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.

翻译：最优莫尔斯匹配（OMM）问题要求在单纯复形上寻找一个能最小化临界单形数量的离散梯度向量场。该问题是NP难的，并已在启发式、近似及参数化复杂度背景下得到广泛研究。以树宽$k$为参数，长期以来已知OMM可在$2^{O(k^2)} n^{O(1)}$时间内求解$3$-流形的三角剖分，且对任意流形的三角剖分可在FPT时间内求解，但其对$k$的精确依赖关系一直是一个开放问题。我们通过为任意有限正则CW复形提供一个新的$2^{O(k \log k)} n$时间算法解决了此问题，并证明除非指数时间假设（ETH）不成立，否则不存在$2^{o(k \log k)} n^{O(1)}$时间算法。