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Tree decompositions meet induced matchings: beyond Max Weight Independent Set

Paloma T. Lima, Martin Milanič, Peter Muršič, Karolina Okrasa, Paweł Rzążewski, Kenny Štorgel

TL;DR

This work studies graphs of bounded induced matching treewidth, a parameter that refines structural width notions and enables efficient algorithms beyond Max Weight Independent Set. It presents a polynomial-time algorithm for Min Weight Feedback Vertex Set (via Max Weight Induced Forest) on such graphs and develops a robust DP framework that leverages a compact signature per bag, along with skeletons and impostors, to handle maximal induced forests. The paper also extends tractability to packing problems and to CMSO2-definable bounded-treewidth subgraphs, including a PTAS for certain induced subgraph packings and a general (r,ψ)-MWIS framework for graphs with bounded tree-independence, linking to a broad class of problems. Structural and complexity results further clarify the landscape: precise bounds, monotonicity under induced minors, and limitations under degree constraints highlight both the potential and the limits of induced matching width as a scalable tool for algorithmic graph problems.

Abstract

For a tree decomposition $\mathcal{T}$ of a graph $G$, by $μ(\mathcal{T})$ we denote the size of a largest induced matching in $G$ all of whose edges intersect one bag of $\mathcal{T}$. Induced matching treewidth of a graph $G$ is the minimum value of $μ(\mathcal{T})$ over all tree decompositions $\mathcal{T}$ of $G$. Yolov [SODA 2018] proved that Max Weight Independent Set can be solved in polynomial time for graphs of bounded induced matching treewidth. In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth. These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO$_2$ formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth. We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth.

Tree decompositions meet induced matchings: beyond Max Weight Independent Set

TL;DR

This work studies graphs of bounded induced matching treewidth, a parameter that refines structural width notions and enables efficient algorithms beyond Max Weight Independent Set. It presents a polynomial-time algorithm for Min Weight Feedback Vertex Set (via Max Weight Induced Forest) on such graphs and develops a robust DP framework that leverages a compact signature per bag, along with skeletons and impostors, to handle maximal induced forests. The paper also extends tractability to packing problems and to CMSO2-definable bounded-treewidth subgraphs, including a PTAS for certain induced subgraph packings and a general (r,ψ)-MWIS framework for graphs with bounded tree-independence, linking to a broad class of problems. Structural and complexity results further clarify the landscape: precise bounds, monotonicity under induced minors, and limitations under degree constraints highlight both the potential and the limits of induced matching width as a scalable tool for algorithmic graph problems.

Abstract

For a tree decomposition of a graph , by we denote the size of a largest induced matching in all of whose edges intersect one bag of . Induced matching treewidth of a graph is the minimum value of over all tree decompositions of . Yolov [SODA 2018] proved that Max Weight Independent Set can be solved in polynomial time for graphs of bounded induced matching treewidth. In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth. These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth. We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth.
Paper Structure (48 sections, 43 theorems, 24 equations, 2 figures)

This paper contains 48 sections, 43 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of our results
  3. Solving Max Weight Induced Forest.
  4. Independent packings of small subgraphs.
  5. Solving $(r,\psi)$-MWIS for bounded tree-independence number.
  6. Complexity of computing induced matching treewidth and bounds
  7. Preliminaries
  8. Tree decompositions.
  9. Tree-independence number and induced matching treewidth.
  10. Solving Max Weight Induced Forest
  11. Proof of Lemma 3.1
  12. What lives in a forest.
  13. Skeletons of maximal induced forests.
  14. Identifying the impostors.
  15. Construction of $\mathbb{F}\xspace_t$.
  16. ...and 33 more sections

Key Result

Theorem 1.1

For every fixed $k$, the MWIS problem on $n$-vertex graphs $G$ with $\mathsf{tree} \textnormal{-} \mu(G) \leqslant k$ can be solved in time $n^{\mathcal{O}(k)}$.

Figures (2)

  • Figure 1: Various subsets of $V(G)$ relevant in the proof of \ref{['lem:ff']}.
  • Figure 2: Relations between some graph width parameters.

Theorems & Definitions (72)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 2.1
  • Lemma 2.2: Simplified statement of \ref{['lem:ff']}
  • Lemma 2.3: A part of the statement of \ref{['lem:H-G']}
  • Corollary 2.3
  • Theorem 2.4: Simplified statement of \ref{['thm:cmsomain-intro']}
  • Proposition 3.1: Yolov DBLP:conf/soda/Yolov18
  • Theorem 3.2: Dallard et al. dallard2022computing
  • Theorem 3.3
  • ...and 62 more