# Collapsibility to a subcomplex of a given dimension is NP-complete

## TL;DR

This paper proves that determining whether a d-dimensional simplicial complex can be collapsed to a k-dimensional subcomplex is NP-complete for most cases, extending previous results and completing the complexity classification.

## Contribution

The paper extends NP-completeness results for (d,k)-collapsibility to all cases with d ≥ k+2, except (2,0), completing the complexity landscape.

## Key findings

- NP-completeness of (d,k)-collapsibility for d ≥ k+2, except (2,0)
- Polynomial algorithms exist for (2,0) and d=k+1 cases
- Complete classification of the complexity of (d,k)-collapsibility

## Abstract

In this paper we extend the works of Tancer and of Malgouyres and Franc\'es, showing that $(d,k)$-collapsibility is NP-complete for $d\geq k+2$ except $(2,0)$. By $(d,k)$-collapsibility we mean the following problem: determine whether a given $d$-dimensional simplicial complex can be collapsed to some $k$-dimensional subcomplex. The question of establishing the complexity status of $(d,k)$-collapsibility was asked by Tancer, who proved NP-completeness of $(d,0)$ and $(d,1)$-collapsibility (for $d\geq 3$). Our extended result, together with the known polynomial-time algorithms for $(2,0)$ and $d=k+1$, answers the question completely.