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22
Graph Homomorphisms with Complex Values: A Dichotomy Theorem
"... Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting ..."
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Cited by 31 (14 self)
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Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting vertex covers and kcolorings. We study the computational complexity of ZA(G) for arbitrary complex valued symmetric matrices A. Building on work by Dyer and Greenhill [6], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg,
On counting homomorphisms to directed acyclic graphs
 ICALP (1). Volume 4051 of Lecture Notes in Computer Science
, 2006
"... It is known that if P and NP are different then there is an infinite hierarchy of different complexity classes which lie strictly between them. Thus, if P ≠ NP, it is not possible to classify NP using any finite collection of complexity classes. This situation has led to attempts to identify smaller ..."
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Cited by 28 (4 self)
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It is known that if P and NP are different then there is an infinite hierarchy of different complexity classes which lie strictly between them. Thus, if P ≠ NP, it is not possible to classify NP using any finite collection of complexity classes. This situation has led to attempts to identify smaller classes of problems within NP where dichotomy results may hold: every problem is either in P or is NPcomplete. A similar situation exists for counting problems. If P ≠ #P, there is an infinite hierarchy in between and it is important to identify subclasses of #P where dichotomy results hold. Graph homomorphism problems are a fertile setting in which to explore dichotomy theorems. Indeed, Feder and Vardi have shown that a dichotomy theorem for the problem of deciding whether there is a homomorphism to a fixed directed acyclic graph would resolve their longstanding dichotomy conjecture for all constraint satisfaction problems. In this paper we give a dichotomy theorem for the problem of counting homomorphisms to directed acyclic graphs. Let H be a fixed directed acyclic graph. The problem is, given an input digraph G, determine how many homomorphisms there are from G to H. We give a graphtheoretic classification, showing that for some digraphs H, the problem is in P and for the rest of the digraphs H the problem is #Pcomplete. An interesting feature of the dichotomy, which is absent from previouslyknown dichotomy results, is that there is a rich supply of tractable graphs H with complex structure.
Nonnegative Weighted #CSPs: An Effective Complexity Dichotomy
"... We prove a complexity dichotomy theorem for all nonnegative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms [19, 8, 18, 12] and the celebrated dichotomy theorem for unwe ..."
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Cited by 17 (10 self)
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We prove a complexity dichotomy theorem for all nonnegative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms [19, 8, 18, 12] and the celebrated dichotomy theorem for unweighted #CSP [6, 4, 21, 22]. Our dichotomy theorem gives a succinct criterion for tractability. If a set F of constraint functions satisfies the criterion, then the counting CSP problem defined by F is solvable in polynomial time; if it does not satisfy the criterion, then the problem is #Phard. We furthermore show that the question of whether F satisfies the criterion is decidable in NP. Surprisingly, our tractability criterion is simpler than the previous criteria for the more restricted classes of problems, although when specialized to those cases, they are logically equivalent. Our proof mainly uses Linear
Approximate counting via correlation decay in spin systems
 In Proceedings of the 23rd Annual ACMSIAM Symposium on Discrete Algorithms
, 2012
"... We give the first deterministic fully polynomialtime approximation scheme (FPTAS) for computing the partition function of a twostate spin system on an arbitrary graph, when the parameters of the system satisfy the uniqueness condition on infinite regular trees. This condition is of physical signif ..."
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Cited by 17 (11 self)
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We give the first deterministic fully polynomialtime approximation scheme (FPTAS) for computing the partition function of a twostate spin system on an arbitrary graph, when the parameters of the system satisfy the uniqueness condition on infinite regular trees. This condition is of physical significance and is believed to be the right boundary between approximable and inapproximable. The FPTAS is based on the correlation decay technique introduced by Bandyopadhyay and Gamarnik [1] and Weitz [61]. The classic correlation decay is defined with respect to graph distance. Although this definition has natural physical meanings, it does not directly support an FPTAS for systems on arbitrary graphs, because for graphs with unbounded degrees, the local computation that provides a desirable precision by correlation decay may take superpolynomial time. We introduce a notion of computationally efficient correlation decay, in which the correlation decay is measured in a refined metric instead of graph distance. We use a potential method to analyze the amortized behavior of this correlation decay and establish a correlation decay that guarantees an inversepolynomial precision by polynomialtime local computation. This gives us an FPTAS for spin systems on arbitrary graphs. This new notion of correlation decay properly reflects the algorithmic aspect of the spin systems, and may be used for designing FPTAS for other counting problems. 1
Counting HColorings of Partial kTrees
"... The problem of counting all Hcolorings of a graph G with n vertices is considered. While the problem is, in general, #Pcomplete, we give linear time algorithms that solve the main variants of this problem when the input graph G is a ktree or, in the case where G is directed, when the underlying g ..."
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Cited by 14 (3 self)
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The problem of counting all Hcolorings of a graph G with n vertices is considered. While the problem is, in general, #Pcomplete, we give linear time algorithms that solve the main variants of this problem when the input graph G is a ktree or, in the case where G is directed, when the underlying graph of G is a ktree. Our algorithms remain polynomial even in the case where k = O(log n) or in the case where the size of H is O(n). Our results are easy to implement and imply the existence of polynomial time algorithms for a series of problems on partial ktrees such as core checking and chromatic polynomial computation.
An effective dichotomy for the counting constraint satisfaction problem. arXiv:1003.3879
, 2010
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The complexity of approximating boundeddegree boolean #CSP
 STACS, VOLUME 5 OF LIPICS
, 2010
"... 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 ..."
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Cited by 8 (1 self)
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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.
On Markov chains for randomly Hcolouring a graph
, 2000
"... Let H = (W; F ) be a graph without multiple edges, but with the possibility of having loops. Let G = (V; E) be a simple graph. A homomorphism c is a map c : V !W with the property that (v; w) 2 E implies (c(v); c(w)) 2 F . We will often refer to c(v) as the colour of v and c as an Hcolouring of ..."
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Cited by 6 (3 self)
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Let H = (W; F ) be a graph without multiple edges, but with the possibility of having loops. Let G = (V; E) be a simple graph. A homomorphism c is a map c : V !W with the property that (v; w) 2 E implies (c(v); c(w)) 2 F . We will often refer to c(v) as the colour of v and c as an Hcolouring of G. We consider the problem of choosing a random Hcolouring of G by Markov Chain Monte Carlo. The probabilistic model we consider includes random proper colourings, random independent sets and the WidomRowlinson and Beach models of Statistical Physics. We prove negative results for uniform sampling and a positive result for weighted sampling when H is a tree. 1 Introduction We consider a class of graph labellings which are the natural generalisation of wellstudied problems such as proper kcolourings and independent sets. School of Mathematical Sciences, University of North London,London N7 8DB,UK y School of Computer Studies, University of Leeds, Leeds LS2 9TJ, UK z Department of...
Holographic Reduction, Interpolation and Hardness
"... We prove a dichotomy theorem for a class of counting problems expressible by Boolean signatures. The proof methods are holographic reductions and interpolations. We show that interpolatability provides a universal strategy to prove #Phardness for this class of problems. For these problems whenever ..."
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Cited by 5 (3 self)
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We prove a dichotomy theorem for a class of counting problems expressible by Boolean signatures. The proof methods are holographic reductions and interpolations. We show that interpolatability provides a universal strategy to prove #Phardness for this class of problems. For these problems whenever holographic reductions followed by interpolations fail to prove #Phardness, we can show that the problems are actually solvable in polynomial time. 1