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

TL;DR

It is shown that no new $2^{O(k \log k)} n$-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails, and it is resolved by giving a new $2^{O(k \log k)} n$-time algorithm for any finite regular CW complex.

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.

Figures (21)

• Figure 1: Discrete Morse theory can simplify a space while preserving its homotopy type.
• Figure 2: Three spaces from left to right, all of relatively low (but increasing) treewidth.
• Figure 3: A graph $G$ (right) and a tree decomposition $T$ (left). Each node of $T$ carries a bag $X_t \subseteq V(G)$, drawn as a blob containing vertices of $G$. Adjacent bags overlap so that every vertex and every edge of $G$ is covered.
• Figure 4: An instance of DFVS, a valid solution of size $2$ and an optimal solution of size $1$.
• Figure 5: A discrete Morse function (top) on the Hasse diagram of a simplicial complex (left) and its geometric realization (right). Below, the induced discrete gradient vector field (Morse matching) is shown both on the Hasse diagram (left) and geometrically as a gradient vector field (right).

Theorems & Definitions (17)

• Remark 2
• Remark 3
• Theorem 5: Bonamy et al. bonamy2018directed
• Theorem 7: Folklore; cf. forman1998morsejoswig2006computingbauer2019hardness
• Theorem 9
• Lemma 11: Matchings vs. orders
• Theorem 13
• Definition 14: Tree decomposition
• Definition 15: Rooted nice tree decomposition
• Definition 16: Feedback Morse order