Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

TL;DR

This paper introduces a faster, more direct algorithm for evaluating the multivariate interlace polynomial of graphs with bounded treewidth, improving upon previous methods by reducing complexity and enabling efficient parallel implementation.

Contribution

The authors develop a new algorithm that evaluates the interlace polynomial more efficiently on graphs of bounded treewidth, surpassing prior logical framework-based approaches.

Findings

Uses 2^{3k^2+O(k)}*n arithmetic operations

Faster than previous algorithms based on logical frameworks

Can be efficiently implemented in parallel

Abstract

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.