Evaluating the Tutte Polynomial for Graphs of Bounded Tree-Width

It is known that evaluating the Tutte polynomial, $T(G; x, y)$, of a graph, $G$, is $\#$P-hard at all but eight specific points and one specific curve of the $(x, y)$-plane. In contrast we show that if $k$ is a fixed constant then for graphs of tree-width at most $k$ there is an algorithm that will evaluate the polynomial at any point using only a linear number of multiplications and additions

Evaluating a weighted graph polynomial for graphs of bounded tree-width

We show that for any $k$ there is a polynomial time algorithm to evaluate the weighted graph polynomial $U$ of any graph with tree-width at most $k$ at any point. For a graph with $n$ vertices, the algorithm requires $O(a_k n^{2k+3})$ arithmetical operations, where $a_k$ depends only on $k$

Evaluating the rank generating function of a graphic 2-polymatroid

We consider the complexity of the two-variable rank generating function, $S$, of a graphic 2-polymatroid. For a graph $G$, $S$ is the generating function for the number of subsets of edges of $G$ having a particular size and incident with a particular number of vertices of $G$. We show that for any $x,y \in \mathbb{Q}$ with $xy \not = 1$, it is $\#$P-hard to evaluate $S$ at $(x,y)$. We also consider the $k$-thickening of a graph and computing $S$ for the $k$-thickening of a graph

A graph polynomial for independent sets of bipartite graphs

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings and the number of perfect matchings. Most importantly, for bipartite graphs the polynomial encodes the number of independent sets (#BIS). We analyze the complexity of exact evaluation of the polynomial at rational points and show that for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial. We conjecture that a natural Markov chain can be used to approximately evaluate the polynomial for a range of parameters. The conjecture, if true, would imply an approximate counting algorithm for #BIS, a problem shown, by [Dyer et al. 2004], to be complete (with respect to, so called, AP-reductions) for a rich logically defined sub-class of #P. We give a mild support for our conjecture by proving that the Markov chain is rapidly mixing on trees. As a by-product we show that the "single bond flip" Markov chain for the random cluster model is rapidly mixing on constant tree-width graphs

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor improvements. 44 pages, 14 figure

Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.Comment: 7 pages, 4 figure

Exact counting of Euler Tours for Graphs of Bounded Treewidth

In this paper we give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian graph of bounded treewidth. The problems of counting ETs are known to be #P-complete for general graphs (Brightwell and Winkler, (Brightwell and Winkler, 2005). To date, no polynomial-time algorithm for counting Euler tours of any class of graphs is known except for the very special case of series-parallel graphs (which have treewidth 2).Comment: 16 pages, draf