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182
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
, 2005
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Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the ..."
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Cited by 50 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
The RandomCluster Model
, 2006
"... Abstract. The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, an ..."
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Cited by 48 (20 self)
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Abstract. The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the randomcluster representation. This systematic summary of randomcluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinitevolume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for twodimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the clusterweighting factor q, and the problem of proving that the critical randomcluster model in two
NormProduct Belief Propagation: PrimalDual MessagePassing for Approximate Inference
, 2008
"... Inference problems in graphical models can be represented as a constrained optimization of a free energy function. In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified me ..."
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Cited by 42 (9 self)
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Inference problems in graphical models can be represented as a constrained optimization of a free energy function. In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified messagepassing algorithm architecture. In particular we generalize the Belief Propagation (BP) algorithms of sumproduct and maxproduct and treerewaighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on ”convexfreeenergy” and LinearProgramming (LP) relaxation as a zerotemprature of a convexfreeenergy. The main idea of this work arises from taking a general perspective on the existing BP and TRBP algorithms while observing that they all are reductions from the basic optimization formula of f + ∑ i hi
Structured prediction models via the matrixtree theorem
 In EMNLPCoNLL
, 2007
"... This paper provides an algorithmic framework for learning statistical models involving directed spanning trees, or equivalently nonprojective dependency structures. We show how partition functions and marginals for directed spanning trees can be computed by an adaptation of Kirchhoff’s MatrixTree ..."
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Cited by 41 (6 self)
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This paper provides an algorithmic framework for learning statistical models involving directed spanning trees, or equivalently nonprojective dependency structures. We show how partition functions and marginals for directed spanning trees can be computed by an adaptation of Kirchhoff’s MatrixTree Theorem. To demonstrate an application of the method, we perform experiments which use the algorithm in training both loglinear and maxmargin dependency parsers. The new training methods give improvements in accuracy over perceptrontrained models. 1
Fixedpoint logics on planar graphs
, 1998
"... We study the expressive power of inflationary fixedpoint logic IFP and inflationary fixedpoint logic with counting IFP+C on planar graphs. We prove the following results:(1) IFP captures polynomial time on 3connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2 ..."
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Cited by 40 (12 self)
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We study the expressive power of inflationary fixedpoint logic IFP and inflationary fixedpoint logic with counting IFP+C on planar graphs. We prove the following results:(1) IFP captures polynomial time on 3connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2) Planar graphs can be characterized up to isomorphism in a logic with finitely many variables and counting. This answers a question of Immerman [7].(3) The class of planar graphs is definable in IFP. This answers a question of Dawar and Grädel [16].
Chromatic roots are dense in the whole complex plane
 In preparation
, 2000
"... to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic ..."
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Cited by 37 (14 self)
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to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc q − 1  < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Pottsmodel partition functions) ZG(q,v) outside the disc q + v  < v. An immediate corollary is that the chromatic roots of notnecessarilyplanar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof. KEY WORDS: Graph, chromatic polynomial, dichromatic polynomial, Whitney rank function, Tutte polynomial, Potts model, Fortuin–Kasteleyn representation,
On the complexity of nonprojective datadriven dependency parsing
 In Proc. IWPT
, 2007
"... In this paper we investigate several nonprojective parsing algorithms for dependency parsing, providing novel polynomial time solutions under the assumption that each dependency decision is independent of all the others, called here the edgefactored model. We also investigate algorithms for nonpro ..."
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Cited by 32 (1 self)
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In this paper we investigate several nonprojective parsing algorithms for dependency parsing, providing novel polynomial time solutions under the assumption that each dependency decision is independent of all the others, called here the edgefactored model. We also investigate algorithms for nonprojective parsing that account for nonlocal information, and present several hardness results. This suggests that it is unlikely that exact nonprojective dependency parsing is tractable for any model richer than the edgefactored model. 1
Probabilistic models of nonprojective dependency trees
 In Proc. EMNLPCoNLL
, 2007
"... A notable gap in research on statistical dependency parsing is a proper conditional probability distribution over nonprojective dependency trees for a given sentence. We exploit the Matrix Tree Theorem (Tutte, 1984) to derive an algorithm that efficiently sums the scores of all nonprojective trees i ..."
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Cited by 29 (9 self)
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A notable gap in research on statistical dependency parsing is a proper conditional probability distribution over nonprojective dependency trees for a given sentence. We exploit the Matrix Tree Theorem (Tutte, 1984) to derive an algorithm that efficiently sums the scores of all nonprojective trees in a sentence, permitting the definition of a conditional loglinear model over trees. While discriminative methods, such as those presented in McDonald et al. (2005b), obtain very high accuracy on standard dependency parsing tasks and can be trained and applied without marginalization, “summing trees ” permits some alternative techniques of interest. Using the summing algorithm, we present competitive experimental results on four nonprojective languages, for maximum conditional likelihood estimation, minimum Bayesrisk parsing, and hidden variable training. 1