19,149 research outputs found

    Minimum Cost Homomorphisms to Reflexive Digraphs

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    For digraphs GG and HH, a homomorphism of GG to HH is a mapping $f:\ V(G)\dom V(H)suchthat such that uv\in A(G)implies implies f(u)f(v)\in A(H).Ifmoreovereachvertex. If moreover each vertex u \in V(G)isassociatedwithcosts is associated with costs c_i(u), i \in V(H),thenthecostofahomomorphism, then the cost of a homomorphism fis is \sum_{u\in V(G)}c_{f(u)}(u).Foreachfixeddigraph. For each fixed digraph H, the {\em minimum cost homomorphism problem} for H,denotedMinHOM(, denoted MinHOM(H),isthefollowingproblem.Givenaninputdigraph), is the following problem. Given an input digraph G,togetherwithcosts, together with costs c_i(u),, u\in V(G),, i\in V(H),andaninteger, and an integer k,decideif, decide if Gadmitsahomomorphismto admits a homomorphism to Hofcostnotexceeding of cost not exceeding k. We focus on the minimum cost homomorphism problem for {\em reflexive} digraphs H(everyvertexof (every vertex of Hhasaloop).ItisknownthattheproblemMinHOM( has a loop). It is known that the problem MinHOM(H)ispolynomialtimesolvableifthedigraph) is polynomial time solvable if the digraph H has a {\em Min-Max ordering}, i.e., if its vertices can be linearly ordered by <sothat so that i<j, s<rand and ir, js \in A(H)implythat imply that is \in A(H)and and jr \in A(H).WegiveaforbiddeninducedsubgraphcharacterizationofreflexivedigraphswithaMinMaxordering;ourcharacterizationimpliesapolynomialtimetestfortheexistenceofaMinMaxordering.Usingthischaracterization,weshowthatforareflexivedigraph. We give a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering; our characterization implies a polynomial time test for the existence of a Min-Max ordering. Using this characterization, we show that for a reflexive digraph H$ which does not admit a Min-Max ordering, the minimum cost homomorphism problem is NP-complete. Thus we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for reflexive digraphs

    Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

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    For digraphs GG and HH, a homomorphism of GG to HH is a mapping $f:\ V(G)\dom V(H)suchthat such that uv\in A(G)implies implies f(u)f(v)\in A(H).If,moreover,eachvertex. If, moreover, each vertex u \in V(G)isassociatedwithcosts is associated with costs c_i(u), i \in V(H),thenthecostofahomomorphism, then the cost of a homomorphism fis is \sum_{u\in V(G)}c_{f(u)}(u).Foreachfixeddigraph. For each fixed digraph H,theminimumcosthomomorphismproblemfor, the minimum cost homomorphism problem for H,denotedMinHOM(, denoted MinHOM(H),canbeformulatedasfollows:Givenaninputdigraph), can be formulated as follows: Given an input digraph G,togetherwithcosts, together with costs c_i(u),, u\in V(G),, i\in V(H),decidewhetherthereexistsahomomorphismof, decide whether there exists a homomorphism of Gto to H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes

    Dichotomies properties on computational complexity of S-packing coloring problems

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    This work establishes the complexity class of several instances of the S-packing coloring problem: for a graph G, a positive integer k and a non decreasing list of integers S = (s\_1 , ..., s\_k ), G is S-colorable, if its vertices can be partitioned into sets S\_i , i = 1,... , k, where each S\_i being a s\_i -packing (a set of vertices at pairwise distance greater than s\_i). For a list of three integers, a dichotomy between NP-complete problems and polynomial time solvable problems is determined for subcubic graphs. Moreover, for an unfixed size of list, the complexity of the S-packing coloring problem is determined for several instances of the problem. These properties are used in order to prove a dichotomy between NP-complete problems and polynomial time solvable problems for lists of at most four integers

    On the minimum and maximum selective graph coloring problems in some graph classes

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    Given a graph together with a partition of its vertex set, the minimum selective coloring problem consists of selecting one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is minimum. The contribution of this paper is twofold. First, we investigate the complexity status of the minimum selective coloring problem in some specific graph classes motivated by some models described in Demange et al. (2015). Second, we introduce a new problem that corresponds to the worst situation in the minimum selective coloring; the maximum selective coloring problem aims to select one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is maximum. We motivat

    The complexity of approximating conservative counting CSPs

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    We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem of counting independent sets in bipartite graphs. #BIS is complete with respect to approximation-preserving reductions for a logically-defined complexity class #RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in difficulty to #SAT, the problem of approximately counting the satisfying assignments of a Boolean formula in conjunctive normal form. We also discuss the algorithmic aspects of our classification.Comment: Minor revisio

    String Matching: Communication, Circuits, and Learning

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    String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings. - Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol. - Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k. - Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem

    The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

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    Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and st-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side - which includes but is not limited to all problems with polynomial time algorithms for satisfiability - is in P for the st-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary
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