Efficient algorithms to decide tightness

TL;DR

This paper introduces new polynomial-time algorithms for deciding tightness in 3-manifolds and fixed parameter tractable algorithms for higher dimensions, advancing the computational understanding of tightness.

Contribution

It presents the first polynomial-time algorithm for tightness in 3-manifolds and fixed parameter tractable algorithms for higher dimensions based on treewidth.

Findings

Polynomial-time decision algorithm for 3-manifolds.

Fixed parameter tractable algorithms for 4-dimensional manifolds.

Algorithm for $ ext{F}_2$-tightness in arbitrary fixed dimensions.

Abstract

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces. In this article, we present a new polynomial time procedure to decide tightness for triangulations of $3$-manifolds -- a problem which previously was thought to be hard. Furthermore, we describe an algorithm to decide general tightness in the case of $4$-dimensional combinatorial manifolds which is fixed parameter tractable in the treewidth of the $1$-skeletons of their vertex links, and we present an algorithm to decide $\mathbb{F}_2$-tightness for weak pseudomanifolds $M$ of arbitrary but fixed dimension which is fixed parameter tractable in the treewidth of the dual graph of $M$.