Faster Computation of Path-Width

TL;DR

This paper introduces a more efficient algorithm for computing path-width and path decompositions in graphs, significantly improving practical usability over previous methods with exponential dependence on the width.

Contribution

It presents a new algorithm that computes the path-width and a corresponding path decomposition in time $2^{O(k^2)} n$, improving upon the classical exponential dependence on $k^3$.

Findings

Algorithm runs in $2^{O(k^2)} n$ time

Computes an optimal path decomposition from a smaller graph

Improves practical applicability of path-width computations

Abstract

Tree-width and path-width are widely successful concepts. Many NP-hard problems have efficient solutions when restricted to graphs of bounded tree-width. Many efficient algorithms are based on a tree decomposition. Sometimes the more restricted path decomposition is required. The bottleneck for such algorithms is often the computation of the width and a corresponding tree or path decomposition. For graphs with $n$ vertices and tree-width or path-width $k$, the standard linear time algorithm to compute these decompositions dates back to 1996. Its running time is linear in $n$ and exponential in $k^3$ and not usable in practice. Here we present a more efficient algorithm to compute the path-width and provide a path decomposition. Its running time is $2^{O(k^2)} n$. In the classical algorithm of Bodlaender and Kloks, the path decomposition is computed from a tree decomposition. Here, an optimal path decomposition is computed from a path decomposition of about twice the width. The latter is computed from a constant factor smaller graph.