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

TL;DR

This paper introduces a quasipolynomial-time approximation scheme for the Maximum Independent Set problem in certain sparse graphs, advancing understanding of algorithms for graphs excluding specific induced minors.

Contribution

It provides a QPTAS for MWIS in graphs with few long induced cycles and extends to finding large bounded-treewidth induced subgraphs, addressing a key conjecture in graph theory.

Findings

QPTAS for MWIS in graphs with bounded long induced cycles

Extension to finding large induced subgraphs with bounded treewidth

Progress towards a conjecture on polynomial algorithms for graphs excluding fixed minors

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 $H$.