounded\nfrom above by 1\n2 once input size is large enough. This gives a polynomial time\nalgorithm that approximates S-MWEB within a factor of λ with probability at\nleast 1\n2.\n⊓⊔\nThis lemma immediately leads to the following corollary.\nCorollary 1. Following the construction in Lemma 3, if S′-MWEB can be ap-\nproximated within a factor of nǫ′, for some ǫ′ > 0, then there exists a polyno-\nmial time algorithm that approximates S-MWEB within a factor of nǫ, where\nǫ = (1 + δ(3−2δ)+3\nδ(1+2δ) )ǫ′, with probability at least\n1\npoly(n). 2\nProof. Let |G| and |GN| be the number of nodes in the S-MWEB and S′-MWEB\nproblem, respectively. Since λ = |GN|ǫ′ ≤|G|(1+ δ(3−2δ)+3\nδ(1+2δ)\n)ǫ′, our claim follows\nfrom Lemma 3.\n⊓⊔\n2.2\n{−1, 0, 1}-MWEB\nIn this section, we prove inapproximability of {−1, 0, 1}-MWEB by giving a\nreduction from CLIQUE; in subsequence sections, we prove inapproximability\nresults for more general S-MWEB by constructing randomized reduction from\n{−1, 0, 1}-MWEB.\nLemma 4. The decision version of the {−1, 0, 1}-MWEB problem is NP-complete.\nProof. We prove this by describing a reduction from CLIQUE. Given a CLIQUE\ninstance G = (V, E), construct G′ = (V ′, E′) such that V ′ = V1∪V2 where V1, V2\nare duplicates of V in that there exist bijections φ1 : V1 →V and φ2 : V2 →V .\nAnd\nE′ = E1 ∪E2 ∪E3\nE1 = {(u, v) | u ∈V1, v ∈V2 and (φ1(u), φ2(v)) ∈E}\nE2 = {(u, v) | u ∈V1, v ∈V2, φ1(u) ̸= φ2(v) and (φ1(u), φ2(v)) /∈E}\nE3 = {(u, v) | u ∈V1, v ∈V2, and φ1(u) = φ2(v)}\nClearly, G′ is a biclique. Now assign weight 0 to edges in E1, −1 to edges in\nE2 and 1 to edges in E3. We then claim that there is a clique of size k in G if\nand only if there is a biclique of total edge weight k in G′.\nFirst consider the case where there is a clique of size k in G, let U be the set\nof vertices of the clique, then taking the subgraph induced by φ−1\n1 (U) × φ−1\n2 (U)\nin G′ gives us a biclique of total weight k.\nNow suppose that there is a biclique U1 ×U2 of total weight k in G′. Without\nloss of generality, assume U1 and U2 correspond to the same subset of vertices in\n2 Note we are slightly abusing notation here by always representing the size of a given\nproblem under discussion by n. Here n refers to the size of S′-MWEB (resp. S-\nMWEB) when we are talking about approximation factor nǫ′ (resp. nǫ). We adopt\nthe same convention in the sequel."},{"paragraph_id":"p16","order":16,"text":"6\nJinsong Tan\nV because if (φ1(U1) −φ2(U2)) ∪(φ2(U2) −φ1(U1)) is not empty, then removing\n(U1 −U2) ∪(U2 −U1) will never decrease the total weight of the solution. Given\nφ1(U1) = φ2(U2), we argue that there is no edge of weight −1 in biclique U1×U2;\nsuppose otherwise there exists a weight −1 edge (i1, j2) (i1 ∈U1, and j2 ∈U2),\nthen the corresponding edge (j1, i2) (j1 ∈U1, and i2 ∈U2) must be of weight\n−1 too and removing i1, i2 from the solution biclique will increase total weight\nby at least 1 because among all edges incident to i1 and i2, (i1, i2) is of weight 1,\n(i1, j2) and (i2, j1) are of weight −1 and the rest are of weights either 0 or −1.\nTherefore, we have shown that if there is a solution U1 × U2 of weight k in\nG′, U1 and U2 correspond to the same set of vertices U ∈V and U is a clique of\nsize k. It is clear that the reduction can be performed in polynomial time and\nthe problem is NP, and thus NP-complete.\n⊓⊔\nGiven Lemma 1, the following corollary follows immediately from the above\nreduction.\nTheorem 1. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate problem {−1, 0, 1}-MWEB within a factor of n1−ǫ unless P = NP.\nProof. It is obvious that the reduction given in the proof of Lemma 4 preserves\ninapproximability exactly, and given that CLIQUE is hard to approximate within\na factor of n1−ǫ unless P = NP, the theorem follows.\n⊓⊔\nTheorem 2. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate {−1, 0, 1}-MWEB within a factor of n1−ǫ with probability at least\n1\npoly(n)\nunless RP = NP.\nProof. If there exists such a randomized algorithm for {−1, 0, 1}-MWEB, com-\nbining it with the reduction given in Lemma 4, we obtain an RP algorithm for\nCLIQUE. This is impossible unless RP = NP.\n⊓⊔\n2.3\n{−1, 1}-MWEB\nLemma 5. If there exists a polynomial time algorithm that approximates {−1, 1}-\nMWEB within a factor of nǫ, then there exists a polynomial time algorithm that\napproximates {−1, 0, 1}-MWEB within a factor of n5ǫ with probability at least\n1\npoly(n).\nProof. We prove this by constructing a {γ, {α, β}}-Product from {−1, 0, 1}-\nMWEB to {−1, 1}-MWEB by setting γ = 0, α = −1 and β = 1. Since δ = 1\n2,\naccording to Corollary 1, it is sufficient to set N = η4 so that the probability of\nobtaining a n5ǫ-approximation for {−1, 0, 1}-MWEB is at least\n1\npoly(n).\n⊓⊔\nTheorem 3. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate {−1, 1}-MWEB within a factor of n\n1\n5 −ǫ with probability at least\n1\npoly(n)\nunless RP = NP.\nProof. This follows directly from Theorem 2 and Lemma 5.\n⊓⊔"},{"paragraph_id":"p17","order":17,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n7\n2.4\n{−η\n1\n2 −δ, 1}-MWEB and {−ηδ−1\n2 , 1}-MWEB\nIn this section, we consider the generalized cases of the S-MWEB problem.\nTheorem 4. For any δ ∈(0, 1\n2], there exists some constant ǫ such that no poly-\nnomial time algorithm can approximate {−η\n1\n2 −δ, 1}-MWEB within a factor of\nnǫ with probability at least\n1\npoly(n) unless RP = NP. The same statement holds\nfor {−ηδ−1\n2 , 1}-MWEB.\nProof. We prove this by first construct a {γ, {α, β}}-Product from {−1, 1}-\nMWEB to {−η\n1\n2 −δ, 1}-MWEB by setting γ = −1, α = −(Nη)\n1\n2 −δ and β = 1. By\nCorollary 1, we know that for any δ ∈(0, 1\n2], if there exists a polynomial time al-\ngorithm that approximates {−η\n1\n2 −δ, 1}-MWEB within a factor of nǫ, then there\nexists a polynomial time algorithm that approximates {−1, 1}-MWEB within a\nfactor of n(1+ δ(3−2δ)+3\nδ(1+2δ)\n)ǫ with probability at least\n1\npoly(n). So invoking the hardness\nresult in Theorem 3 gives the desired hardness result for {−η\n1\n2 −δ, 1}-MWEB.\nThe same conclusion applies to {−1, η\n1\n2 −δ}-MWEB by setting γ = 1, α = −1\nand β = (Nη)\n1\n2 −δ. Since η is a constant for any given graph, we can simply divide\neach weight in {−1, η\n1\n2 −δ} by η\n1\n2 −δ.\n⊓⊔\nTheorem 4 leads to the following general statement.\nTheorem 5. For any small constant δ ∈(0, 1\n2], if\n min S\nmax S\n ∈Ω(ηδ−1/2)∩O(η1/2−δ),\nthen there exists some constant ǫ such that no polynomial time algorithm can ap-\nproximate S-MWEB within a factor of nǫ with probability at least\n1\npoly(n) unless\nRP = NP.\n3\nTwo Applications\nIn this section, we describe two applications of the results establish in Sect. 3 by\nproving hardness and inapproximability of problems found in practice.\n3.1\nSAMBA Model is Hard\nMicroarray technology has been the latest technological breakthrough in biolog-\nical and biomedical research; in many applications, a key step in analyzing gene\nexpression data obtained through microarray is the identification of a bicluster\nsatisfying certain properties and with largest area (see the survey [13] for a fairly\nextensive discussion on this).\nIn particular, Tanay et. al. [18] considered the Statistical-Algorithmic Method\nfor Bicluster Analysis (SAMBA) model. In their formulation, a complete bipar-\ntite graph is given where one side corresponds to genes and the other size cor-\nresponds to conditions. An edges (u, v) is assigned a real weight which could be\neither positive or negative, depending on the expression level of gene u in condi-\ntion v, in a way such that heavy subgraphs corresponds to statistically significant"},{"paragraph_id":"p18","order":18,"text":"8\nJinsong Tan\nbiclusters. Two weight-assigning schemes are considered in their paper. In the\nfirst, or simple statistical model, a tight upper-bound on the probability of an\nobserved biclusters in computed; in the second, or refined statistical model, the\nweights are assigned in a way such that a maximum weight biclique subgraph\ncorresponds to a maximum likelihood bicluster.\nThe Simple SAMBA Statistical Model: Let H = (V ′\n1, V ′\n2, E′) be a subgraph\nof G = (V1, V2, E), E′ = {V ′\n1 × V ′\n2} −E′ and p =\n|E|\n|V1||V2|. The simple statistical\nmodel assumes that edges occur independently and identically at random with\nprobability p. Denote by BT (k, p, n) the probability of observing k or more\nsuccesses in n binomial trials, the probability of observing a graph at least as\ndense as H is thus p(H) = BT (|E′|, p, |V ′\n1||V ′\n2|). This model assumes p < 1\n2 and\n|V ′\n1||V ′\n2| ≪|V1||V2|, therefore p(H) is upper bounded by\np∗(H) = 2|V ′\n1 ||V ′\n2|p|E′|(1 −p)|V ′\n1 ||V ′\n2|−|E′|\nThe goal of this model is thus to find a subgraph H with the smallest p∗(H).\nThis is equivalent to maximizing\n−log p∗(H) = |E′|(−1 −log p) + (|V ′\n1||V ′\n2| −|E′|)(−1 −log (1 −p))\nwhich is essentially solving a S-MWEB problem that assigns either positive\nweight (−1 −log p) or negative weight (−1 −log (1 −p)) to an edge (u, v), de-\npending on whether gene u express or not in condition v, respectively. The\nsummation of edge weights over H is defined as the statistical significance of H.\nSince\n1\nη2 ≤p < 1\n2, asymptotically we have −1−log (1−p)\n−1−log p\n∈Ω(\n1\nlog η) ∩O(1).\nInvoking Theorem 5 gives the following.\nTheorem 6. For the Simple SAMBA Statistical model, there exists some ǫ > 0\nsuch that no polynomial time algorithm, possibly randomized, can find a bicluster\nwhose statistical significance is within a factor of nǫ of optimal unless RP = NP.\nThe Refined SAMBA Statistical Model: In the refined model, each edge\n(u, v) is assumed to take an independent Bernoulli trial with parameter pu,v,\ntherefore p(H) = (Q\n(u,v)∈E′ pu,v)(Q\n(u,v)∈E′(1 −pu,v)) is the probability of ob-\nserving a subgraph H. Since p(H) generally decreases as the size of H increases,\nTanay et al. aims to find a bicluster with the largest (normalized) likelihood ra-\ntio L(H) =\n(Q\n(u,v)∈E′ pc)(Q\n(u,v)∈E′(1 −pc))\np(H)\n, where pc > max(u,v)∈E pu,v is a\nconstant probability and chosen with biologically sound assumptions. Note this\nis equivalent to maximizing the log-likelihood ratio\nlog L(H) =\nX\n(u,v)∈E′\nlog pc\npu,v\n+\nX\n(u,v)∈E′\nlog 1 −pc\n1 −pu,v\nWith this formulation, each edge is assigned weight either log\npc\npu,v\n> 0 or\nlog\n1−pc\n1−pu,v < 0 and finding the most statistically significant bicluster is equiva-\nlent to solving S-MWEB with S = {log\n1−pc\n1−pu,v , log\npc\npu,v }. Since pc is a constant"},{"paragraph_id":"p19","order":19,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n9\nand\n1\nη2 ≤pu,v < pc, we have log (1−pc)−log (1−pu,v)\nlog pc−log pu,v\n∈Ω(\n1\nlog η) ∩O(1). Invoking\nTheorem 5 gives the following.\nTheorem 7. For the Refined SAMBA Statistical model, there exists some ǫ > 0\nsuch that no polynomial time algorithm, possibly randomized, can find a bicluster\nwhose log-likelihood is within a factor of nǫ of optimal unless RP = NP.\n3.2\nMinimum Description Length with Holes (MDLH) is Hard\nBu et. al [4] considered the Minimum Description Length with Holes problem\n(defined in the following); the 2-dimensional case is claimed NP-hard in this\npaper and the proof is referred to [3]. However, the proof given in [3] suffers\nfrom an error in its reduction3, thus whether MDLH is NP-complete remains\nunsettled. In this section, by employing the results established in the previous\nsections, we show that no polynomial time algorithm exists for MDLH, under\nthe slightly weaker (than P ̸= NP) but widely believed assumption RP ̸= NP.\nWe first briefly describe the Minimum Description Length summarization\nwith Holes problem; for a detailed discussion of the subject, we refer the readers\nto [3,4].\nSuppose one is given a k-dimensional binary matrix M, where each entry is\nof value either 1, which is of interest, or of value 0, which is not of interest. Be-\nsides, there are also k hierarchies (trees) associated with each dimension, namely\nT1, T2, ..., Tk, each of height l1, l2, ..., lk respectively. Define level l = maxi(li).\nFor each Ti, there is a bijection between its leafs and the ’hyperplanes’ in the\nith dimension (e.g. in a 2-dimensional matrix, these hyperplanes corresponds to\nrows and columns). A region is a tuple (x1, x2, ..., xk), where xi is a leaf node\nor an internal node in hierarchy Ti. Region (x1, x2, ..., xk) is said to cover cell\n(c1, c2, ..., ck) if ci is a descendant of xi, for all 1 ≤i ≤k. A k-dimensional l-level\nMDLH summary is defined as two sets S and H, where 1) S is a set of regions\ncovering all the 1-entries in M; and 2) H is the set of 0-entries covered (unde-\nsirably) by S and to be excluded from the summary. The length of a summary\nis defined as |S| + |H|, and the MDLH problem asks the question if there exists\na MDLH summary of length at most K, for a given K > 0.\nIn an effort to establish hardness of MDLH, we first define the following\nproblem, which serves as an intermediate problem bridging {−1, 1}-MWEB and\nMDLH.\nDefinition 3. (Problem P)\nInstance: A complete bipartite graph G = (V1, V2, E) where each edge takes on\na value in {−1, 1}, and a positive integer k.\nQuestion:\nDoes there exist an induced subgraph (a biclique U1 × U2) whose\ntotal weight of edges is ω, such that |U1| + |U2| + ω ≥k.\nLemma 6. No polynomial time algorithm exists for Problem P unless RP = NP.\n3 In Lemma 3.2.1 of [3], the reduction from CLIQUE to CEW is incorrect."},{"paragraph_id":"p20","order":20,"text":"10\nJinsong Tan\nProof. We prove this by constructing a reduction from {−1, 1}-MWEB to Prob-\nlem P as follows: for the given input biclique G = (V1, V2, E), make N duplicates\nof V1 and N duplicates of V2, where N = (|V1| + |V2|)2. Connect each copy of\nV1 to each copy of V2 in a way that is identical to the input biclique, we then\nclaim that there is a size k solution to {−1, 1}-MWEB if and only if there is a\nsize N 2k solution to Problem P.\nIf there is a size k solution to {−1, 1}-MWEB, then it is straightforward that\nthere is a solution to Problem P of size at least N 2k. For the reverse direction, we\nshow that if no solution to {−1, 1}-MWEB is of size at least k, then the maximum\nsolution to Problem P is strictly less than N 2k. Note a solution U N\n1 × U N\n2\nto\nProblem P consists of at most N 2 (not necessarily all distinct) solutions to\n{−1, 1}-MWEB, and each of them can contribute at most (k −1) in weight to\nU N\n1 × U N\n2 , so the total weight gained from edges is at most N 2(k −1). And note\nthe total weight gained from vertices is at most N(|V1|+|V2|) = N\n√\nN, therefore\nthe weight is upper bounded by N\n√\nN + N 2(k −1) < N 2k and this completes\nthe proof.\nAs a conclusion, we have a polynomial time reduction from {−1, 1}-MWEB\nto Problem P. Since no polynomial time algorithm exists for {−1, 1}-MWEB\nunless RP = NP, the same holds for Problem P.\n⊓⊔\nTheorem 8. No polynomial time algorithm exists for MDLH summarization,\neven in the 2-dimension 2-level case, unless RP = NP.\nProof. We prove this by showing that Problem P is a complementary problem\nof 2-dimensional 2-level MDLH.\nLet the input 2D matrix M be of size n1×n2, with a tree of height 2 associated\nwith each dimension. Without loss of generality, we only consider the ’sparse’\ncase where the number of 1-entries is less than the number of 0-entries by at\nleast 2 so that the optimal solution will never contain the whole matrix as one\nof its regions. Let S be the set of regions in a solution. Let R and C be the set\nof rows and columns not included in S. Let Z be the set of all zero entries in M.\nLet z be the total number of zero entries in the R × C ’leftover’ matrix and let\nw be the total number of 1-entries in it. MDLH tries to minimize the following:\n(n1 −|R|) + (n2 −|C|) + (|Z| −z) + w = (n1 + n2 + |Z|) −(|R| + |C| + z −w)\nSince (n1 + n2 + |Z|) is a fixed quantity for any given input matrix, the 2-\ndimensional 2-level MDLH problem is equivalent to maximizing (|R|+|C|+z−w),\nwhich is precisely the definition of Problem P.\nTherefore, 2-dimensional 2-level MDLH is a complementary problem to Prob-\nlem P and by Lemma 6 we conclude that no polynomial time algorithm exists\nfor 2-dimensional 2-level MDLH unless RP = NP.\n⊓⊔\n4\nConcluding Remarks\nMaximum weighted edge biclique and its variants have received much atten-\ntion in recently years because of it wide range of applications in various fields"},{"paragraph_id":"p21","order":21,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n11\nincluding machine learning, database, and particularly bioinformatics and com-\nputational biology, where many computational problems for the analysis of mi-\ncroarray data are closely related. To tackle these applied problems, various kinds\nof heuristics are proposed and experimented and it is not known whether these\nalgorithms give provable approximations. In this work, we answer this question\nby showing that it is highly unlikely (under the assumption RP ̸= NP) that good\npolynomial time approximation algorithm exists for maximum weighted edge\nbiclique for a wide range of choices of weight; and we further give specific appli-\ncations of this result to two applied problems. We conclude our work by listing\na few open questions.\n1. We have shown that {Θ(−ηδ), 1}-MWEB is nǫ-inapproximable for δ ∈\n(−1\n2, 1\n2); also it is easy to see that (i) the problem is in P when δ ≤−1, where\nthe entire input graph is the optimal solution; (ii) for any δ ≥1, the problem is\nequivalent to MEB, which is conjectured to be nǫ-inapproximable [8]. Therefore\nit is natural to ask what is the approximability of the {−nδ, 1}-MWEB problem\nwhen δ ∈(−1, −1\n2] and δ ∈[ 1\n2, 1]. In particular, can this be answered by a better\nanalysis of Lemma 3?\n2. We are especially interested in {−1, 1}-MWEB, which is closely related\nto the formulations of many natural problems [1,3,4,18]. We have shown that\nno polynomial time algorithm exists for this problem unless RP = NP, and we\nbelieve this problem is NP-complete, however a proof has eluded us so far.\nReferences\n1. N. Bansal, A. Blum, and S. Chawla. Correlation clustering, Machine Learning,\n56:89-113, 2004.\n2. A. Ben-Dor, B. Chor, R. Karp, and Z. Yakhini. Discovering local structure in\ngene expression data: The Order-Preserving Submatrix Problem. In Proceedings of\nRECOMB’02, 49-57, 2002.\n3. S. Bu. The summarization of hierarchical data with exceptions. Master The-\nsis, Department of Computer Science, University of British Columbia, 2004.\nhttp://www.cs.ubc.ca/grads/resources/thesis/Nov04/Shaofeng Bu.pdf\n4. S. Bu, L. V. S. Lakshmanan, R. T. Ng. MDL Summarization with Holes. In Pro-\nceedings of VLDB’05, 433-444, 2005.\n5. Y. Cheng, and G. Church. Biclustering of expression data. In Proceedings of\nISMB’00, 93-103. AAAI Press, 2000.\n6. M. Dawande, P. Keskinocak, J. M. Swaminathan, and S. Tayur. On Bipartite and\nmultipartite clique problems. Journal of Algorithms, 41(2):388-403, 2001.\n7. U. Feige. Relations between average case complexity and approximation complex-\nity. In Proceedings of STOC’02, 534-543, 2002.\n8. U. Feige and S. Kogan. Hardness of approximation of the Balanced Complete\nBipartite Subgraph problem. Technical Report MCS04-04, The Weizmann Institute\nof Science, 2004.\n9. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the\nTheory of NP-completeness. Freeman, San Francisco, 1979.\n10. P. Fontana, S. Guha and J. Tan. Recursive MDL Summarization and Approxima-\ntion Algorithms. Preprint, 2007."},{"paragraph_id":"p22","order":22,"text":"12\nJinsong Tan\n11. J. H ̊astad. Clique is hard to approximate within n1−ǫ. Acta Mathematica, 182:105-\n142, 1999.\n12. S. Khot. Ruling out PTAS for Graph Min-Bisection, Densest Subgraph and Bipar-\ntite Clique. In Proceedings of FOCS’04, 136-145, 2004.\n13. S. C. Madeira, and A. L. Oliveira. Biclustering algorithms for biological data anal-\nysis: a survey. IEEE/ACM Transactions on Computational Biology and Bioinfor-\nmatics, 1:24-45, 2004.\n14. N. Mishra, D. Ron, and R. Swaminathan. On finding large conjunctive clusters. In\nProceedings of COLT’03, 448-462, 2003.\n15. R. Peeters. The maximum edge biclique problem is NP-complete. Discrete Applied\nMathematics, 131:651-654, 2003.\n16. J. M. Swaminathan and S. Tayur. Managing Broader Product Lines Through De-\nlayed Differentiation Using Vanilla Boxes. Management Science, 44:161-172, 1998.\n17. J. Tan, K. Chua, L. Zhang, and S. Zhu. Algorithmic and Complexity Issues of\nThree Clustering Methods in Microarray Data Analysis Algorithmica, 48(2): 203-\n219, 2007.\n18. A. Tanay, R. Sharan, and R. Shamir. Discovering statistically significant biclusters\nin gene expression data. Bioinformatics, 18, Supplement 1:136-144, 2002.\n19. L. Zhang, and S. Zhu. A New Clustering Method for Microarray Data Analysis.\nIn Proceedings of CSB’02, 268-275, 2002.\n20. D. Zuckerman. Linear Degree Extractors and the Inapproximability of Max Clique\nand Chromatic Number. In Proceedings of STOC’06, 681-690, 2006."}],"pages":[{"page":1,"text":"arXiv:0704.0468v2 [cs.CC] 23 Mar 2009\nInapproximability of Maximum Weighted Edge\nBiclique and Its Applications\nJinsong Tan\nDepartment of Computer and Information Science\nSchool of Engineering and Applied Science\nUniversity of Pennsylvania, Philadelphia, PA 19104, USA\njinsong@seas.upenn.edu\nAbstract. Given a bipartite graph G = (V1, V2, E) where edges take\non both positive and negative weights from set S, the maximum weighted\nedge biclique problem, or S-MWEB for short, asks to find a bipartite sub-\ngraph whose sum of edge weights is maximized. This problem has various\napplications in bioinformatics, machine learning and databases and its\n(in)approximability remains open. In this paper, we show that for a wide\nrange of choices of S, specifically when\n ̨ ̨ min S\nmax S\n ̨ ̨ ∈Ω(ηδ−1/2) ∩O(η1/2−δ)\n(where η = max{|V1|, |V2|}, and δ ∈(0, 1/2]), no polynomial time algo-\nrithm can approximate S-MWEB within a factor of nǫ for some ǫ > 0\nunless RP = NP. This hardness result gives justification of the heuristic\napproaches adopted for various applied problems in the aforementioned\nareas, and indicates that good approximation algorithms are unlikely to\nexist. Specifically, we give two applications by showing that: 1) finding\nstatistically significant biclusters in the SAMBA model, proposed in [18]\nfor the analysis of microarray data, is nǫ-inapproximable; and 2) no poly-\nnomial time algorithm exists for the Minimum Description Length with\nHoles problem [4] unless RP = NP.\n1\nIntroduction\nLet G = (V1, V2, E) be an undirected bipartite graph. A biclique subgraph in G\nis a complete bipartite subgraph of G and maximum edge biclique (MEB) is the\nproblem of finding a biclique subgraph with the most number of edges. MEB is\na well-known problem and received much attention in recent years because of\nits wide range of applications in areas including machine learning [14], manage-\nment science [16] and bioinformatics, where it is found particularly relevant in\nthe formulation of numerous biclustering problems for biological data analysis\n[5,2,18,19,17], and we refer readers to the survey by Madeira and Oliveira [13]\nfor a fairly extensive discussion on this. Maximum edge biclique is shown to be\nNP-hard by Peeters [15] via a reduction from 3SAT. Its approximability status,\non the other hand, remains an open question despite considerable efforts [7,8,12]\n1. In particular, Feige and Kogan [8] conjectured that maximum edge biclique\n1 Note it might be easy to confuse the MEB problem with the Bipartite Clique problem\ndiscussed by Khot in [12]. Bipartite Clique, which also known as Balanced Complete"},{"page":2,"text":"2\nJinsong Tan\nis hard to approximate within a factor of nǫ for some ǫ > 0. In this paper, we\nconsider a weighted formulation of this problem defined as follows\nDefinition 1. S-Maximum Weighted Edge Biclique (S-MWEB)\nInstance: A complete bipartite graph G = (V1, V2, E) (throughout the paper, let\nη = max{|V1|, |V2|} and n = |V1| + |V2|), a weight function wG : E →S, where\nS is a set consisting of both positive and negative integers.\nQuestion: Find a biclique subgraph of G where the sum of weights on edges is\nmaximized.\nA few comments are in order. First note it is not a lose of generality but a\ntechnical convenience to require the graph be complete, one can always think of\nan incomplete bipartite graph as complete where non-edges are assigned weight\n0. Also note we require that both positive and negative weights be in S at the\nsame time because otherwise S-MWEB becomes a trivial problem.\nOur study of S-MWEB is motivated by the problem of finding statistically\nsignificant biclusters in microarray data analysis in the SAMBA model [18]\nand the Minimum Description Length with Holes (MDLH) problem [3,4,10];\ndetailed discussion of the two problems can be found in Sect. 4. Our main\ntechnical contribution of this paper is to show that if S satisfies the condition\n| min S\nmax S | ∈Ω(ηδ−1/2) ∩O(η1/2−δ), where δ > 0 is any arbitrarily small constant,\nthen no polynomial time algorithm can approximate S-MWEB within a factor\nof nǫ for some ǫ > 0 unless RP = NP. This result enables us to answer open\nquestions regarding the hardness of the SAMBA model and the MDLH prob-\nlem. Since maximum edge biclique can be characterized as a special case of\nS-MWEB with S = {−η, 1}, the nǫ-inapproximability result also provides inter-\nesting insights into the conjectured nǫ-inapproximability [8] of maximum edge\nbiclique.\nThe rest of the paper is organized in three sections. In Sect. 2, we present\nthe main technical result by proving the aforementioned inapproximability of S-\nMWEB. We give applications of this by answering hardness questions regarding\ntwo applied problems in Sect. 3. We conclude this work by raising a few open\nproblems in the last section.\n2\nApproximating S-Maximum Edge Biclique is Hard\nWe start this section by giving two lemmas about CLIQUE, which will be used\nin establishing inapproximability for the biclique problems we consider later.\nLemma 1 is a recent result by Zuckerman [20], obtained by a derandomization\nof results of H ̊astad [11]; Lemma 2 follows immediately from Lemma 1.\nLemma 1. ([20]) It is NP-hard to approximate CLIQUE within a factor of\nn1−ǫ, for any ǫ > 0.\nBipartite Subgraph [8], aims to maximize the number of vertices of a balanced sub-\ngraph whereas MEB aims to maximize the total weights on edges in a (not necessarily\nbalanced) subgraph."},{"page":3,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n3\nLemma 2. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate CLIQUE within a factor of n1−ǫ with probability at least\n1\npoly(n) unless\nRP = NP.\n2.1\nA Technical Lemma\nWe first describe the construction of a structure called {γ, {α, β}}-Product,\nwhich will be used in the proof of our main technical lemma.\nDefinition 2. ({γ, {α, β}}-Product)\nInput: An instance of S-MWEB on complete bipartite graph G = V1×V2, where\nγ ∈S and α < γ < β; an integer N.\nOutput: Complete bipartite graph GN = V N\n1\n× V N\n2\nconstructed as follows: V N\n1\nand V N\n2\nare N duplicates of V1 and V2, respectively. For each edge (i, j) ∈GN,\nlet (φ(i), φ(j)) be the corresponding edge in G. If wG(φ(i), φ(j)) = γ, assign\nweight α or β to (i, j) independently and identically at random with expectation\nbeing γ, denote the weight by random variable X. If wG(φ(i), φ(j)) ̸= γ, then\nkeep the weight unchanged. Call the weight function constructed this way w(·).\nFor any subgraph H of GN, denote by wγ(H) (resp., w−γ(H)) the total\nweight of H contributed by former-γ-edges (resp., other edges). Clearly, w(H) =\nwγ(H) + w−γ(H).\nWith a graph product constructed in this randomized fashion, we have the fol-\nlowing lemma.\nLemma 3. Given an S-MWEB instance G = (V1, V2, E) where γ ∈S, and a\nnumber δ ∈(0, 1\n2]; let η = max (|V1|, |V2|), N = η\nδ(3−2δ)+3\nδ(1+2δ) , GN = (V N\n1 , V N\n2 , E)\nbe the {γ, {α, β}}-product of G and S′ = (S ∪{α, β}) −{γ}. If\n1. |β −α| = O((Nη)\n1\n2 −δ); and\n2. there is a polynomial time algorithm that approximates the S′-MWEB\ninstance within a factor of λ, where λ is some arbitrary function in the size of\nthe S′-MWEB instance\nthen there exists a polynomial time algorithm that approximates the S-MWEB\ninstance within a factor of λ, with probability at least\n1\npoly(n).\nProof. For notational convenience, we denote η\n1\n2 −δ by f(η) throughout the proof.\nDefine random variable Y = X −γ, clearly E[Y ] = 0. Suppose there is a poly-\nnomial time algorithm A that approximates S′-MWEB within a factor of λ, we\ncan then run A on GN, the output biclique G∗\nB corresponds to N 2 bicliques in\nG (not necessarily all distinct). Let G∗\nA be the most weighted among these N 2\nsubgraphs of G, in the rest of the proof we show that with high probability, G∗\nA\nis a λ-approximation of S-MWEB on G.\nDenote by E1 the event that G∗\nB does not imply a λ-approximation on G.\nLet H be the set of subgraphs of GN that do not imply a λ-approximation on G,"},{"page":4,"text":"4\nJinsong Tan\nclearly, |H| ≤4Nη. Let H′ be an arbitrary element in H, we have the following\ninequalities\nPr {E1} ≤Pr\n \nat least one element in H is a λ-approximation of GN \n≤4Nη · Pr\n \nH′ is a λ-approximation of GN \n= 4Nη · Pr{E2}\nwhere E2 is the event that H′ is a λ-approximation of GN.\nLet the weight of an optimal solution U1×U2 of G be K, denote by U N\n1 ×U N\n2\nthe corresponding N 2-duplication in GN. Let x1 and x2 be the number of former-\nγ-edges in H′ and U N\n1 × U N\n2 , respectively. Suppose E2 happens, then we must\nhave\nw−γ(H′) + x1γ ≤N 2( K\nλ −1)\nw−γ(H′) + wγ(H′) ≥1\nλ(w−γ(U N\n1 × U N\n2 ) + wγ(U N\n1 × U N\n2 ))\nwhere the first inequality follows from the fact that we only consider integer\nweights. Since w−γ(U N\n1 × U N\n2 ) = N 2K −x2γ, it implies\n(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2\nso we have the following statement on probability\nPr{E2} ≤Pr\n \n(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2 \nLet z1 (resp., z2 and z3) be the number of edges in E(H′) −E(U N\n1 × U N\n2 )\n( resp., E(U N\n1 × U N\n2 ) −E(H′) and E(U N\n1 × U N\n2 ) ∩E(H′) ) transformed from\nformer-γ-edges in G. We have\nPr\n \n(wγ(H′) −x1γ) −1\nλ(wγ(U N\n1 × U N\n2 ) −x2γ) ≥N 2 \n= Pr\nnPz1\ni=1 Yi −1\nλ\nPz2\nj=1 Yj + λ−1\nλ\nPz3\nk=1 Yk ≥N 2o\n= Pr\nnPz1\ni=1 Yi + 1\nλ\nPz2\nj=1 (−Yj) + λ−1\nλ\nPz3\nk=1 Yk ≥N 2o\n≤Pr\nnPz1\ni=1 Yi ≥N 2\n3\no\n+ Pr\nn\n1\nλ\nPz2\nj=1 (−Yj) ≥N 2\n3\no\n+ Pr\nn\nλ−1\nλ\nPz3\nk=1 Yk ≥N 2\n3\no\n≤Pr\nnPz1\ni=1 Yi ≥N 2\n3\no\n+ Pr\nnPz2\nj=1 (−Yj) ≥N 2\n3\no\n+ Pr\nnPz3\nk=1 Yk ≥N 2\n3\no\n≤P\ni∈{1,2,3}\n \nexp\n \n−2zi\n \nN 2\n3zi(c1f(Nη))\n 2 \n(Hoeffding bound)\n≤3 · exp\n \n−c2 · N 1+2δ\nη3−2δ\n \n(zi ≤η2N 2)\nwhere c1, c2 are constants (c2 > 0). Now if we set N = η\n3−2δ\n1+2δ +θ for some θ, we\nhave\nPr {E1} ≤4Nη · Pr {E2} ≤3 · exp\n \nln 4 · η\n4\n(1+2δ) +θ −c2 · η(1+2δ)θ \nFor this probability to be bounded by 1\n2 as η is large enough, we need to have\n4\n1+2δ +θ < (1+2δ)θ. Solving this inequality gives θ >\n2\nδ(1+2δ). Therefore, for any\nδ ∈(0, 1\n2], by setting N = η\nδ(3−2δ)+3\nδ(1+2δ) , we have Pr{E1}, i.e. the probability that"},{"page":5,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n5\nthe solution returned by A does not imply a λ-approximation of G, is bounded\nfrom above by 1\n2 once input size is large enough. This gives a polynomial time\nalgorithm that approximates S-MWEB within a factor of λ with probability at\nleast 1\n2.\n⊓⊔\nThis lemma immediately leads to the following corollary.\nCorollary 1. Following the construction in Lemma 3, if S′-MWEB can be ap-\nproximated within a factor of nǫ′, for some ǫ′ > 0, then there exists a polyno-\nmial time algorithm that approximates S-MWEB within a factor of nǫ, where\nǫ = (1 + δ(3−2δ)+3\nδ(1+2δ) )ǫ′, with probability at least\n1\npoly(n). 2\nProof. Let |G| and |GN| be the number of nodes in the S-MWEB and S′-MWEB\nproblem, respectively. Since λ = |GN|ǫ′ ≤|G|(1+ δ(3−2δ)+3\nδ(1+2δ)\n)ǫ′, our claim follows\nfrom Lemma 3.\n⊓⊔\n2.2\n{−1, 0, 1}-MWEB\nIn this section, we prove inapproximability of {−1, 0, 1}-MWEB by giving a\nreduction from CLIQUE; in subsequence sections, we prove inapproximability\nresults for more general S-MWEB by constructing randomized reduction from\n{−1, 0, 1}-MWEB.\nLemma 4. The decision version of the {−1, 0, 1}-MWEB problem is NP-complete.\nProof. We prove this by describing a reduction from CLIQUE. Given a CLIQUE\ninstance G = (V, E), construct G′ = (V ′, E′) such that V ′ = V1∪V2 where V1, V2\nare duplicates of V in that there exist bijections φ1 : V1 →V and φ2 : V2 →V .\nAnd\nE′ = E1 ∪E2 ∪E3\nE1 = {(u, v) | u ∈V1, v ∈V2 and (φ1(u), φ2(v)) ∈E}\nE2 = {(u, v) | u ∈V1, v ∈V2, φ1(u) ̸= φ2(v) and (φ1(u), φ2(v)) /∈E}\nE3 = {(u, v) | u ∈V1, v ∈V2, and φ1(u) = φ2(v)}\nClearly, G′ is a biclique. Now assign weight 0 to edges in E1, −1 to edges in\nE2 and 1 to edges in E3. We then claim that there is a clique of size k in G if\nand only if there is a biclique of total edge weight k in G′.\nFirst consider the case where there is a clique of size k in G, let U be the set\nof vertices of the clique, then taking the subgraph induced by φ−1\n1 (U) × φ−1\n2 (U)\nin G′ gives us a biclique of total weight k.\nNow suppose that there is a biclique U1 ×U2 of total weight k in G′. Without\nloss of generality, assume U1 and U2 correspond to the same subset of vertices in\n2 Note we are slightly abusing notation here by always representing the size of a given\nproblem under discussion by n. Here n refers to the size of S′-MWEB (resp. S-\nMWEB) when we are talking about approximation factor nǫ′ (resp. nǫ). We adopt\nthe same convention in the sequel."},{"page":6,"text":"6\nJinsong Tan\nV because if (φ1(U1) −φ2(U2)) ∪(φ2(U2) −φ1(U1)) is not empty, then removing\n(U1 −U2) ∪(U2 −U1) will never decrease the total weight of the solution. Given\nφ1(U1) = φ2(U2), we argue that there is no edge of weight −1 in biclique U1×U2;\nsuppose otherwise there exists a weight −1 edge (i1, j2) (i1 ∈U1, and j2 ∈U2),\nthen the corresponding edge (j1, i2) (j1 ∈U1, and i2 ∈U2) must be of weight\n−1 too and removing i1, i2 from the solution biclique will increase total weight\nby at least 1 because among all edges incident to i1 and i2, (i1, i2) is of weight 1,\n(i1, j2) and (i2, j1) are of weight −1 and the rest are of weights either 0 or −1.\nTherefore, we have shown that if there is a solution U1 × U2 of weight k in\nG′, U1 and U2 correspond to the same set of vertices U ∈V and U is a clique of\nsize k. It is clear that the reduction can be performed in polynomial time and\nthe problem is NP, and thus NP-complete.\n⊓⊔\nGiven Lemma 1, the following corollary follows immediately from the above\nreduction.\nTheorem 1. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate problem {−1, 0, 1}-MWEB within a factor of n1−ǫ unless P = NP.\nProof. It is obvious that the reduction given in the proof of Lemma 4 preserves\ninapproximability exactly, and given that CLIQUE is hard to approximate within\na factor of n1−ǫ unless P = NP, the theorem follows.\n⊓⊔\nTheorem 2. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate {−1, 0, 1}-MWEB within a factor of n1−ǫ with probability at least\n1\npoly(n)\nunless RP = NP.\nProof. If there exists such a randomized algorithm for {−1, 0, 1}-MWEB, com-\nbining it with the reduction given in Lemma 4, we obtain an RP algorithm for\nCLIQUE. This is impossible unless RP = NP.\n⊓⊔\n2.3\n{−1, 1}-MWEB\nLemma 5. If there exists a polynomial time algorithm that approximates {−1, 1}-\nMWEB within a factor of nǫ, then there exists a polynomial time algorithm that\napproximates {−1, 0, 1}-MWEB within a factor of n5ǫ with probability at least\n1\npoly(n).\nProof. We prove this by constructing a {γ, {α, β}}-Product from {−1, 0, 1}-\nMWEB to {−1, 1}-MWEB by setting γ = 0, α = −1 and β = 1. Since δ = 1\n2,\naccording to Corollary 1, it is sufficient to set N = η4 so that the probability of\nobtaining a n5ǫ-approximation for {−1, 0, 1}-MWEB is at least\n1\npoly(n).\n⊓⊔\nTheorem 3. For any constant ǫ > 0, no polynomial time algorithm can approx-\nimate {−1, 1}-MWEB within a factor of n\n1\n5 −ǫ with probability at least\n1\npoly(n)\nunless RP = NP.\nProof. This follows directly from Theorem 2 and Lemma 5.\n⊓⊔"},{"page":7,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n7\n2.4\n{−η\n1\n2 −δ, 1}-MWEB and {−ηδ−1\n2 , 1}-MWEB\nIn this section, we consider the generalized cases of the S-MWEB problem.\nTheorem 4. For any δ ∈(0, 1\n2], there exists some constant ǫ such that no poly-\nnomial time algorithm can approximate {−η\n1\n2 −δ, 1}-MWEB within a factor of\nnǫ with probability at least\n1\npoly(n) unless RP = NP. The same statement holds\nfor {−ηδ−1\n2 , 1}-MWEB.\nProof. We prove this by first construct a {γ, {α, β}}-Product from {−1, 1}-\nMWEB to {−η\n1\n2 −δ, 1}-MWEB by setting γ = −1, α = −(Nη)\n1\n2 −δ and β = 1. By\nCorollary 1, we know that for any δ ∈(0, 1\n2], if there exists a polynomial time al-\ngorithm that approximates {−η\n1\n2 −δ, 1}-MWEB within a factor of nǫ, then there\nexists a polynomial time algorithm that approximates {−1, 1}-MWEB within a\nfactor of n(1+ δ(3−2δ)+3\nδ(1+2δ)\n)ǫ with probability at least\n1\npoly(n). So invoking the hardness\nresult in Theorem 3 gives the desired hardness result for {−η\n1\n2 −δ, 1}-MWEB.\nThe same conclusion applies to {−1, η\n1\n2 −δ}-MWEB by setting γ = 1, α = −1\nand β = (Nη)\n1\n2 −δ. Since η is a constant for any given graph, we can simply divide\neach weight in {−1, η\n1\n2 −δ} by η\n1\n2 −δ.\n⊓⊔\nTheorem 4 leads to the following general statement.\nTheorem 5. For any small constant δ ∈(0, 1\n2], if\n min S\nmax S\n ∈Ω(ηδ−1/2)∩O(η1/2−δ),\nthen there exists some constant ǫ such that no polynomial time algorithm can ap-\nproximate S-MWEB within a factor of nǫ with probability at least\n1\npoly(n) unless\nRP = NP.\n3\nTwo Applications\nIn this section, we describe two applications of the results establish in Sect. 3 by\nproving hardness and inapproximability of problems found in practice.\n3.1\nSAMBA Model is Hard\nMicroarray technology has been the latest technological breakthrough in biolog-\nical and biomedical research; in many applications, a key step in analyzing gene\nexpression data obtained through microarray is the identification of a bicluster\nsatisfying certain properties and with largest area (see the survey [13] for a fairly\nextensive discussion on this).\nIn particular, Tanay et. al. [18] considered the Statistical-Algorithmic Method\nfor Bicluster Analysis (SAMBA) model. In their formulation, a complete bipar-\ntite graph is given where one side corresponds to genes and the other size cor-\nresponds to conditions. An edges (u, v) is assigned a real weight which could be\neither positive or negative, depending on the expression level of gene u in condi-\ntion v, in a way such that heavy subgraphs corresponds to statistically significant"},{"page":8,"text":"8\nJinsong Tan\nbiclusters. Two weight-assigning schemes are considered in their paper. In the\nfirst, or simple statistical model, a tight upper-bound on the probability of an\nobserved biclusters in computed; in the second, or refined statistical model, the\nweights are assigned in a way such that a maximum weight biclique subgraph\ncorresponds to a maximum likelihood bicluster.\nThe Simple SAMBA Statistical Model: Let H = (V ′\n1, V ′\n2, E′) be a subgraph\nof G = (V1, V2, E), E′ = {V ′\n1 × V ′\n2} −E′ and p =\n|E|\n|V1||V2|. The simple statistical\nmodel assumes that edges occur independently and identically at random with\nprobability p. Denote by BT (k, p, n) the probability of observing k or more\nsuccesses in n binomial trials, the probability of observing a graph at least as\ndense as H is thus p(H) = BT (|E′|, p, |V ′\n1||V ′\n2|). This model assumes p < 1\n2 and\n|V ′\n1||V ′\n2| ≪|V1||V2|, therefore p(H) is upper bounded by\np∗(H) = 2|V ′\n1 ||V ′\n2|p|E′|(1 −p)|V ′\n1 ||V ′\n2|−|E′|\nThe goal of this model is thus to find a subgraph H with the smallest p∗(H).\nThis is equivalent to maximizing\n−log p∗(H) = |E′|(−1 −log p) + (|V ′\n1||V ′\n2| −|E′|)(−1 −log (1 −p))\nwhich is essentially solving a S-MWEB problem that assigns either positive\nweight (−1 −log p) or negative weight (−1 −log (1 −p)) to an edge (u, v), de-\npending on whether gene u express or not in condition v, respectively. The\nsummation of edge weights over H is defined as the statistical significance of H.\nSince\n1\nη2 ≤p < 1\n2, asymptotically we have −1−log (1−p)\n−1−log p\n∈Ω(\n1\nlog η) ∩O(1).\nInvoking Theorem 5 gives the following.\nTheorem 6. For the Simple SAMBA Statistical model, there exists some ǫ > 0\nsuch that no polynomial time algorithm, possibly randomized, can find a bicluster\nwhose statistical significance is within a factor of nǫ of optimal unless RP = NP.\nThe Refined SAMBA Statistical Model: In the refined model, each edge\n(u, v) is assumed to take an independent Bernoulli trial with parameter pu,v,\ntherefore p(H) = (Q\n(u,v)∈E′ pu,v)(Q\n(u,v)∈E′(1 −pu,v)) is the probability of ob-\nserving a subgraph H. Since p(H) generally decreases as the size of H increases,\nTanay et al. aims to find a bicluster with the largest (normalized) likelihood ra-\ntio L(H) =\n(Q\n(u,v)∈E′ pc)(Q\n(u,v)∈E′(1 −pc))\np(H)\n, where pc > max(u,v)∈E pu,v is a\nconstant probability and chosen with biologically sound assumptions. Note this\nis equivalent to maximizing the log-likelihood ratio\nlog L(H) =\nX\n(u,v)∈E′\nlog pc\npu,v\n+\nX\n(u,v)∈E′\nlog 1 −pc\n1 −pu,v\nWith this formulation, each edge is assigned weight either log\npc\npu,v\n> 0 or\nlog\n1−pc\n1−pu,v < 0 and finding the most statistically significant bicluster is equiva-\nlent to solving S-MWEB with S = {log\n1−pc\n1−pu,v , log\npc\npu,v }. Since pc is a constant"},{"page":9,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n9\nand\n1\nη2 ≤pu,v < pc, we have log (1−pc)−log (1−pu,v)\nlog pc−log pu,v\n∈Ω(\n1\nlog η) ∩O(1). Invoking\nTheorem 5 gives the following.\nTheorem 7. For the Refined SAMBA Statistical model, there exists some ǫ > 0\nsuch that no polynomial time algorithm, possibly randomized, can find a bicluster\nwhose log-likelihood is within a factor of nǫ of optimal unless RP = NP.\n3.2\nMinimum Description Length with Holes (MDLH) is Hard\nBu et. al [4] considered the Minimum Description Length with Holes problem\n(defined in the following); the 2-dimensional case is claimed NP-hard in this\npaper and the proof is referred to [3]. However, the proof given in [3] suffers\nfrom an error in its reduction3, thus whether MDLH is NP-complete remains\nunsettled. In this section, by employing the results established in the previous\nsections, we show that no polynomial time algorithm exists for MDLH, under\nthe slightly weaker (than P ̸= NP) but widely believed assumption RP ̸= NP.\nWe first briefly describe the Minimum Description Length summarization\nwith Holes problem; for a detailed discussion of the subject, we refer the readers\nto [3,4].\nSuppose one is given a k-dimensional binary matrix M, where each entry is\nof value either 1, which is of interest, or of value 0, which is not of interest. Be-\nsides, there are also k hierarchies (trees) associated with each dimension, namely\nT1, T2, ..., Tk, each of height l1, l2, ..., lk respectively. Define level l = maxi(li).\nFor each Ti, there is a bijection between its leafs and the ’hyperplanes’ in the\nith dimension (e.g. in a 2-dimensional matrix, these hyperplanes corresponds to\nrows and columns). A region is a tuple (x1, x2, ..., xk), where xi is a leaf node\nor an internal node in hierarchy Ti. Region (x1, x2, ..., xk) is said to cover cell\n(c1, c2, ..., ck) if ci is a descendant of xi, for all 1 ≤i ≤k. A k-dimensional l-level\nMDLH summary is defined as two sets S and H, where 1) S is a set of regions\ncovering all the 1-entries in M; and 2) H is the set of 0-entries covered (unde-\nsirably) by S and to be excluded from the summary. The length of a summary\nis defined as |S| + |H|, and the MDLH problem asks the question if there exists\na MDLH summary of length at most K, for a given K > 0.\nIn an effort to establish hardness of MDLH, we first define the following\nproblem, which serves as an intermediate problem bridging {−1, 1}-MWEB and\nMDLH.\nDefinition 3. (Problem P)\nInstance: A complete bipartite graph G = (V1, V2, E) where each edge takes on\na value in {−1, 1}, and a positive integer k.\nQuestion:\nDoes there exist an induced subgraph (a biclique U1 × U2) whose\ntotal weight of edges is ω, such that |U1| + |U2| + ω ≥k.\nLemma 6. No polynomial time algorithm exists for Problem P unless RP = NP.\n3 In Lemma 3.2.1 of [3], the reduction from CLIQUE to CEW is incorrect."},{"page":10,"text":"10\nJinsong Tan\nProof. We prove this by constructing a reduction from {−1, 1}-MWEB to Prob-\nlem P as follows: for the given input biclique G = (V1, V2, E), make N duplicates\nof V1 and N duplicates of V2, where N = (|V1| + |V2|)2. Connect each copy of\nV1 to each copy of V2 in a way that is identical to the input biclique, we then\nclaim that there is a size k solution to {−1, 1}-MWEB if and only if there is a\nsize N 2k solution to Problem P.\nIf there is a size k solution to {−1, 1}-MWEB, then it is straightforward that\nthere is a solution to Problem P of size at least N 2k. For the reverse direction, we\nshow that if no solution to {−1, 1}-MWEB is of size at least k, then the maximum\nsolution to Problem P is strictly less than N 2k. Note a solution U N\n1 × U N\n2\nto\nProblem P consists of at most N 2 (not necessarily all distinct) solutions to\n{−1, 1}-MWEB, and each of them can contribute at most (k −1) in weight to\nU N\n1 × U N\n2 , so the total weight gained from edges is at most N 2(k −1). And note\nthe total weight gained from vertices is at most N(|V1|+|V2|) = N\n√\nN, therefore\nthe weight is upper bounded by N\n√\nN + N 2(k −1) < N 2k and this completes\nthe proof.\nAs a conclusion, we have a polynomial time reduction from {−1, 1}-MWEB\nto Problem P. Since no polynomial time algorithm exists for {−1, 1}-MWEB\nunless RP = NP, the same holds for Problem P.\n⊓⊔\nTheorem 8. No polynomial time algorithm exists for MDLH summarization,\neven in the 2-dimension 2-level case, unless RP = NP.\nProof. We prove this by showing that Problem P is a complementary problem\nof 2-dimensional 2-level MDLH.\nLet the input 2D matrix M be of size n1×n2, with a tree of height 2 associated\nwith each dimension. Without loss of generality, we only consider the ’sparse’\ncase where the number of 1-entries is less than the number of 0-entries by at\nleast 2 so that the optimal solution will never contain the whole matrix as one\nof its regions. Let S be the set of regions in a solution. Let R and C be the set\nof rows and columns not included in S. Let Z be the set of all zero entries in M.\nLet z be the total number of zero entries in the R × C ’leftover’ matrix and let\nw be the total number of 1-entries in it. MDLH tries to minimize the following:\n(n1 −|R|) + (n2 −|C|) + (|Z| −z) + w = (n1 + n2 + |Z|) −(|R| + |C| + z −w)\nSince (n1 + n2 + |Z|) is a fixed quantity for any given input matrix, the 2-\ndimensional 2-level MDLH problem is equivalent to maximizing (|R|+|C|+z−w),\nwhich is precisely the definition of Problem P.\nTherefore, 2-dimensional 2-level MDLH is a complementary problem to Prob-\nlem P and by Lemma 6 we conclude that no polynomial time algorithm exists\nfor 2-dimensional 2-level MDLH unless RP = NP.\n⊓⊔\n4\nConcluding Remarks\nMaximum weighted edge biclique and its variants have received much atten-\ntion in recently years because of it wide range of applications in various fields"},{"page":11,"text":"Inapproximability of Maximum Weighted Edge Biclique and Its Applications\n11\nincluding machine learning, database, and particularly bioinformatics and com-\nputational biology, where many computational problems for the analysis of mi-\ncroarray data are closely related. To tackle these applied problems, various kinds\nof heuristics are proposed and experimented and it is not known whether these\nalgorithms give provable approximations. In this work, we answer this question\nby showing that it is highly unlikely (under the assumption RP ̸= NP) that good\npolynomial time approximation algorithm exists for maximum weighted edge\nbiclique for a wide range of choices of weight; and we further give specific appli-\ncations of this result to two applied problems. We conclude our work by listing\na few open questions.\n1. We have shown that {Θ(−ηδ), 1}-MWEB is nǫ-inapproximable for δ ∈\n(−1\n2, 1\n2); also it is easy to see that (i) the problem is in P when δ ≤−1, where\nthe entire input graph is the optimal solution; (ii) for any δ ≥1, the problem is\nequivalent to MEB, which is conjectured to be nǫ-inapproximable [8]. Therefore\nit is natural to ask what is the approximability of the {−nδ, 1}-MWEB problem\nwhen δ ∈(−1, −1\n2] and δ ∈[ 1\n2, 1]. In particular, can this be answered by a better\nanalysis of Lemma 3?\n2. We are especially interested in {−1, 1}-MWEB, which is closely related\nto the formulations of many natural problems [1,3,4,18]. We have shown that\nno polynomial time algorithm exists for this problem unless RP = NP, and we\nbelieve this problem is NP-complete, however a proof has eluded us so far.\nReferences\n1. N. Bansal, A. Blum, and S. Chawla. Correlation clustering, Machine Learning,\n56:89-113, 2004.\n2. A. Ben-Dor, B. Chor, R. Karp, and Z. Yakhini. Discovering local structure in\ngene expression data: The Order-Preserving Submatrix Problem. In Proceedings of\nRECOMB’02, 49-57, 2002.\n3. S. Bu. The summarization of hierarchical data with exceptions. Master The-\nsis, Department of Computer Science, University of British Columbia, 2004.\nhttp://www.cs.ubc.ca/grads/resources/thesis/Nov04/Shaofeng Bu.pdf\n4. S. Bu, L. V. S. Lakshmanan, R. T. Ng. MDL Summarization with Holes. In Pro-\nceedings of VLDB’05, 433-444, 2005.\n5. Y. Cheng, and G. Church. Biclustering of expression data. In Proceedings of\nISMB’00, 93-103. AAAI Press, 2000.\n6. M. Dawande, P. Keskinocak, J. M. Swaminathan, and S. Tayur. On Bipartite and\nmultipartite clique problems. Journal of Algorithms, 41(2):388-403, 2001.\n7. U. Feige. Relations between average case complexity and approximation complex-\nity. In Proceedings of STOC’02, 534-543, 2002.\n8. U. Feige and S. Kogan. Hardness of approximation of the Balanced Complete\nBipartite Subgraph problem. Technical Report MCS04-04, The Weizmann Institute\nof Science, 2004.\n9. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the\nTheory of NP-completeness. Freeman, San Francisco, 1979.\n10. P. Fontana, S. Guha and J. Tan. Recursive MDL Summarization and Approxima-\ntion Algorithms. Preprint, 2007."},{"page":12,"text":"12\nJinsong Tan\n11. J. H ̊astad. Clique is hard to approximate within n1−ǫ. Acta Mathematica, 182:105-\n142, 1999.\n12. S. Khot. Ruling out PTAS for Graph Min-Bisection, Densest Subgraph and Bipar-\ntite Clique. In Proceedings of FOCS’04, 136-145, 2004.\n13. S. C. Madeira, and A. L. Oliveira. Biclustering algorithms for biological data anal-\nysis: a survey. IEEE/ACM Transactions on Computational Biology and Bioinfor-\nmatics, 1:24-45, 2004.\n14. N. Mishra, D. Ron, and R. Swaminathan. On finding large conjunctive clusters. In\nProceedings of COLT’03, 448-462, 2003.\n15. R. Peeters. The maximum edge biclique problem is NP-complete. Discrete Applied\nMathematics, 131:651-654, 2003.\n16. J. M. Swaminathan and S. Tayur. Managing Broader Product Lines Through De-\nlayed Differentiation Using Vanilla Boxes. Management Science, 44:161-172, 1998.\n17. J. Tan, K. Chua, L. Zhang, and S. Zhu. Algorithmic and Complexity Issues of\nThree Clustering Methods in Microarray Data Analysis Algorithmica, 48(2): 203-\n219, 2007.\n18. A. Tanay, R. Sharan, and R. Shamir. Discovering statistically significant biclusters\nin gene expression data. Bioinformatics, 18, Supplement 1:136-144, 2002.\n19. L. Zhang, and S. Zhu. A New Clustering Method for Microarray Data Analysis.\nIn Proceedings of CSB’02, 268-275, 2002.\n20. D. Zuckerman. Linear Degree Extractors and the Inapproximability of Max Clique\nand Chromatic Number. In Proceedings of STOC’06, 681-690, 2006."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"Abstract. Given a bipartite graph G = (V1, V2, E) where edges take","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"(where η = max{|V1|, |V2|}, and δ ∈(0, 1/2]), no polynomial time algo-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"unless RP = NP. This hardness result gives justification of the heuristic","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"Holes problem [4] unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"Let G = (V1, V2, E) be an undirected bipartite graph. A biclique subgraph in G","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"Instance: A complete bipartite graph G = (V1, V2, E) (throughout the paper, let","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"η = max{|V1|, |V2|} and n = |V1| + |V2|), a weight function wG : E →S, where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"of nǫ for some ǫ > 0 unless RP = NP. This result enables us to answer open","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"S-MWEB with S = {−η, 1}, the nǫ-inapproximability result also provides inter-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"Input: An instance of S-MWEB on complete bipartite graph G = V1×V2, where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"Output: Complete bipartite graph GN = V N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"let (φ(i), φ(j)) be the corresponding edge in G. If wG(φ(i), φ(j)) = γ, assign","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"being γ, denote the weight by random variable X. If wG(φ(i), φ(j)) ̸= γ, then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"weight of H contributed by former-γ-edges (resp., other edges). Clearly, w(H) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"Lemma 3. Given an S-MWEB instance G = (V1, V2, E) where γ ∈S, and a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"2]; let η = max (|V1|, |V2|), N = η","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"δ(1+2δ) , GN = (V N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"be the {γ, {α, β}}-product of G and S′ = (S ∪{α, β}) −{γ}. If","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"1. |β −α| = O((Nη)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"Define random variable Y = X −γ, clearly E[Y ] = 0. Suppose there is a poly-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"= 4Nη · Pr{E2}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"2 ) = N 2K −x2γ, it implies","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"= Pr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"i=1 Yi −1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"j=1 Yj + λ−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"k=1 Yk ≥N 2o","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"= Pr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"i=1 Yi + 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"j=1 (−Yj) + λ−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"k=1 Yk ≥N 2o","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"i=1 Yi ≥N 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"j=1 (−Yj) ≥N 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"k=1 Yk ≥N 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"i=1 Yi ≥N 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"j=1 (−Yj) ≥N 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"k=1 Yk ≥N 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"where c1, c2 are constants (c2 > 0). Now if we set N = η","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"2], by setting N = η","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"problem, respectively. Since λ = |GN|ǫ′ ≤|G|(1+ δ(3−2δ)+3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"instance G = (V, E), construct G′ = (V ′, E′) such that V ′ = V1∪V2 where V1, V2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"E′ = E1 ∪E2 ∪E3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"E1 = {(u, v) | u ∈V1, v ∈V2 and (φ1(u), φ2(v)) ∈E}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"E2 = {(u, v) | u ∈V1, v ∈V2, φ1(u) ̸= φ2(v) and (φ1(u), φ2(v)) /∈E}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"E3 = {(u, v) | u ∈V1, v ∈V2, and φ1(u) = φ2(v)}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"φ1(U1) = φ2(U2), we argue that there is no edge of weight −1 in biclique U1×U2;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"imate problem {−1, 0, 1}-MWEB within a factor of n1−ǫ unless P = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"a factor of n1−ǫ unless P = NP, the theorem follows.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"CLIQUE. This is impossible unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"MWEB to {−1, 1}-MWEB by setting γ = 0, α = −1 and β = 1. Since δ = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"according to Corollary 1, it is sufficient to set N = η4 so that the probability of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"poly(n) unless RP = NP. The same statement holds","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"2 −δ, 1}-MWEB by setting γ = −1, α = −(Nη)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"2 −δ and β = 1. By","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"2 −δ}-MWEB by setting γ = 1, α = −1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"and β = (Nη)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"The Simple SAMBA Statistical Model: Let H = (V ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"of G = (V1, V2, E), E′ = {V ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"2} −E′ and p =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"dense as H is thus p(H) = BT (|E′|, p, |V ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"p∗(H) = 2|V ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"−log p∗(H) = |E′|(−1 −log p) + (|V ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"whose statistical significance is within a factor of nǫ of optimal unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"therefore p(H) = (Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"tio L(H) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"log L(H) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"lent to solving S-MWEB with S = {log","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"whose log-likelihood is within a factor of nǫ of optimal unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"the slightly weaker (than P ̸= NP) but widely believed assumption RP ̸= NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"T1, T2, ..., Tk, each of height l1, l2, ..., lk respectively. Define level l = maxi(li).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"Instance: A complete bipartite graph G = (V1, V2, E) where each edge takes on","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"Lemma 6. No polynomial time algorithm exists for Problem P unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"lem P as follows: for the given input biclique G = (V1, V2, E), make N duplicates","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"of V1 and N duplicates of V2, where N = (|V1| + |V2|)2. Connect each copy of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"the total weight gained from vertices is at most N(|V1|+|V2|) = N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"unless RP = NP, the same holds for Problem P.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"even in the 2-dimension 2-level case, unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"(n1 −|R|) + (n2 −|C|) + (|Z| −z) + w = (n1 + n2 + |Z|) −(|R| + |C| + z −w)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"for 2-dimensional 2-level MDLH unless RP = NP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"by showing that it is highly unlikely (under the assumption RP ̸= NP) that good","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"no polynomial time algorithm exists for this problem unless RP = NP, and we","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":30596,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}