ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes

TL;DR

This paper establishes the precise computational complexity of the Optimal Morse Matching problem on complexes with bounded treewidth, providing a tight algorithm and proving optimality under ETH.

Contribution

It introduces a new $2^{O(k \, log k)} n$-time algorithm for the problem and proves that no significantly faster algorithm exists unless ETH fails.

Findings

New $2^{O(k \, log k)} n$-time algorithm for OMM

Lower bound matching the algorithm's complexity under ETH

Clarifies the complexity landscape of OMM on bounded-treewidth complexes

Abstract

The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth $k$, OMM has long been known to be solvable on triangulations of $3$-manifolds in $2^{O(k^2)} n^{O(1)}$ time and in FPT time for triangulations of arbitrary manifolds, but the exact dependence on $k$ has remained an open question. We resolve this by giving a new $2^{O(k \log k)} n$-time algorithm for any finite regular CW complex, and show that no $2^{o(k \log k)} n^{O(1)}$-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails.