Parametrized Complexity of Expansion Height

TL;DR

This paper investigates the computational complexity of transforming 2-dimensional simplicial complexes into 1-dimensional complexes via simple-homotopy equivalences, establishing W[P]-completeness for the problem parameterized by the number of expansions.

Contribution

It introduces and proves the W[P]-completeness of the erasability expansion height problem, a novel complexity classification for simple-homotopy transformations.

Findings

The problem is W[P]-complete when parameterized by the number of expansions.

Transforming complexes via simple-homotopy equivalences is computationally complex.

The study advances understanding of the complexity of topological simplification processes.

Abstract

Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p.