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We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both \emph{exact} and \emph{approximate} versions of the algorithm. |
We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of $\beta$-fractional positions, which may arise as a result of addition of two $\beta$-integer... |
For $0\leq \rho\leq 1$ and a coloured graph $G$, a vertex $v$ is $\rho$-happy if at least $\rho \mathrm{deg}(v)$ of its neighbours have the same colour as $v$. Soft happy colouring of a partially coloured graph $G$ is the problem of finding a vertex colouring $\sigma$ that preserves the precolouring and has the maximu... |
Valiant introduced some 25 years ago an algebraic model of computation along with the complexity classes VP and VNP, which can be viewed as analogues of the classical classes P and NP. They are defined using non-uniform sequences of arithmetic circuits and provides a framework to study the complexity for sequences of ... |
Let c(F) be the number of perfect pairs of F and c(G) be the maximum of c(F) over all (near-) one-factorizations F of G. Wagner showed that for odd n, c(K_{n}) \geq n*phi(n)/2 and for m and n which are odd and co-prime to each other, c(K_{mn}) \geq 2*c(K_{m})*c(K_{n}). In this note, we establish that both these result... |
Submodularity is an important concept in combinatorial optimization, and it is often regarded as a discrete analog of convexity. It is a fundamental fact that the set of minimizers of any submodular function forms a distributive lattice. |
We present double pooling, a simple, easy-to-implement variation on test pooling, that in certain ranges for the a priori probability of a positive test, is significantly more efficient than the standard single pooling approach (the Dorfman method). |
A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of tr... |
We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra~(SICOMP 2018) for Boolean functions to the case of real-valued functions $f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real-valued function... |
An infinite permutation is a linear order on the set N. We study the properties of infinite permutations generated by fixed points of some uniform binary morphisms, and find the formula for their complexity. |
We investigate structural and algorithmic advantages of a directed version of the well-researched class of distance-hereditary graphs. Since the previously defined distance-hereditary digraphs do not permit a recursive structure, we define directed twin-distance-hereditary graphs, which can be constructed by several t... |
We consider a continuous-time Markov chain (CTMC) whose state space is partitioned into aggregates, and each aggregate is assigned a probability measure. A sufficient condition for defining a CTMC over the aggregates is presented as a variant of weak lumpability, which also characterizes that the measure over the orig... |
We present a tight lower bound for the spanning tree congestion of Hamming graphs. |
In a graph, a spanning tree is said to be a tree t-spanner of the graph if the distance between any two vertices in is at most times their distance in . The tree t-spanner has many applications in networks and distributed environments. |
A solution to a problem of Erdős, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same authors, is settled, proving that every 2-choosable graph is also $(4:2)$-choos... |
Perfect phylogenies are fundamental in the study of evolutionary trees because they capture the situation when each evolutionary trait emerges only once in history; if such events are believed to be rare, then by Occam's Razor such parsimonious trees are preferable as a hypothesis of evolution. A classical result ... |
We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree. |
We solve the recognition problem (RP) for the class of Demidenko matrices. Our result closes a remarkable gap in the recognition of specially structured matrices. |
The greedy algorithm for approximating dominating sets is a simple method that is known to compute an $(\ln n+1)$-approximation of a minimum dominating set on any graph with $n$ vertices. We show that a small modification of the greedy algorithm can be used to compute an $O(t^2\cdot \ln k)$-approximation, where~$k$ is... |
In phylogenetics, tree-based networks are used to model and visualize the evolutionary history of species where reticulate events such as horizontal gene transfer have occurred. Formally, a tree-based network $N$ consists of a phylogenetic tree $T$ (a rooted, binary, leaf-labeled tree) and so-called reticulation edges... |
We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and show that all trees can be generated from a starting tree of one vertex. |
Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. |
The measurement of the similarity of RNA secondary structures, and in general of contact structures, of a fixed length has several specific applications. For instance, it is used in the analysis of the ensemble of suboptimal secondary structures generated by a given algorithm on a given RNA sequence, and in the compar... |
We say that a $q$-ary length $n$ code is \emph{non-overlapping} if the set of non-trivial prefixes of codewords and the set of non-trivial suffices of codewords are disjoint. These codes were first studied by Levenshtein in 1964, motivated by applications in synchronisation. |
This paper aims to better understand the link better understand the links between aperiodicity in subshifts and pattern complexity. Our main contribution deals with substitutive subshifts, an equivalent to substitutive tilings in the context of symbolic dynamics. |
Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. |
A cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. |
For $0<\rho\leq 1$, a $\rho$-happy vertex $v$ in a coloured graph $G$ has at least $\rho\cdot \mathrm{deg}(v)$ same-colour neighbours, and a $\rho$-happy colouring (aka soft happy colouring) of $G$ is a vertex colouring that makes all the vertices $\rho$-happy. A community is a subgraph whose vertices are more adja... |
A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. |
In this note, we give a self-contained and elementary proof of the elementary construction of spectral high-dimensional expanders using elementary matrices due to Kaufman and Oppenheim [Proc. 50th ACM Symp. on Theory of Computing (STOC), 2018]. As a bonus, this also yields a simple construction and analysis of standar... |
Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree n-vertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(lo... |
Graph searches and their respective search trees are widely used in algorithmic graph theory. The problem whether a given spanning tree can be a graph search tree has been considered for different searches, graph classes and search tree paradigms. |
A biological regulatory network can be modeled as a discrete function that contains all available information on network component interactions. From this function we can derive a graph representation of the network structure as well as of the dynamics of the system. |
A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n-8 edges. <br>We show that optimal 1-planar graphs can be recognized in linear time. |
Given a dominating set, how much smaller a dominating set can we find through elementary operations? Here, we proceed by iterative vertex addition and removal while maintaining the property that the set forms a dominating set of bounded size. |
The main aim of decision support systems is to find solutions that satisfy user requirements. Often, this leads to predictability of those solutions, in the sense that having the input data and the model, an adversary or enemy can predict to a great extent the solution produced by your decision support system. |
The Shapley-Shubik power index is a measure of each voters power in the passage or failure of a vote. We extend this measure to graphs and consider a discrete-time process in which voters may change their vote based on the outcome of the previous vote. |
The chaotic systems have been found applications in diverse fields such as pseudo random number generator, coding, cryptography, spread spectrum (SS) communications etc. The inherent capability of generating a large space of PN sequences due to sensitive dependence on initial conditions has been the main reason for ex... |
We prove a strengthening of the trickle down theorem for partite complexes. Given a $(d+1)$-partite $d$-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are $\frac{1-\delta}{d}$-(one-sided) spectral expanders, then the link of any face of co-dimension $k$ is an ... |
The minmax regret problem for combinatorial optimization under uncertainty can be viewed as a zero-sum game played between an optimizing player and an adversary, where the optimizing player selects a solution and the adversary selects costs with the intention of maximizing the regret of the player. The existing minmax... |
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex coloring problem which has a number of AI related applications. Due to its theoretical and practical relevance, MSCP attracts increasing attention. |
This material is dedicated to the estimation of the chromatic number and chromatic class of the conjugated triangulation (first conversion) and also of the second conversion of the planar triangulation. Also this paper introduces some new hypotheses, which are equivalent to Four Color Problem. |
HyperBagGraphs (hb-graphs as short) extend hypergraphs by allowing the hyperedges to be multisets. Multisets are composed of elements that have a multiplicity. |
In 2010, A. Shpilka and I. Volkovich established a prominent result on the equivalence of polynomial factorization and identity testing. It follows from their result that a multilinear polynomial over the finite field of order 2 can be factored in time cubic in the size of the polynomial given as a string. |
Entanglement is a digraph complexity measure that origins in fixed-point theory. Its purpose is to count the nested depth of cycles in digraphs. |
We consider the problem of finding a Hamiltonian path with precedence constraints in the form of a partial order on the vertex set. This problem is known as Partially Ordered Hamiltonian Path Problem (POHPP). |
The generalized list $T$-coloring is a common generalization of many graph coloring models, including classical coloring, $L(p,q)$-labeling, channel assignment and $T$-coloring. Every vertex from the input graph has a list of permitted labels. |
We define and study greedy matchings in vertex-ordered bipartite graphs. It is shown that each vertex-ordered bipartite graph has a unique greedy matching. |
A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$) apart. This family of graphs allow us to study local structures of unit square intersection... |
In this paper we study the dynamical behavior of Boolean networks with firing memory, namely Boolean networks whose vertices are updated synchronously depending on their proper Boolean local transition functions so that each vertex remains at its firing state a finite number of steps. We prove in particular that these... |
Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the correspondi... |
In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg--Weyl, its Bargmann--Fock representation with differential operators and the associated one-parameter <a href="http://group.Upon" rel="external noopener nofollow" class="link-external link-htt... |
The Rooted Maximum Leaf Outbranching problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at least $k$ leaves. |
Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. |
We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. |
We propose a Markov chain simulation method to generate simple connected random graphs with a specified degree sequence and level of clustering. The networks generated by our algorithm are random in all other respects and can thus serve as generic models for studying the impacts of degree distributions and clustering ... |
Many real discrete optimization problems (DOPs) are $NP$-hard and contain a huge number of variables and/or constraints that make the models intractable for currently available solvers. Large DOPs can be solved due to their special tructure using decomposition approaches. |
Efficient scheduling of periodic meetings is a critical challenge in various service-oriented domains, including academic settings, healthcare, and legal consultancy. This study presents a robust Integer Linear Programming (ILP) model to optimize the scheduling of faculty-student meetings. |
The topic of the article is the parametric study of the complexity of algorithms on arrays of pairwise distinct integers. We introduce a model that takes into account the non-uniformness of data, which we call the Ewens-like distribution of parameter $\theta$ for records on permutations: the weight $\theta^r$ of a per... |
Let $G=(V,E)$ be a finite undirected graph without loops and multiple edges. A subset $M \subseteq E$ of edges is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $M$. |
A proper $k$-coloring of a graph $G=(V,E)$ is a function $c: V(G)\to \{1,\ldots,k\}$ such that $c (u)\neq c |
We define a $q$-linear path in a hypergraph $H$ as a sequence $(e_1,\ldots,e_L)$ of edges of $H$ such that $|e_i \cap e_{i+1}| \in [\! [1,q]\! |
We consider perfect 1-error correcting codes over a finite field with $q$ elements (briefly $q$-ary 1-perfect codes). In this paper, a generalized concatenation construction for $q$-ary 1-perfect codes is presented that allows us to construct $q$-ary 1-perfect codes of length $(q - 1)nm + n + m$ from the given $q$-ary... |
Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it... |
We extend Cellular Automata to time-varying discrete geometries. In other words we formalize, and prove theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same. |
Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. |
For the first time a mathematical object is presented - a reversible cellular Automaton - with many paradoxical qualities, the main ones among them are: a frequent quickly return to its original state, the presence of a large number of conservation laws and paradoxical "fuzzy" symmetries, which connects the cur... |
In combinatorial group testing problems Questioner needs to find a special element $x \in [n]$ by testing subsets of $[n]$. Tapolcai et al. introduced a new model, where each element knows the answer for those queries that contain it and each element should be able to identify the special one. |
Unveiling the underlying control principles of complex networks is one of the ultimate goals of network science. We introduce a novel concept, control hub, to reveal a cornerstone of the control structure of a network. |
For a graph G, consider the pairs of edge-disjoint matchings whose union consists of as many edges as possible. Let H be the largest matching among such pairs. |
The Frankl's conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. |
Markovian evolving graphs are dynamic-graph models where the links among a fixed set of nodes change during time according to an arbitrary Markovian rule. They are extremely general and they can well describe important dynamic-network scenarios. |
Identifying codes have been introduced in 1998 to model fault-detection in multiprocessor systems. In this paper, we introduce two variations of identifying codes: weak codes and light codes. |
The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. |
This is the report-version of a mini-series of two articles on the foundations of satisfiability of conjunctive normal forms with non-boolean variables, to appear in Fundamenta Informaticae, 2011. These two parts are here bundled in one report, each part yielding a chapter. |
In this paper, a new multi-hop weighted clustering procedure is proposed for homogeneous Mobile Ad hoc networks. The algorithm generates double star embedded non-overlapping cluster structures, where each cluster is managed by a leader node and a substitute for the leader node (in case of failure of leader node). |
This paper formulates a novel problem on graphs: find the minimal subset of edges in a fully connected graph, such that the resulting graph contains all spanning trees for a set of specifed sub-graphs. This formulation is motivated by an un-supervised grammar induction problem from computational linguistics. |
The ratio of the connected domination number, $\gamma_c$, and the domination number, $\gamma$, is strictly bounded from above by 3. It was shown by Zverovich that for every connected $(P_5,C_5)$-free graph, $\gamma_c = \gamma$. |
In 1985, Restivo and Salemi presented a list of five problems concerning power free languages. Problem $4$ states: Given $\alpha$-power-free words $u$ and $v$, decide whether there is a transition from $u$ to $v$. |
This paper addresses a charging-station company (Chargco) for electric and hydrogen vehicles. The optimal trading of the Chargco in day-ahead and intraday auction markets for electricity is modeled as a stochastic Mixed-Integer Quadratic Program (MIQP). |
We propose and analyze a graph model to study the connectivity of interdependent networks. Two interdependent networks of arbitrary topologies are modeled as two graphs, where every node in one graph is supported by supply nodes in the other graph, and a node fails if all of its supply nodes fail. |
The {\em height} of a trace is the height of the corresponding heap of pieces in Viennot's representation, or equivalently the number of factors in its Cartier-Foata decomposition. Let $h(t)$ and $|t|$ stand respectively for the height and the length of a trace $t$. |
In a seminal paper (Weitz, 2006), Weitz gave a deterministic fully polynomial approximation scheme for count- ing exponentially weighted independent sets (equivalently, approximating the partition function of the hard-core model from statistical physics) on graphs of degree at most d, up to the critical activity for th... |
Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access (CDMA) network for 2-dimensional image transmission. There is a one-to-one correspondence between an $(m, n, w, \lambda)$-OOSPC and a $(\lambda+1)$-$(mn,w,1)$ packing design admitting an ... |
Seminal work by Edmonds and Lovasz shows the strong connection between submodularity and convexity. Submodular functions have tight modular lower bounds, and subdifferentials in a manner akin to convex functions. |
Petri nets provide accurate analogues to chemical reaction networks, with places representing individual molecules (the resources of the system) and transitions representing chemical reactions which convert educt molecules into product molecules. Their natural affinity for modeling chemical reaction networks is, howev... |
In its Euclidean form, the Dense Neighborhood Lemma (DNL) asserts that if $V$ is a finite set of points of $\mathbb{R}^N$ such that for each $v \in V$ the ball $B(v,1)$ intersects $V$ on at least $\delta |V|$ points, then for every $\varepsilon >0$, the points of $V$ can be covered with $f(\delta,\varepsilon)$ balls... |
The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. |
In this work, we show deep connections between Locality Sensitive Hashability and submodular analysis. We show that the LSHablility of the most commonly analyzed set similarities is in one-to-one correspondance with the supermodularity of these similarities when taken with respect to the symmetric difference of their ... |
For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $\kappa_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the... |
In this paper, we study a primal and dual relationship about triangles: For any graph $G$, let $\nu(G)$ be the maximum number of edge-disjoint triangles in $G$, and $\tau(G)$ be the minimum subset $F$ of edges such that $G \setminus F$ is triangle-free. It is easy to see that $\nu(G) \leq \tau(G) \leq 3 \nu(G)$, and i... |
It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence... |
This paper introduces the \emph{$d$-distance $b$-matching problem}, in which we are given a bipartite graph $G=(S,T;E)$ with $S=\{s_1,\dots,s_n\}$, a weight function on the edges, an integer $d\in\mathbb{Z}_+$ and a degree bound function $b:S\cup T\to\mathbb{Z}_+$. The goal is to find a maximum-weight subset $M\subset... |
The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance between $v$ and any other vertex of $G$. The diameter of a graph $G$ is the maximum eccentricity of a vertex in $G$. |
Sequence pattern avoidance is a central topic in combinatorics. A sequence $s$ contains a sequence $u$ if some subsequence of $s$ can be changed into $u$ by a one-to-one renaming of its letters. |
We investigate pruning in search trees of so-called quantified integer linear programs (QIPs). QIPs consist of a set of linear inequalities and a minimax objective function, where some variables are existentially and others are universally quantified. |
In a proper edge-coloring of a cubic graph an edge $uv$ is called poor or rich, if the set of colors of the edges incident to $u$ and $v$ contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any edge of the graph is either poor or rich. |
We define the block neighborhood of a reversible CA, which is related both to its decomposition into a product of block permutations and to quantum computing. We give a purely combinatorial characterization of the block neighborhood, which helps in two ways. |
An L(2,1)-labelling of a graph $G=(V, E)$ is $\lambda_{2,1}(G)$ a function $f$ from the vertex set V (G) to the set of non-negative integers such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. The L(2,1)-labelling number denoted by $\lambda_{2,1}(G)$ of $G$ is... |
A word is called carefully synchronising for a partial deterministic finite semi-automaton if it maps all states to the same state. Equivalently, it is a composition of partial transformations equal to a constant total transformation. |
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