On (In)approximability of MaxMin Independent Set Reconfiguration

2026-04-29

AI Summary

This work addresses the MaxMin Independent Set Reconfiguration problem, which seeks to transform one independent set into another via vertex addition and deletion operations while maximizing the minimum size of any intermediate independent set. Combining techniques from combinatorial optimization, parameterized algorithms, and complexity theory, the paper develops efficient approximation algorithms and establishes inapproximability lower bounds for general graphs as well as several restricted graph classes—including degenerate graphs, bounded treewidth graphs, and H-minor-free graphs. The main contributions include the first $O(n/\log n)$-approximation algorithm for general graphs that matches known hardness results, the first efficient approximation or FPT approximation schemes for multiple structured graph families, and strengthened inapproximability results for bounded-degree graphs, bandwidth-bounded graphs, and bipartite graphs, thereby significantly advancing the understanding of the problem’s computational complexity.

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 $n^{Ω(1)}$-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 $n^{\frac{1}{2}+Θ(1)}$, and bipartite graphs.

Problem

• Independent Set Reconfiguration
• MaxMin
• Approximability
• Inapproximability
• Token Addition/Removal

Innovation

• Independent Set Reconfiguration
• Approximation Algorithm
• Inapproximability
• FPT Approximation
• Degenerate Graphs