Fast and Practical Single-Exponential Algorithms for Branchwidth

2026-05-17

AI Summary

Branchwidth computation is an NP-hard combinatorial optimization problem. This work proposes single-exponential-time exact algorithms based on recursive enumeration over vertex subsets, tailored separately for graphs and hypergraphs. For hypergraphs, the study presents the first exact algorithm running in O*(4ⁿ) time. For graphs, it introduces an improved algorithm with a running time of O(3.293ⁿ), surpassing the previous best bound of O(3.4652ⁿ) and demonstrating substantially better practical performance than existing state-of-the-art methods. The algorithms integrate dynamic programming with refined pruning techniques and instance-specific optimizations, achieving notable advances both theoretically and empirically.

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.

Problem

• branchwidth
• single-exponential algorithms
• hypergraphs
• exact algorithms
• graph algorithms

Innovation

• branchwidth
• single-exponential algorithm
• exact exponential algorithm
• hypergraph
• graph algorithms