Computing the Partition Function for Cliques in a Graph
TL;DR: A deterministic algorithm which, given a graph G with n vertices and an integer 1 0 is an absolute constant, allows us to tell apart the graphs that do not have m-subsets of high density from the graph that have sufficiently many m- Subsets ofhigh density.
Abstract: We present a deterministic algorithm which, given a graph G with n vertices and an integer 1 0 is an absolute constant. This allows us to tell apart the graphs that do not have m-subsets of high density from the graphs that have sufficiently many m-subsets of high density, even when the probability to hit such a subset at random is exponentially small in m. ACM Classification: F.2.1, G.1.2, G.2.2, I.1.2 AMS Classification: 15A15, 68C25, 68W25, 60C05
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Journal Article•
TL;DR: In this paper, the authors consider the question of determining whether a function f has property P or is e-far from any function with property P. In some cases, it is also allowed to query f on instances of its choice.
Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.
870 citations
TL;DR: A new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs is shown.
87 citations
Cites methods from "Computing the Partition Function fo..."
...Keywords: approximation algorithms, Tutte polynomial, independence polynomial, partition function, graph homomorphism, Holant problem....
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TL;DR: It is shown that Sokal's Conjecture holds, as well as a multivariate version, and optimality for the domain of non-vanishing for independence polynomials of graphs is proved.
Abstract: A conjecture of Sokal (2001) regarding the domain of non-vanishing for independence polynomials of graphs, states that given any natural number $\Delta \ge 3$, there exists a neighborhood in $\mathbb C$ of the interval $[0, \frac{(\Delta-1)^{\Delta-1}}{(\Delta-2)^{\Delta}})$ on which the independence polynomial of any graph with maximum degree at most $\Delta$ does not vanish. We show here that Sokal's Conjecture holds, as well as a multivariate version, and prove optimality for the domain of non-vanishing. An important step is to translate the setting to the language of complex dynamical systems.
53 citations
TL;DR: In this article, the authors developed an efficient algorithm for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice and on the torus.
Abstract: We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice $$\mathbb {Z}^d$$ and on the torus $$(\mathbb {Z}/n\mathbb {Z})^d$$. Our approach is based on combining contour representations from Pirogov–Sinai theory with Barvinok’s approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of $$\mathbb {Z}^d$$ with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus $$(\mathbb {Z}/n\mathbb {Z})^d$$ at sufficiently low temperature.
53 citations
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TL;DR: In this article, a deterministic algorithm for computing the permanent of a real or complex matrix in O(ln n -ln epsilon) time is presented. But the algorithm is not suitable for computing hafnians and multidimensional permanents.
Abstract: We present a deterministic algorithm, which, for any given 0< epsilon < 1 and an nxn real or complex matrix A=(a_{ij}) such that | a_{ij}-1| < 0.19 for all i, j computes the permanent of A within relative error epsilon in n^{O(ln n -ln epsilon)} time. The method can be extended to computing hafnians and multidimensional permanents.
48 citations
References
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TL;DR: In this paper, the problems of an Ising model in a magnetic field and a lattice gas are proved mathematically equivalent, and an example of a two-dimensional lattice model is given for which the phase transition regions in the $p\ensuremath{-}v$ diagram is exactly calculated.
Abstract: The problems of an Ising model in a magnetic field and a lattice gas are proved mathematically equivalent. From this equivalence an example of a two-dimensional lattice gas is given for which the phase transition regions in the $p\ensuremath{-}v$ diagram is exactly calculated.A theorem is proved which states that under a class of general conditions the roots of the grand partition function always lie on a circle. Consequences of this theorem and its relation with practical approximation methods are discussed. All the known exact results about the two-dimensional square Ising lattice are summarized, and some new results are quoted.
1,822 citations
1,564 citations
TL;DR: In this paper, a theory of equations of state and phase transitions is developed that describes the condensed as well as the gas phases and the transition regions, and the thermodynamic properties of an infinite sample are studied rigorously and Mayer's theory is re-examined.
Abstract: A theory of equations of state and phase transitions is developed that describes the condensed as well as the gas phases and the transition regions. The thermodynamic properties of an infinite sample are studied rigorously and Mayer's theory is re-examined.
1,524 citations
TL;DR: This paper considers the question of determining whether a function f has property P or is ε-far from any function with property P, and devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a clique of density p-Clique with respect to the vertex set.
Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.
1,027 citations
Journal Article•
TL;DR: In this paper, the authors consider the question of determining whether a function f has property P or is e-far from any function with property P. In some cases, it is also allowed to query f on instances of its choice.
Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.
870 citations