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87
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 82 (17 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...
Set partitioning via inclusionexclusion
 SIAM J. Comput
"... Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of t ..."
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Cited by 59 (7 self)
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Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion–exclusion and the zeta transform. In effect we get exact algorithms in 2nnO(1) time for several wellstudied partition problems including Domatic Number, Chromatic Number, Maximum kCut, Bin Packing, List Colouring, and the Chromatic Polynomial. We also have applications to Bayesian learning with decision graphs and to modelbased data clustering. If only polynomial space is available, our algorithms run in time 3nnO(1) if membership in F can be decided in polynomial time. We solve Chromatic Number in O(2.2461n) time and Domatic Number in O(2.8718n) time. Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and d(1 + )χ(G)e in time O(1.2209n + 2.2461e−n). 1. Introduction. Graph colouring, domatic partitioning, weighted kcut, and a
Accidental Algorithms
 In Proc. 47th Annual IEEE Symposium on Foundations of Computer Science 2006
, 2004
"... Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in onetoone, or possibly onetomany, fashion. In this paper we propose a new ..."
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Cited by 57 (2 self)
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Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in onetoone, or possibly onetomany, fashion. In this paper we propose a new kind of reduction that allows for gadgets with manytomany correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a #Pcomplete problem then polynomial systems can be obtained the solvability of which would imply P #P = NC2. 1
The complexity of the counting constraint satisfaction problem
 In ICALP (1
, 2008
"... The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can ..."
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Cited by 45 (7 self)
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The Counting Constraint Satisfaction Problem (#CSP(H)) over a finite relational structureH can be expressed as follows: given a relational structure G over the same vocabulary, determine the number of homomorphisms from G toH. In this paper we characterize relational structuresH for which#CSP(H) can be solved in polynomial time and prove that for all other structures the problem is #Pcomplete. 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 40 (15 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
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 23 (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
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 22 (9 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
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 21 (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.
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 21 (5 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...