Hardness of Approximation for Morse Matching

Abstract

Discrete Morse theory has emerged as a powerful tool for a wide range of problems, including the computation of (persistent) homology. In this context, discrete Morse theory is used to reduce the problem of computing a topological invariant of an input simplicial complex to computing the same topological invariant of a (significantly smaller) collapsed cell or chain complex. Consequently, devising methods for obtaining gradient vector fields on complexes to reduce the size of the problem instance has become an emerging theme over the last decade. While computing the optimal gradient vector field on a simplicial complex is NP-hard, several heuristics have been observed to compute near-optimal gradient vector fields on a wide variety of datasets. Understanding the theoretical limits of these strategies is therefore a fundamental problem in computational topology.In this paper, we consider the approximability of maximization and minimization variants of the Morse matching problem. We establish hardness results for Max-Morse matching and Min-Morse matching, settling an open problem posed by Joswig and Pfetsch [19]. In particular, we show that, for a simplicial complex of dimension d ≥ 3 with n simplices, it is NP-hard to approximate Min-Morse matching within a factor of O(n 1−ǫ ), for any ǫ > 0. Moreover, we establish hardness of approximation results for Max-Morse matching for simplicial complexes of dimension d ≥ 2, using an L-reduction from Degree 3 Max-Acyclic Subgraph to Max-Morse matching.

Parametrized Complexity of Expansion Height

“…This particular triangulation of the modified dunce hat uses only 7 vertices, 19 edges, and 13 triangles. The modified dunce hat has previously been used as a gadget to show hardness of approximation for Morse matchings [3].…”

“…However, unlike in [6], the use of combinatorial and topological properties of the dunce hat is a key ingredient of the reduction used in this paper. In particular, there is only one gadget in the reductiona subdivision of the so-called modified dunce hat [3], see Figure 2. In this sense the techniques used in this paper are also related to recent work by the first and second author in [3], where they show hardness of approximation for some Morse matching problems.…”

Multivariate Central Limit Theorems for Random Clique Complexes

“…Thus, one naturally seeks an optimal acyclic partial matching on L which admits the fewest possible critical simplices. Unfortunately, the optimal matching problem is computationally intractable to solve [18] even approximately [3] for large L. Our goal in this paper is to simultaneously quantify the expected benefit of using discrete Morse theoretic reductions on a variety of random simplical complexes and to provide a robust null model by which to measure their efficacy on general (i.e., not necessarily random) choices of simplicial complex L. We accomplish these tasks by carefully analysing the distribution of critical simplex counts of a specific (lexicographical) type of acyclic partial matching defined on clique complexes X(n, p) of Bernoulli random graphs G(n, p).…”

Morse Theoretic Templates for High Dimensional Homology Computation

“…Hence one wants to minimize the cardinality of A(w). However, it is known that in general computing an acyclic partial matching which minimizes A(w), i.e., one that is optimal, is NP-hard [10], and can even be NP-hard to approximate [3]. In this paper, we instead leverage one of the strengths of discrete Morse theory: given a fixed cell complex X , there are typically many admissible (acyclic) partial matchings on X .…”