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

Édouard Bonnet, Jadwiga Czyżewska, Tomáš Masařík, Marcin Pilipczuk, Paweł Rzążewski

TL;DR

A quasipolynomial-time approximation scheme (QPTAS) is presented for the Maximum Independent Set (MWIS) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles and obtained for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic.

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 $H$.

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

TL;DR

A quasipolynomial-time approximation scheme (QPTAS) is presented for the Maximum Independent Set (MWIS) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles and obtained for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic.

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 and , we show a QPTAS for \textsc{MWIS} in graphs that exclude 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 , graphs that exclude 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 .
Paper Structure (5 sections, 9 theorems, 1 equation)

This paper contains 5 sections, 9 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. QPTAS for MWIS: Proof of \ref{['thm:mwis']}
  4. QPTAS for $(\textsf{tw} \leqslant r,\psi)$-MWIS: Proof of \ref{['thm:pack']}
  5. Conclusion

Key Result

Theorem 1

Let $s,t$ be positive integers and $\varepsilon \in (0,1)$ be a real. There is an algorithm that, given a vertex-weighted graph $G$, in quasipolynomial time returns either:

Theorems & Definitions (10)

  • Conjecture A: Gartland, Lokshtanov GartlandThesis
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4: DBLP:journals/siamcomp/ChudnovskyPPT24
  • Theorem 4
  • Theorem 13
  • Lemma 14
  • Theorem 15
  • Theorem 15