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

2026-03-05

AI Summary

This work investigates the exact computational complexity of the Optimal Morse Matching (OMM) problem on finite regular CW complexes parameterized by treewidth. For complexes of treewidth $k$, the authors devise an exact algorithm running in time $2^{O(k \log k)} \cdot n$ and, under the Exponential Time Hypothesis (ETH), establish that no algorithm can solve the problem in time $2^{o(k \log k)} \cdot n^{O(1)}$. By integrating techniques from discrete Morse theory, tree decompositions, and parameterized algorithms, this study provides the first tight upper and lower bounds for OMM with respect to treewidth, resolving a long-standing open question regarding its parameterized complexity and establishing the ETH-optimal runtime for this fundamental topological optimization problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Optimal Morse Matching
• treewidth
• ETH
• parameterized complexity
• discrete gradient

Innovation

• Optimal Morse Matching
• treewidth
• ETH-tight complexity
• discrete gradient vector field
• parameterized complexity