Homology Localization Through the Looking-Glass of Parameterized Complexity Theory

TL;DR

This paper investigates the computational complexity of homology localization, proving its hardness in approximation and offering two fixed-parameter tractable algorithms based on treewidth, with practical performance differences.

Contribution

It introduces the first fixed-parameter algorithms for homology localization using treewidth and establishes their tight complexity bounds.

Findings

Homology localization is W[1]-hard to approximate when parameterized by solution size.

Two FPT algorithms based on treewidth are designed and implemented.

One algorithm outperforms the other in practical scenarios.

Abstract

Finding a cycle of lowest weight that represents a homology class in a simplicial complex is known as homology localization (HL). Here we address this NP-complete problem using parameterized complexity theory. We show that it is W[1]-hard to approximate the HL problem when it is parameterized by solution size. We have also designed and implemented two algorithms based on treewidth solving the HL problem in FPT-time. Both algorithms are ETH-tight but our results shows that one outperforms the other in practice.