Fast and Practical Single-Exponential Algorithms for Branchwidth

TL;DR

This paper introduces new exact exponential algorithms for computing branchwidth in hypergraphs and graphs, achieving faster theoretical and practical performance than previous methods.

Contribution

The paper presents the first single-exponential time algorithm for hypergraphs and improves the running time for graph branchwidth algorithms, with practical performance benefits.

Findings

Hypergraph algorithm runs in *(4^n) time.

Graph algorithm improves previous best to (3.293^n).

Experimental results show significant practical performance improvements.

Abstract

In this paper, we present exact exponential algorithms for computing branchwidth that are fast both in theory and in practice. The running times of these algorithms are single-exponential in the number of vertices. Our basic algorithm is based on a conceptually simple recurrence on vertex sets and computes the branchwidth of an $n$-vertex hypergraph in time $\mathcal{O}^*(4^n)$. This is the first single-exponential time algorithm for hypergraphs. We have two algorithms tailored specifically for graphs. The first algorithm runs in time $\mathcal{O}(3.293^n)$, improving upon the previously best-known running time of $\mathcal{O}(3.4652^n)$ [Fomin-Mazoit-Todinca, DAM 2009]. Moreover, our computational experiment shows that it overwhelmingly outperforms state-of-the-art practical algorithms for computing branchwidth. The second algorithm is a candidate for a theoretical improvement: we conjecture that it runs in time $\mathcal{O}(c^n)$ for some constant $c$ that is smaller than 3.293. In practice, it performs significantly better on some instances that are hard for the first algorithm.