Homology localization through the looking-glass of parameterized complexity theory
Copyright policy )Homology localization through the looking-glass of parameterized complexity theory
Abstract Homology localization means finding a cycle of lowest weight that represents a homology class in a simplicial complex. It is an NP-complete problem, which this paper addresses using parameterized complexity theory. We prove that to find even a constant factor approximation to this problem is W[1]-hard when solution size is used as a parameter. We have also designed and implemented two new algorithms that are fixed parameter tractable when parameterized by the treewidth of graphs associated to the simplicial complex. The running time of both algorithms matches the lower bounds we obtain from the exponential time hypothesis. We analysed the performance of the two algorithms experimentally and found that one algorithm is significantly faster than the other.
Computational Geometry (cs.CG), FOS: Computer and information sciences, tree decomposition, Parameterized complexity, tractability and kernelization, Computational Complexity (cs.CC), F.2.2; F.1.3, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, Computational methods for problems pertaining to algebraic topology, parameterized complexity, dynamic programming, computational topology, Persistent homology and applications, topological data analysis, Computer Science - Computational Complexity, homology localization, 55-08 (primary) 68Q27, 68Q17 (Secondary), Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.), simplicial complex, Computer Science - Computational Geometry, F.2.2, F.1.3
Computational Geometry (cs.CG), FOS: Computer and information sciences, tree decomposition, Parameterized complexity, tractability and kernelization, Computational Complexity (cs.CC), F.2.2; F.1.3, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, Computational methods for problems pertaining to algebraic topology, parameterized complexity, dynamic programming, computational topology, Persistent homology and applications, topological data analysis, Computer Science - Computational Complexity, homology localization, 55-08 (primary) 68Q27, 68Q17 (Secondary), Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.), simplicial complex, Computer Science - Computational Geometry, F.2.2, F.1.3
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