Parameterized Complexity of Discrete Morse Theory

TL;DR

This paper explores the computational complexity of discrete Morse theory problems on 3-manifolds, establishing hardness results and proposing fixed-parameter tractable algorithms based on treewidth parameters.

Contribution

It refines the complexity analysis of Morse matching problems, proving W[P]-completeness for erasability and introducing FPT algorithms parameterized by treewidth.

Findings

Erasability problem is W[P]-complete.

Proposed FPT algorithm based on treewidth.

Results extend to dual graph treewidth and various triangulation types.

Abstract

Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand we prove that the erasability problem is W[P]-complete on the natural parameter. On the other hand we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplexes. This algorithm also shows fixed parameter tractability for problems such as erasability and maximum alternating cycle-free matching. We further show that these results are also true when the treewidth of the dual graph of the triangulated 3-manifold is bounded. Finally, we investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.