Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

TL;DR

This paper investigates two NP-hard problems in simplicial complexes related to finding minimal chains, and demonstrates they are fixed-parameter tractable when considering the treewidth of the complex's 1-skeleton.

Contribution

The paper proves that the Optimal Bounded Chain and Homologous Chain problems are fixed-parameter tractable based on the treewidth of the 1-skeleton, despite their NP-hardness.

Findings

Problems are NP-hard and hard to approximate within any constant factor.

Both problems are fixed-parameter tractable with respect to treewidth of the 1-skeleton.

Results provide new algorithmic insights for chain problems in simplicial complexes.

Abstract

We consider two problems on simplicial complexes: the Optimal Bounded Chain Problem and the Optimal Homologous Chain Problem. The Optimal Bounded Chain Problem asks to find the minimum weight $d$-chain in a simplicial complex $K$ bounded by a given $(d{-}1)$-chain, if such a $d$-chain exists. The Optimal Homologous Chain problem asks to find the minimum weight $(d{-}1)$-chain in $K$ homologous to a given $(d{-}1)$-chain. Both of these problems are NP-hard and hard to approximate within any constant factor assuming the Unique Games Conjecture. We prove that these problems are fixed-parameter tractable with respect to the treewidth of the 1-skeleton of $K$.