Parameterized Complexity of Discrete Morse Theory

Abstract

Optimal Morse matchings reveal essential structures of cell complexes that 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-simplices. 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 discuss the topological significance of the chosen parameters and investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.

Hardness of Approximation for Morse Matching

“…Most methods so far have relied on effective heuristics [19,14,3,16,25]. The first theoretical result in context of Morse matchings was established by Burton et al [5], who developed a fixed parameter tractable algorithm for computing optimal Morse functions on 3-manifolds. More recently, Rathod et al [34] proposed the first approximation algorithms for MaxMM on simplicial complexes that provide constant factor approximation bounds for fixed dimension.…”

“…We shall use the terms gradient vector fields and Morse matchings interchangeably. In the next subsection, we will elaborate on the practical interest [1,2,4,5,13,14,23] in computing gradient vector fields on simplicial complexes with (near-)optimal number of critical simplices (unmatched nodes in the Hasse diagram).…”

Analysis of Morse Matchings: Parameterized Complexity and Stable Matching

“…The results from Chapters 3 and 4 are published [14] and are joint work with Jonathan Spreer and Benjamin Burton at University of Queensland.…”

“…Our first main result shows that the Erasability problem is W [P ]-complete, i.e. in the worst category of the W -hierarchy [14] (Theorems 3.13 and 3.16), when the parameter is the natural parameter -the number of cells that have to be removed. In other words, we prove that the Erasability problem is fixed-parameter intractable in this parameter.…”

“…This chapter is a joint work with Jonathan Spreer and Benjamin Burton[14] PUC-Rio -Certificação Digital Nº 1021470/CA…”