Parameterized inapproximability of Morse matching

TL;DR

This paper investigates the computational hardness of minimizing critical simplices in Morse matchings, establishing strong inapproximability results and providing an approximation algorithm for 2-complexes.

Contribution

It proves inapproximability and W[P]-hardness for Min-Morse Matching, and offers an $O(n/\, ext{log} )$ approximation for 2-complexes, advancing understanding of its complexity.

Findings

Inapproximability within $2^{ ext{log}^{(1- ext{epsilon})}n}$

W[P]-hardness with respect to standard parameter

An $O(n/ ext{log} n)$ approximation algorithm for 2-complexes

Abstract

We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of $2^{\log^{(1-\epsilon)}n}$. Our second result shows that Min-Morse Matching is ${\bf W{[P]}}$-hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor $\rho$. The above hardness results are applicable to complexes of dimension $\ge 2$. On the positive side, we provide a factor $O(\frac{n}{\log n})$ approximation algorithm for Min-Morse Matching on $2$-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa-Farber model of random complexes.