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(k²) 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.

Directed Edge Geography and Undirected Edge Geography are classical PSPACE-complete two-player graph games in which players alternately make moves along edges, deleting each one after use; the first player unable to move loses. We prove that both problems are XNLP-hard when parameterized by pathwidth, addressing a question raised by Bodlaender over 30 years ago. On the positive side, we observe that Directed Edge Geography is fixed-parameter tractable when parameterized by treewidth and maximum degree. We also prove that both problems are in XP on simple graphs when parameterized by tree-partition width. These results develop modern lower-bound and decomposition-based algorithmic methods for width-based questions in PSPACE-complete graph games.

Finding the smallest d-chain with a specific (d−1)-boundary in a simplicial complex is known as the Minimum Bounded Chain problem (MBCd). MBCd is NP-hard for all d≥2. In this paper, we prove that it is also W[1]-hard for all d≥2, if we parameterize the problem by solution size. We also give an algorithm solving MBC1 in polynomial time and introduce and implement two fixed parameter tractable (FPT) algorithms solving MBCd for all d. The first algorithm uses a shortest path approach and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

Given a simplicial complex with $n$ simplices, we consider the Connected Subsurface Recognition (c-SR) problem of finding a subcomplex that is homeomorphic to a given connected surface with a fixed boundary. We also study the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead search for a surface whose boundary, number of connected components, and total genus are given. For both of these problems, we give parameterized algorithms with respect to the treewidth $k$ of the Hasse diagram that run in $2^{O(k \log k)}n^{O(1)}$ time. For the SoG problem, we also prove that our algorithm is optimal assuming the exponential-time hypothesis. In fact, we prove the stronger result that our algorithm is ETH-tight even without restriction on the total genus.