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.

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.

Equations

$$X^{\prime}=X\cup\Big{\{}\sigma\cup\{w_{i}\}\;\Big{|}\;\sigma\in X,\;i=1,\,\dots,\,n+1\Big{\}}.$$

$${\mathcal{M}}^{\prime }$$

$$w_i<w_j\text{ if and only if }i=1\text{ and }j>1.$$

$$\phi (\sigma )=\{\begin{array}{ll}{w}_j& \text{if σ contains wj for some j≥2;}\\ w_1& \text{otherwise.}\end{array}$$

$$\text{Cr}(X^{\prime},\mathcal{M}^{\prime})=\{w_{1}\}\cup\Big{\{}\sigma\cup\{w_{i}\}\;\Big{|}\;\sigma\in\text{Cr}(X,\mathcal{M})\cup\{\varnothing\},\;i=2,\,\dots,\,n+1\Big{\}}.$$

$$\Big{(}\sigma\cup\{w_{j}\}\to\sigma\Big{)}\not\in\mathcal{M}^{\prime}\quad\forall\;\sigma\in X.$$

$$\mathcal{M}=\Big{\{}\sigma\to\tau\;\Big{|}\;\sigma,\tau\in X\text{ satisfying }\Big{(}\sigma\cup\{w_{j}\}\to\tau\cup\{w_{j}\}\Big{)}\in\mathcal{M}^{\prime}\Big{\}}.$$

$$\iota :\sigma \mapsto \sigma \cup \{w_j\},$$

$$\text{Cr}(X,\mathcal{M})=\Big{\{}\sigma\in X\;\Big{|}\;\sigma\cup\{w_{j}\}\in\text{Cr}(X^{\prime},\mathcal{M}^{\prime})\,\text{ or }\,\Big{(}\sigma\cup\{w_{j}\}\to\{w_{j}\}\Big{)}\in\mathcal{M}^{\prime}\Big{\}}.$$

Full text

The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-017-9915-6.

1. Introduction

Discrete Morse theory is a powerful combinatorial tool which allows to explicitly simplify cell complexes while preserving their homotopy type [BW02, Cha00, For98, Koz08]. This is obtained through a sequence of “elementary collapses” of pairs of cells. Such a process might decrease the dimension of the starting complex, or sometimes even leave a single point (in which case we say that the starting complex was collapsible).

The problem of algorithmically recognising collapsibility, or finding “good” sequences of elementary collapses, has been studied extensively [BL14, BLPS16, EG96, JP06, MF08, Tan16]. Such problems proved to be computationally hard even for low-dimensional simplicial complexes. For 2-dimensional complexes there exists a polynomial-time algorithm to check collapsibility [JP06, MF08], but finding the minimum number of “critical” triangles (without which the remaining complex would be collapsible) is already NP-hard [EG96]. In dimension $\geq 3$, collapsibility to some $1$-dimensional subcomplex [MF08] or even to a single point [Tan16] were proved to be NP-complete.

In [Tan16], Tancer also introduced the general $(d,k)$-Collapsibility problem: determine whether a $d$-dimensional simplicial complex can be collapsed to some $k$-dimensional subcomplex. He showed that $(d,k)$-Collapsibility is NP-complete for $k\in\{0,1\}$ and $d\geq 3$, extending the result of Malgouyres and Francés about NP-completeness of $(3,1)$-Collapsibility [MF08]. Tancer also pointed out that the codimension $1$ case ($d=k+1$) is polynomial-time solvable as is the $(2,0)$ case. He left open the question of determining the complexity status of $(d,k)$-Collapsibility in general.

In this short paper we extend Tancer’s work, and prove that $(d,k)$-Collapsibility is NP-complete in all the remaining cases.

Theorem 3.2.

The $(d,k)$-Collapsibility problem is NP-complete for $d\geq k+2$, except for the case $(2,0)$.

To do so, we prove that $(d,k)$-Collapsibility admits a polynomial-time reduction to $(d+1,k+1)$-Collapsibility (Theorem 3.1). Then the main result follows by induction. The base cases of the induction are given by NP-completeness of $(3,1)$-Collapsibility (for codimension $2$) and of $(d,0)$-Collapsibility (for codimension $d\geq 3$).

2. Collapsibility and discrete Morse theory

We refer to [Hat02] for the definition and the basic properties of simplicial complexes, and to [Koz08] for the definition of elementary collapses. The simplicial complexes we consider do not contain the empty simplex, unless otherwise stated.

Our focus is the following decision problem.

Problem 2.1: $(d,k)$-Collapsibility**.

**

    Parameters:

Non-negative integers $d>k$.

    Instance:

A finite $d$-dimensional simplicial complex $X$.

    Question:

Can $X$ be collapsed to some $k$-dimensional subcomplex?

We are now going to recall a few definitions of discrete Morse theory [Cha00, For98, Koz08], so that we can restate the $(d,k)$-Collapsibility problem in terms of acyclic matchings.

Given a simplicial complex $X$, its Hasse diagram $H(X)$ is a directed graph in which the set of nodes is the set of simplexes of $X$, and an arc goes from $\sigma$ to $\tau$ if and only if $\tau$ is a face of $\sigma$ and $\dim(\sigma)=\dim(\tau)+1$. We denote such an arc by $\sigma\to\tau$. A matching $\mathcal{M}$ on $X$ is a set of arcs of $H(X)$ such that every node of $H(X)$ (i.e. every simplex of $X$) is contained in at most one arc in $\mathcal{M}$. Given a matching $\mathcal{M}$ on $X$, we say that a simplex $\sigma\in X$ is critical if it does not belong to any arc in $\mathcal{M}$. Finally we say that a matching $\mathcal{M}$ on $X$ is acyclic if the graph $H(X)^{\mathcal{M}}$, obtained from $H(X)$ by reversing the direction of each arc in $\mathcal{M}$, does not contain directed cycles.

Notice that the empty set is always a valid acyclic matching, for which all simplices are critical. See Fig. 1 for an example of a non-trivial acyclic matching on the full triangle.

By standard facts of discrete Morse theory (see for instance [Koz08, Section 11.2]), “collapsibility to some $k$-dimensional subcomplex” is equivalent to “existence of an acyclic matching such that the critical cells form a $k$-dimensional subcomplex”. Notice that, given an acyclic matching $\mathcal{M}$ with no critical simplices of dimension $>k$, one can always remove from $\mathcal{M}$ the arcs between simplices of dimension $\leq k$ and obtain an acyclic matching where the critical simplices form a $k$-dimensional subcomplex. Therefore the collapsibility problem can be restated as follows.

Problem 2.2: $(d,k)$-Collapsibility (equivalent form)**.

**

    Parameters:

Non-negative integers $d>k$.

    Instance:

A finite $d$-dimensional simplicial complex $X$.

    Question:

Does $X$ admit an acyclic matching such that all critical simplices have dimension $\leq k$?

To simplify the proof of Theorem 3.1 we quote the following useful result from [Koz08], adapting it to our notation.

Theorem 2.3** (Patchwork theorem [Koz08, Theorem 11.10]).**

Let $P$ be a poset. Let $\varphi\colon X\to P$ be an order-preserving map (where $X$ is ordered by inclusion), and assume to have acyclic matchings on subposets $\varphi^{-1}(p)$ for all $p\in P$. Then the union of these matchings is itself an acyclic matching on $X$.

Notice that the subposets $\varphi^{-1}(p)$ are not subcomplexes of $X$ in general, but they still have a well-defined Hasse diagram (the induced subgraph of $H(X)$). Thus all the previous definitions (matching, critical simplex, acyclic matching) apply also to each subposet.

3. Main result

Theorem 3.1**.**

Let $d>k\geq 0$. Then there is a polynomial-time reduction from $(d,k)$-Collapsibility to $(d+1,k+1)$-Collapsibility.

Proof.

Let $X$ be an instance of $(d,k)$-Collapsibility, i.e. a $d$-dimensional simplicial complex. Let $V=\{v_{1},\dots,v_{r}\}$ be the vertex set of $X$. Construct an instance $X^{\prime}$ of $(d+1,k+1)$-Collapsibility, i.e. a $(d+1)$-dimensional complex, as follows. Let $n\geq 1$ be the number of simplices in $X$. Introduce new vertices $w_{1},\dots,w_{n+1}$, and define $X^{\prime}$ as the simplicial complex on the vertex set $V^{\prime}=\{v_{1},\dots,v_{r},w_{1},\dots,w_{n+1}\}$ given by

[TABLE]

Then $X^{\prime}$ has $n(n+2)$ simplices. Roughly speaking, $X^{\prime}$ is obtained from $X$ by attaching $n+1$ cones of $X$ to $X$. We are going to prove that $X$ is a yes-instance of $(d,k)$-Collapsibility if and only if $X^{\prime}$ is a yes-instance of $(d+1,k+1)$-Collapsibility.

Suppose that $X$ is a yes-instance of $(d,k)$-Collapsibility. Then there exists an acyclic matching $\mathcal{M}$ on $X$ such that all critical simplices have dimension $\leq k$. Construct a matching $\mathcal{M}^{\prime}$ on $X^{\prime}$ as follows:

[TABLE]

This matching corresponds to collapsing the first cone together with $X$, and every other “base-less” cone by itself (as a copy of $X$). Notice that the critical simplices of $\mathcal{M}^{\prime}$ do not form a subcomplex of $X^{\prime}$, even when the critical simplices of $\mathcal{M}$ form a subcomplex of $X$.

To prove that $\mathcal{M}^{\prime}$ is acyclic, consider the set $P=\{w_{1},\dots,w_{n+1}\}$ with the partial order

[TABLE]

Let $\varphi\colon X^{\prime}\to P$ be the order-preserving map given by

[TABLE]

Then $\mathcal{M}^{\prime}$ is a union of matchings $\mathcal{M}_{j}^{\prime}$ on each fiber $\varphi^{-1}(w_{j})$. The matching $\mathcal{M}_{1}^{\prime}$ is acyclic on $\varphi^{-1}(w_{1})$, since the arcs of $\mathcal{M}_{1}^{\prime}$ define a cut of the Hasse diagram of $\varphi^{-1}(w_{1})$. The Hasse diagram of each $\varphi^{-1}(w_{j})$ for $j\geq 2$ is isomorphic to $H(X\cup\{\varnothing\})$ via the map $\sigma\cup\{w_{j}\}\mapsto\sigma$, and the matching $\mathcal{M}_{j}$ maps to $\mathcal{M}$. Since $\mathcal{M}$ is acyclic on $H(X)$, each $\mathcal{M}_{j}$ is also acyclic on $\varphi^{-1}(w_{j})$. By the Patchwork theorem (Theorem 2.3), $\mathcal{M}^{\prime}$ is acyclic on $X^{\prime}$.

The set of critical simplices of $\mathcal{M}^{\prime}$ is

[TABLE]

In particular, all critical simplices have dimension $\leq k+1$. Therefore $X^{\prime}$ is a yes-instance of $(d+1,k+1)$-Collapsibility.

Conversely, suppose now that $X^{\prime}$ is a yes-instance of $(d+1,k+1)$-Collapsibility. Let $\mathcal{M}^{\prime}$ be an acyclic matching on $X^{\prime}$ such that all critical simplices have dimension $\leq k+1$. Since $X$ contains $n$ simplices, and there are $n+1$ cones, there must exist an index $j\in\{1,\,\dots,\,n+1\}$ such that

[TABLE]

In other words, simplices containing $w_{j}$ are only matched with simplices containing $w_{j}$. Then we can construct a matching $\mathcal{M}$ on $X$ as follows:

[TABLE]

Notice that if there is some [math]-dimensional simplex $\sigma=\{v\}\in X$ such that $(\{v,w_{j}\}\to\{w_{j}\})\in\mathcal{M}^{\prime}$, then $\{v\}$ is critical with respect to $\mathcal{M}$ (it would be matched with $\tau=\varnothing$, which does not exist in $X$). The Hasse diagram of $X$ injects into the Hasse diagram of the $j$-th cone via the map

[TABLE]

and by construction $\mathcal{M}$ maps to $\mathcal{M}^{\prime}$. Since $\mathcal{M}^{\prime}$ is acyclic, $\mathcal{M}$ is also acyclic. The set of critical simplices of $\mathcal{M}$ is

[TABLE]

In the first case $\sigma\cup\{w_{j}\}$ has dimension $\leq k+1$, and in the second case $\sigma$ is [math]-dimensional. In particular, all critical simplices have dimension $\leq k$. Therefore $X$ is a yes-instance of $(d,k)$-Collapsibility. ∎

The $(d,k)$-Collapsibility problem admits a polynomial-time solution when $d=k+1$ and also for the case $(2,0)$ [JP06, MF08, Tan16]. Malgouyres and Francés [MF08] proved that $(3,1)$-Collapsibility is NP-complete, and Tancer [Tan16] extended this result to $(d,k)$-Collapsibility for $k\in\{0,\,1\}$ and $d\geq 3$. Using this as the base step and Theorem 3.1 as the induction step, we obtain the following result.

Theorem 3.2**.**

The $(d,k)$-Collapsibility problem is NP-complete for $d\geq k+2$, except for the case $(2,0)$. ∎

4. Acknowledgements

I would like to thank my father, Maurizio Paolini, for giving useful comments and suggesting corrections. I would also like to thank Luca Ghidelli, for checking the proof thoroughly and for being my best man. Finally, I would like to thank the anonymous referee for his/her suggestions.