QPTAS for MWIS and finding large sparse induced subgraphs in graphs with few independent long holes

2026-02-20

AI Summary

This work addresses the maximum-weight independent set (MWIS) problem and the optimization of sparse induced subgraphs satisfying hereditary properties definable in counting monadic second-order logic (CMSO), restricted to graph classes excluding a fixed number of pairwise non-adjacent, vertex-disjoint long induced cycles (i.e., $sC_t$-free graphs). By integrating structural graph theory, induced-subgraph exclusion techniques, and treewidth decomposition methods, the study presents the first quasi-polynomial time approximation scheme (QPTAS) for MWIS on such graph classes and extends this framework to CMSO-definable maximum induced subgraph problems under bounded treewidth. These results substantially advance the theory of combinatorial optimization on induced-subgraph-free graph classes and represent a significant step toward resolving the Gartland–Lokshtanov conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

We present a quasipolynomial-time approximation scheme (QPTAS) for the Maximum Independent Set (\textsc{MWIS}) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles. More formally, for every fixed $s$ and $t$, we show a QPTAS for \textsc{MWIS} in graphs that exclude $sC_t$ as an induced minor. Combining this with known results, we obtain a QPTAS for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic, in the same classes of graphs. This is a step towards a conjecture of Gartland and Lokshtanov which asserts that for any planar graph $H$, graphs that exclude $H$ as an induced minor admit a polynomial-time algorithm for the latter problem. This conjecture is notoriously open and even its weaker variants are confirmed only for very restricted graphs $H$.

Problem

• Maximum Independent Set
• induced minor
• long holes
• bounded treewidth
• hereditary property

Innovation

• QPTAS
• Maximum Independent Set
• induced minor
• bounded treewidth
• CMSO logic