31 research outputs found
Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”
We establish for which weighted graphs H homomorphism functions from multigraphs
G to H are specializations of the Tutte polynomial of G, answering a question
of Freedman, Lov´asz and Schrijver.
We introduce a new property of graphs called “q-state Potts uniqueness” and relate
it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”,
recently studied by Duan, Wu and Yu.Ministerio de Educación y Ciencia MTM2005-08441-C02-0
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte
uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and
Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164
Homomorphisms and polynomial invariants of graphs
This paper initiates a general study of the connection between graph homomorphisms and the Tutte
polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte
polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials.
As an application, we describe in terms of homomorphism counting some fundamental evaluations of the
Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a
homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de Andalucía PAI-FQM-0164Junta de Andalucía P06-FQM-0164
Counting Constraint Satisfaction Problems
This chapter surveys counting Constraint Satisfaction Problems (counting CSPs, or #CSPs) and their computational complexity. It aims to provide an introduction to the main concepts and techniques, and present a representative selection of results and open problems. It does not cover holants, which are the subject of a separate chapter
A Little Statistical Mechanics for the Graph Theorist
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.
This paper is an elementary general audience survey, intended to popularize
the area and provide an accessible first point of entry for further
exploration.Comment: 30 pages, 3 figure
Two-state spin systems with negative interactions
We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2 × 2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2 × 2 interaction matrices with real entries. We show that in some regions of the parameter space, it’s #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions
Algorithmic Pirogov-Sinai theory
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 and on the torus
. 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 with
appropriate boundary conditions and an efficient sampling algorithm for the
ferromagnetic Potts model on the discrete torus at
sufficiently low temperature
On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs
For a graph , , and a complex number the
partition function of the univariate Potts model is defined as where . In this paper we give zero-free regions for the
partition function of the anti-ferromagnetic Potts model on bounded degree
graphs. In particular we show that for any and any
, there exists an open set in the complex plane that
contains the interval such that for any
and any graph of maximum degree at most . (Here denotes the
base of the natural logarithm.) For small values of we are able to
give better results.
As an application of our results we obtain improved bounds on for the
existence of deterministic approximation algorithms for counting the number of
proper -colourings of graphs of small maximum degree.Comment: In this version the constant 3.02 has been improved to e(=2.71). As a
result the entire paper has undergone some changes to accomodate for this
improvement. We note that the proofs have in essence not changed much. 22
pages; 2 figure