Computing the Tutte Polynomial on Graphs of Bounded Clique-Width

Giménez, Omer; Hliněný, Petr; Noy, Marc

doi:10.1007/11604686_6

Cited by 18 publications

(21 citation statements)

References 12 publications

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“…For example, Noble [Nob98] has shown that the Tutte polynomial may be computed in polynomial time (in fact requires only a linear number of multiplications and additions) for rational points on graphs with bounded tree width, and Makowsky [Mak05] and Traldi [Tra06] have extended this result to the colored Tutte polynomial. Gimenez, Hlineny and Noy [GHN06] and Makowsky, Rotics, Averbouch and Godlin [MRAG06] provide similar results for bounded clique-width (a notion with significant computational complexity consequences analogous to those for bounded tree-width; see Oum and Seymour [OS06]).…”

Section: Computational Complexity Connectionsmentioning

confidence: 76%

A little statistical mechanics for the graph theorist

Beaudin

Ellis-Monaghan

Pangborn

et al. 2010

Discrete Mathematics

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In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches.

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Section: Computational Complexity Connectionsmentioning

confidence: 76%

A little statistical mechanics for the graph theorist

Beaudin

Ellis-Monaghan

Pangborn

et al. 2010

Discrete Mathematics

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“…Likewise, the Tutte polynomial is known to be polynomial-time computable on graphs of bounded tree-width: see [2,21]. In contrast, for graphs of bounded clique-width, a width notion which generalizes tree-width, the best algorithm known for the Tutte polynomial is subexponential: see [12]. We show the following.…”

Section: Theorem 12 (Hardness Of the Trivariate Ising Polynomial)mentioning

confidence: 93%

Complexity of Ising Polynomials

Kotek¹

2012

Combinator. Probab. Comp.

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This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomialZ(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson.Z(G;x,y,z) generalizes a bivariate polynomialZ(G;t,y), which was studied in by Andrén and Markström.We consider the complexity ofZ(Gt,y) andZ(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show thatZ(G;x,y,z) is #P-hard to evaluate at all points in3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETHa dichotomy theorem stating that evaluations ofZ(G;t,y) either take exponential time in the number of vertices ofGto compute, or can be done in polynomial time. Finally, we give an algorithm for computingZ(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

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“…Since graph 3-colouring corresponds to evaluating T at (−2, 0), the exponential time complexity for T (G; −2, 0) was thereby already understood. In particular, computing T (G; x, y) for input G and (x, y) requires vertex-exponential time, an observation that is already made in [GHN06] without explicit reference to ETH.…”

Section: Computing the Tutte Polynomialmentioning

confidence: 99%

Exponential Time Complexity of the Permanent and the Tutte Polynomial

Dell

Husfeldt

Marx

et al. 2014

ACM Trans. Algorithms

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We show conditional lower bounds for well-studied #P-hard problems:• The number of satisfying assignments of a 2-CNF formula with n variables cannot be computed in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph.• The permanent of an n × n matrix with entries 0 and 1 cannot be computed in time exp(o(n)).• The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x, y) in the case of multigraphs, and it cannot be computed in time exp(o(n/ poly log n)) in the case of simple graphs.Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting.

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Computing the Tutte Polynomial on Graphs of Bounded Clique-Width

Cited by 18 publications

References 12 publications

A little statistical mechanics for the graph theorist

A little statistical mechanics for the graph theorist

Complexity of Ising Polynomials

Exponential Time Complexity of the Permanent and the Tutte Polynomial

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