Approximation Algorithms for Max-Morse Matching

TL;DR

This paper proves that the Max-Morse Matching problem is approximable and introduces two algorithms with provable approximation ratios for different classes of simplicial complexes, aiding efficient homology computation.

Contribution

It provides the first approximation algorithms for Max-Morse Matching, including ratios for general complexes and manifolds, solving an open problem.

Findings

Algorithms achieve near-optimal results in experiments.

Approximation ratios are proven for different complex types.

Efficient homology computation is facilitated by these algorithms.

Abstract

In this paper, we prove that the Max-Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch. We describe two different approximation algorithms for the Max-Morse Matching Problem. For $D$-dimensional simplicial complexes, we obtain a $\frac{(D+1)}{(D^2+D+1)}$-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. Our second result is an algorithm that provides a $\frac{2}{D}$-factor approximation for simplicial manifolds by processing the simplices in increasing order of dimension. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results.