Results 1  10
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56
On Markov chains for independent sets
 Journal of Algorithms
, 1997
"... Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. A new rapidly mixing Markov chain for independent sets is defined in this paper. We show that it is rapidly mixing for a wider range of values of the parameter than the LubyVigoda chain, ..."
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Cited by 66 (16 self)
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Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. A new rapidly mixing Markov chain for independent sets is defined in this paper. We show that it is rapidly mixing for a wider range of values of the parameter than the LubyVigoda chain, the best previously known. Moreover the new chain is apparently more rapidly mixing than the LubyVigoda chain for larger values of (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov ch...
Random walks on combinatorial objects
 Surveys in Combinatorics 1999
, 1999
"... Summary Approximate sampling from combinatoriallydefined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we reexamine the unde ..."
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Cited by 23 (8 self)
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Summary Approximate sampling from combinatoriallydefined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we reexamine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined. In perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area. These are results which imply that there are no polynomial time algorithms of a particular type for a particular problem. 1
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 19 (10 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
An Algorithm for Counting Maximum Weighted Independent Sets and its Applications
 IN 13TH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA 2002), ACM AND SIAM, 2002
, 2002
"... We present an O(1.3247^n) algorithm for counting the number of independent sets with maximum weight in graphs. We show how this algorithm can be used for solving a number of different counting problems: counting exact covers, exact hitting sets, weighted set packing and satisfying assignments in 1i ..."
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Cited by 18 (3 self)
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We present an O(1.3247^n) algorithm for counting the number of independent sets with maximum weight in graphs. We show how this algorithm can be used for solving a number of different counting problems: counting exact covers, exact hitting sets, weighted set packing and satisfying assignments in 1ink SAT.
Algorithms for counting 2SAT solutions and colorings with applications
 TR05033, Electronic Colloquium on Computational Complexity
, 2005
"... An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2Cnf formula. The worst case running time of ..."
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Cited by 18 (2 self)
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An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2Cnf formula. The worst case running time of
The Complexity of Counting Graph Homomorphisms
 In 11th ACM/SIAM Symposium on Discrete Algorithms
, 1999
"... The problem of counting graph homomorphisms is considered. We show that the counting problem corresponding to a given graph is #Pcomplete unless every connected component of the graph is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite gra ..."
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Cited by 15 (4 self)
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The problem of counting graph homomorphisms is considered. We show that the counting problem corresponding to a given graph is #Pcomplete unless every connected component of the graph is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite graph. 1 Introduction Many combinatorial counting problems on graphs can be restated as the problem of counting the number of homomorphisms to a particular graph H. The vertices of H correspond to colours, and the edges show which colours may be adjacent. The graph H may contain loops. Specifically, let C be a set of k colours, where k is a constant. Let H = (C; EH ) be a graph with vertex set C. Given a graph G = (V; E) with vertex set V , a map X : V 7! C is called a Hcolouring if fX(v); X(w)g 2 EH for all fv; wg 2 E: In other words, X is a homomorphism from G to H. Let\Omega H (G) denote the set of all Hcolourings of G. Two wellknown combinatorial counting problems which can be c...
Complexity of Inference in Graphical Models
"... It is wellknown that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with u ..."
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Cited by 13 (0 self)
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It is wellknown that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with unbounded treewidth in which inference is tractable? Subject to a combinatorial hypothesis due to Robertson et al. (1994), we show that low treewidth is indeed the only structural restriction that can ensure tractability. Thus, even for the “best case” graph structure, there is no inference algorithm with complexity polynomial in the treewidth. 1
The Complexity of Planar Counting Problems
, 1998
"... . We prove the #Phardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include 3Sat, 13Sat, 1Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Ve ..."
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Cited by 13 (0 self)
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.<F3.862e+05> We prove the #Phardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include<F3.771e+05> 3Sat, 13Sat, 1Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover.<F3.862e+05> We also prove the<F4.039e+05><F3.862e+05> NPcompleteness of the<F3.771e+05> Ambiguous Satisfiability<F3.862e+05> problems [J. B. Saxe,<F3.783e+05> Two Papers on Graph Embedding<F3.862e+05> Problems, Tech. Report CMUCS80102, Dept. of Computer Science, Carnegie Mellon Univ., Pittsburgh, PA, 1980] and the<F4.039e+05> D<F2.539e+05> P<F3.862e+05> completeness (with respect to random polynomial reducibility) of the unique satisfiability problems [L. G. Valiant and V. V. Vazirani,<F3.783e+05> NP is as easy as detecting unique<F3.862e+05> solutions, in Proc. 17th ACM Symp. on Theory of Computing, 1985, pp. 458463] associ...
Power and stability in connectivity games
 In AAMAS (2
, 2008
"... We consider computational aspects of a game theoretic approach to network reliability. Consider a network where failure of one node may disrupt communication between two other nodes. We model this network as a simple coalitional game, called the vertex Connectivity Game (CG). In this game, each agen ..."
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Cited by 12 (5 self)
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We consider computational aspects of a game theoretic approach to network reliability. Consider a network where failure of one node may disrupt communication between two other nodes. We model this network as a simple coalitional game, called the vertex Connectivity Game (CG). In this game, each agent owns a vertex, and controls all the edges going to and from that vertex. A coalition of agents wins if it fully connects a certain subset of vertices in the graph, called the primary vertices. We show that power indices, which express an agent’s ability to affect the outcome of the vertex connectivity game, can be used to identify significant possible points of failure in the communication network, and can thus be used to increase network reliability. We show that in general graphs, calculating the Banzhaf power index is #Pcomplete, but suggest a polynomial algorithm for calculating this index in trees. We also show a polynomial algorithm for computing the core of a CG, which allows a stable division of payments to coalition agents.
On the Optimal Approximation of Queries Using Tractable Propositional Languages
"... This paper investigates the problem of approximating conjunctive queries without selfjoins on probabilistic databases by lower and upper bounds that can be computed more efficiently. We study this problem via an indirection: Given a propositional formula Φ, find formulas in a more restricted langua ..."
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Cited by 9 (4 self)
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This paper investigates the problem of approximating conjunctive queries without selfjoins on probabilistic databases by lower and upper bounds that can be computed more efficiently. We study this problem via an indirection: Given a propositional formula Φ, find formulas in a more restricted language that are greatest lower bound and least upper bound, respectively, ofΦ. We studyboundsin the languages of readonce formulas, where every variable occurs at most once, and of readonce formulas in disjunctive normal form. We show equivalences of syntactic and modeltheoretic characterisations of optimal bounds for unate formulas, and present algorithms that can enumerate them with polynomial delay. Such bounds can be computed by queries expressed using firstorder queries extended with transitive closure and a special choice construct. Besides probabilistic databases, theseresults can also benefit the problem of approximate query evaluation in relational databases, since the bounds expressed by queries can be computed in polynomial combined complexity. Categories andSubject Descriptors H.2.4 [Database Management]: Systems—Query Processing