Publications (161)81.21 Total impact
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ABSTRACT: We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. We ask: for which classes of graphs is the Markov chain ergodic and for which is it rapidly mixing? We provide a precise answer to the ergodicity question and close bounds on the mixing question. We show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs, a class that includes examples of interest in the statistical setting. 
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ABSTRACT: Given a symmetric matrix $M\in \{0,1,*\}^{D\times D}$, an $M$partition of a graph $G$ is a function from $V(G)$ to $D$ such that no edge of $G$ is mapped to a $0$ of $M$ and no nonedge to a $1$. We give a computerassisted proof that, when $D=4$, the problem of counting the $M$partitions of an input graph is either in FP or is #Pcomplete. Tractability is proved by reduction to the related problem of counting list $M$partitions; intractability is shown using a gadget construction and interpolation. We use a computer program to determine which of the two cases holds for all but a small number of matrices, which we resolve manually to establish the dichotomy. We conjecture that that the dichotomy also holds for $D>4$. More specifically, we conjecture that, for any symmetric matrix $M\in\{0,1,*\}^{D\times D}$, the complexity of counting $M$partitions is the same as the related problem of counting list $M$partitions.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that heatbath chains (which we define in a general setting) have no negative eigenvalues. Two applications of this result are presented: one to singlesite heatbath chains for spin systems and one to a heatbath Markov chain for sampling contingency tables. Some implications of our main result for the analysis of the mixing time of heatbath Markov chains are discussed. We also prove an alternative characterisation of heatbath chains, and consider possible generalisations.  [Show abstract] [Hide abstract]
ABSTRACT: 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 logmodularity and weak logsupermodularity. We show that if F is weakly logmodular, 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 approximationpreserving reductions for a logicallydefined complexity class #RHPi1, and is believed to be intractable. We further subdivide the #BIShard case. If F is weakly logsupermodular, then we show that #CSP(F) is as easy as a (Boolean) logsupermodular weighted #CSP. Otherwise, we show that it is NPhard to approximate. Finally, we give a full trichotomy for the arity2 case, where #CSP(F) is in FP, or is #BISequivalent, 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.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of kcolouring a random runiform hypergraph with n vertices and cn edges, where k, r, c remain constant as n tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case r=2, must have one of two easily computable values as n tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.  [Show abstract] [Hide abstract]
ABSTRACT: An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of logsupermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial nonlsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function "implies". As a consequence, in the conservative case, all nontrivial counting CSPs are as hard as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexitytheoretic results, it is natural to ask whether the "implies" clone is equivalent to the clone of lsm functions. We use the Mobius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available. 
Conference Paper: PairwiseInteraction Games
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ABSTRACT: We study the complexity of computing Nash equilibria in games where players arranged as the vertices of a graph play a symmetric 2player game against their neighbours. We call this a pairwiseinteraction game. We analyse this game for n players with a fixed number of actions and show that (1) a mixed Nash equilibrium can be computed in constant time for any game, (2) a pure Nash equilibrium can be computed through Nash dynamics in polynomial time for games with a symmetrisable payoff matrix, (3) determining whether a pure Nash equilibrium exists for zerosum games is NPcomplete, and (4) counting pure Nash equilibria is #Pcomplete even for 2strategy games. In proving (3), we define a new defective graph colouring problem called Nash colouring, which is of independent interest, and prove that its decision version is NPcomplete. Finally, we show that pairwiseinteraction games form a proper subclass of the usual graphical games. 
Article: The #CSP dichotomy is decidable
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ABSTRACT: Bulatov (2008) and Dyer and Richerby (2010) have established the following dichotomy for the counting constraint satisfaction problem (\#\csp): for any constraint language $\Gamma\!$, the problem of computing the number of satisfying assignments to constraints drawn from $\Gamma$ is either in \fp{} or is \numpc{}, depending on the structure of $\Gamma\!$. The principal question left open by this research was whether the criterion of the dichotomy is decidable. We show that it is; in fact, it is in \np{}.  [Show abstract] [Hide abstract]
ABSTRACT: In evolutionary game theory, repeated twoplayer games are used to study strategy evolution in a population under natural selection. As the evolution greatly depends on the interaction structure, there has been growing interests in studying the games on graphs. In this setting, players occupy the vertices of a graph and play the game only with their immediate neighbours. Various evolutionary dynamics have been studied in this setting for different games. Due to the complexity of the analysis, however, most of the work in this area is experimental. This paper aims to contribute to a more complete understanding, by providing rigorous analysis. We study the imitation dynamics on two classes of graph: cycles and complete graphs. We focus on three well known social dilemmas, namely the Prisoner's Dilemma, the Stag Hunt and the Snowdrift Game. We also consider, for completeness, the socalled Harmony Game. Our analysis shows that, on the cycle, all four games converge fast, either to total cooperation or total defection. On the complete graph, all but the Snowdrift game converge fast, either to cooperation or defection. The Snowdrift game reaches a metastable state fast, where cooperators and defectors coexist. It will converge to cooperation or defection only after spending time in this state which is exponential in the size, n, of the graph. In exceptional cases, it will remain in this state indefinitely. Our theoretical results are supported by experimental investigations.  [Show abstract] [Hide abstract]
ABSTRACT: Pavlov, a wellknown strategy in game theory, has been shown to have some advantages in the Iterated Prisoner's Dilemma (IPD) game. However, this strategy can be exploited by inveterate defectors. We modify this strategy to mitigate the exploitation. We call the resulting strategy Rational Pavlov. This has a parameter p which measures the "degree of forgiveness" of the players. We study the evolution of cooperation in the IPD game, when n players are arranged in a cycle, and all play this strategy. We examine the effect of varying p on the convergence rate and prove that the convergence rate is fast, O(n log n) time, for high values of p. We also prove that the convergence rate is exponentially slow in n for small enough p. Our analysis leaves a gap in the range of p, but simulations suggest that there is, in fact, a sharp phase transition. 
Conference Paper: Networks of random cycles
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ABSTRACT: We present a family of peertopeer network protocols that yield regular graph topologies having known Hamilton cycles. These topologies are equivalent, in a welldefined sense, to the random regular graph. As a consequence, we have connectivity deterministically, and logarithmic diameter and expansion properties with high probability. We study the efficacy of certain simple topologyaltering operations, designed to introduce randomness. These operations enable the network to selfstabilise when damaged. They resemble the operations used by Cooper, Dyer and Greenhill (2007) for a similar purpose in the case of random regular graphs. There is a link between our protocols and certain combinatorial structures which have been studied previously, in particular discordant permutations and Latin rectangles. We give the first rigorous polynomial mixingtime bounds for natural Markov chains that sample these objects at random. We do so by developing a novel extension to the canonical path technique for bounding mixing times: routing via a random destination. This resembles a technique used by Valiant (1982) for lowcongestion routing in hypercubes. 
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ABSTRACT: Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituting Post's lattice. One of these clones is the collection of logsupermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In our study, we assume that nonnegative unary functions (weights) are available. Given this, we prove that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial nonlsm function is computationally as hard as any problem in #P. Furthermore, any nontrivial functional clone (in a sense that will be made precise below) contains the binary function "implies". As a consequence, all nontrivial counting CSPs (with nonnegative unary weights assumed to be available) are computationally at least as difficult as #BIS, the problem of counting independent sets in a bipartite graph. There is empirical evidence that #BIS is hard to solve, even approximately. Finally, we investigate functional clones in which only restricted unary functions (either favouring 0 or 1) are available.  [Show abstract] [Hide abstract]
ABSTRACT: We give some reductions among problems in (nonnegative) weighted #CSP which restrict the class of functions that needs to be considered in computational complexity studies. Our reductions can be applied to both exact and approximate computation. In particular, we show that a recent dichotomy for unweighted #CSP can be extended to rationalweighted #CSP.  [Show abstract] [Hide abstract]
ABSTRACT: We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the coclone IM"2 from Post's lattice, then the problem of counting satisfying assignments is complete with respect to approximationpreserving reductions for the complexity class #RH@P"1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respect to approximationpreserving reductions, meaning that there is no fully polynomial randomised approximation scheme for counting satisfying assignments unless NP=RP.  [Show abstract] [Hide abstract]
ABSTRACT: Bulatov (2008) gave a dichotomy for the counting constraint satisfaction problem #CSP. A problem from #CSP is characterised by a constraint language, which is a fixed, finite set of relations over a finite domain D. An instance of the problem uses these relations to constrain the variables in a larger set. Bulatov showed that the problem of counting the satisfying assignments of instances of any problem from #CSP is either in polynomial time (FP) or is #Pcomplete. His proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field. We give an elementary proof of Bulatov's dichotomy, based on succinct representations, which we call frames, of a class of highly structured relations, which we call strongly rectangular. We show that these are precisely the relations which are invariant under a Mal'tsev polymorphism. En route, we give a simplification of a decision algorithm for strongly rectangular constraint languages, due to Bulatov and Dalmau (2006). We establish a new criterion for the #CSP dichotomy, which we call strong balance, and we prove that this property is decidable. In fact, we establish membership in NP. Thus, we show that the dichotomy is effective, resolving the most important open question concerning the #CSP dichotomy. 
Conference Paper: The Complexity of Approximating BoundedDegree Boolean #CSP
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ABSTRACT: The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with boundeddegree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum degree is at least 25 we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomialtime if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP = RP. For lower degree bounds, additional cases arise in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.  [Show abstract] [Hide abstract]
ABSTRACT: The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with boundeddegree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum degree is at least 25 we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomialtime if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs. Comment: 12page conference version for STACS 2010 
Conference Paper: On the complexity of #CSP
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ABSTRACT: Bulatov (2008) has given a dichotomy for the counting constraint satisfaction problem, #CSP. A problem from #CSP is characterized by a constraint language γ, which is a fixed, finite set of relations over a finite domain. An instance of the problem uses these relations to constrain the values taken by a finite set of variables. Bulatov showed that, for any fixed γ, the problem of counting the satisfying assignments of instances of any problem from #CSP is either in polynomial time (FP) or #Pcomplete, according on the structure of the constraint language γ. His proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field. We give an elementary proof of Bulatov's dichotomy, based on succinct representations, which we call frames, of a class of highly structured relations, which we call strongly rectangular. We show that these are precisely the relations that are invariant under a Mal'tsev polymorphism. En route, we give a simplification of a decision algorithm for strongly rectangular constraint languages due to Bulatov and Dalmau (2006). Out proof uses no universal algebra, except for the straightforward concept of the Mal'tsev polymorphism and is accessible to readers with little background in #CSP.
Publication Stats
4k  Citations  
81.21  Total Impact Points  
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19882013

University of Leeds
 School of Computing
Leeds, England, United Kingdom


1995

Carnegie Mellon University
 Department of Mathematical Sciences
Pittsburgh, Pennsylvania, United States
