On (In)approximability of MaxMin Independent Set Reconfiguration

TL;DR

This paper investigates the computational difficulty of approximating the MaxMin Independent Set Reconfiguration problem, providing algorithms for certain graph classes and establishing hardness results for others.

Contribution

It introduces approximation algorithms for general and restricted graph classes and proves inapproximability results for bounded-degree, bandwidth, and bipartite graphs.

Findings

Polynomial-time $(n / \log n)$-factor approximation for general graphs.

Approximation algorithms for degenerate, bounded-treewidth, and $H$-minor-free graphs.

Hardness results extend to bounded-degree, bandwidth, and bipartite graphs.

Abstract

In the Independent Set Reconfiguration problem under the Token Addition/Removal rule, given a graph $G$ and two independent sets $I$ and $J$ of $G$, we want to transform $I$ into $J$ by adding and removing vertices, such that all the sets throughout the process are independent sets. Its approximate version called MaxMin Independent Set Reconfiguration aims to maximise the minimum size of the independent sets in the process above. We study the (in)approximability of this problem for general graphs as well as restricted graph classes. Firstly, on general graphs, we obtain a polynomial-time $(n / \log n)$-factor approximation algorithm, complementing the $\mathsf{PSPACE}$-hardness of $n^{\Omega(1)}$-factor approximation due to Hirahara and Ohsaka [STOC 2024, ICALP 2024] and the $\mathsf{NP}$-hardness of $n^{1-\varepsilon}$-factor approximation due to Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [TCS 2011]. Secondly, we present a polynomial-time approximation algorithm for degenerate graphs as well as $\mathsf{FPT}$-approximation schemes for bounded-treewidth graphs and $H$-minor-free graphs. Lastly, we extend the above inapproximability results to bounded-degree graphs, graphs of bandwidth $n^{\frac{1}{2}+\Theta(1)}$, and bipartite graphs.