ETH-tight algorithms for finding surfaces in simplicial complexes of bounded treewidth

TL;DR

This paper introduces parameterized algorithms for recognizing surfaces within simplicial complexes of bounded treewidth, providing optimal solutions under the exponential-time hypothesis.

Contribution

It presents the first fixed-parameter algorithms for the Connected Subsurface Recognition and Sum-of-Genus Subsurface Recognition problems based on treewidth, with proven optimality under ETH.

Findings

Algorithms run in $2^{O(k \log k)}n^{O(1)}$ time

Proves the optimality of the algorithms assuming ETH

Establishes ETH-tight bounds even without genus restrictions

Abstract

Given a simplicial complex with $n$ simplices, we consider the Connected Subsurface Recognition (c-SR) problem of finding a subcomplex that is homeomorphic to a given connected surface with a fixed boundary. We also study the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead search for a surface whose boundary, number of connected components, and total genus are given. For both of these problems, we give parameterized algorithms with respect to the treewidth $k$ of the Hasse diagram that run in $2^{O(k \log k)}n^{O(1)}$ time. For the SoG problem, we also prove that our algorithm is optimal assuming the exponential-time hypothesis. In fact, we prove the stronger result that our algorithm is ETH-tight even without restriction on the total genus.