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43
A COMPLEXITY DICHOTOMY FOR PARTITION FUNCTIONS WITH MIXED SIGNS
, 2009
"... Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of kcolourings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physi ..."
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Cited by 32 (7 self)
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Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of kcolourings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill [7] and Bulatov and Grohe [6], we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #Pcomplete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #Pcomplete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices — these turn out to be central in our proofs — we obtain a simple algebraic tractability criterion, which says that the tractable cases are those “representable” by a quadratic polynomial over the field F2.
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,
An approximation trichotomy for Boolean #CSP
, 2007
"... 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 ..."
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Cited by 25 (7 self)
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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 IM2 from Post’s lattice, then the problem of counting satisfying assignments is complete with respect to approximationpreserving reductions in the complexity class #RHΠ1. 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. 1
Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP
"... Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a un ..."
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Cited by 17 (7 self)
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Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #Phard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary realvalued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #Phard on general graphs but tractable on planar graphs, or (3) those which are #Phard even on planar graphs. The classification criteria
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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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
Dichotomy for Holant Problems of Boolean Domain
"... Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and taking comp ..."
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Cited by 13 (4 self)
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Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and taking complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #Phard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems over the complex domain, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (nonzero) vector whose inner product with itself is zero.
The Complexity of Symmetric Boolean Parity Holant Problems (Extended Abstract)
"... Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizatio ..."
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Cited by 12 (3 self)
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Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizations such as Constraint Satisfaction Problems, and directed and undirected Graph Homomorphism Problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant Problems. These additionally allow for the expression of matching problems, which have had pivotal roles in complexity theory. As our main result we prove the dichotomy theorem that, for the class ⊕P, every set of boolean symmetric Holant signatures of any arities that is not polynomial time computable is ⊕Pcomplete. The result exploits some special properties of the class ⊕P and characterizes four distinct tractable subclasses within ⊕P. It leaves open the corresponding questions for NP, #P and #kP for k ̸ = 2. 1
The Complexity of Weighted Boolean #CSP with Mixed Signs
, 2009
"... We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set Γ of rational functions, each of which assigns a weight to each variable assignment. Our dichotomy extends previous work ..."
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We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set Γ of rational functions, each of which assigns a weight to each variable assignment. Our dichotomy extends previous work in which the weight functions were restricted to being nonnegative. We represent a weight function as a product of the form (−1) s g, where the polynomial s determines the sign of the weight and the nonnegative function g determines its magnitude. We show that the problem of computing the partition function (the sum of the weights of all possible variable assignments) is computable in polynomial time if either every function in Γ can be defined by a “pure affine ” magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of “product type” with a linear sign polynomial. In all other cases, computing the partition function is FP #Pcomplete.