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Max Weight Independent Set in sparse graphs with no long claws

Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Marcin Pilipczuk, Paweł Rzążewski

TL;DR

This work advances the tractability frontier for MWIS by combining a streamlined Border MWIS framework with extended strip decompositions. It delivers a faster $n^{\mathcal{O}(\Delta^2)}$-time algorithm for bounded-degree graphs excluding a fixed subdivided-claw as an induced subgraph and then extends to polynomial-time solvability for graphs excluding a fixed $H \in \mathcal{S}$ along with a fixed biclique, using tree decompositions to handle sparsity and structure. The results hinge on sophisticated decomposition techniques that reduce MWIS to manageable subproblems on particles and border instances, enabling recursive solving with carefully bounded terminal sets. The paper also proposes a conjecture to further relax conditions (e.g., combining $S_{t,t,t}$-freeness with $K_r$-freeness) while preserving polynomial-time solvability, inviting future work on the boundary between tractable and intractable MWIS instances.

Abstract

We revisit the recent polynomial-time algorithm for the MAX WEIGHT INDEPENDENT SET (MWIS) problem in bounded-degree graphs that do not contain a fixed graph whose every component is a subdivided claw as an induced subgraph [Abrishami, Dibek, Chudnovsky, Rzążewski, SODA 2022]. First, we show that with an arguably simpler approach we can obtain a faster algorithm with running time $n^{\mathcal{O}(Δ^2)}$, where $n$ is the number of vertices of the instance and $Δ$ is the maximum degree. Then we combine our technique with known results concerning tree decompositions and provide a polynomial-time algorithm for MWIS in graphs excluding a fixed graph whose every component is a subdivided claw as an induced subgraph, and a fixed biclique as a subgraph.

Max Weight Independent Set in sparse graphs with no long claws

TL;DR

This work advances the tractability frontier for MWIS by combining a streamlined Border MWIS framework with extended strip decompositions. It delivers a faster -time algorithm for bounded-degree graphs excluding a fixed subdivided-claw as an induced subgraph and then extends to polynomial-time solvability for graphs excluding a fixed along with a fixed biclique, using tree decompositions to handle sparsity and structure. The results hinge on sophisticated decomposition techniques that reduce MWIS to manageable subproblems on particles and border instances, enabling recursive solving with carefully bounded terminal sets. The paper also proposes a conjecture to further relax conditions (e.g., combining -freeness with -freeness) while preserving polynomial-time solvability, inviting future work on the boundary between tractable and intractable MWIS instances.

Abstract

We revisit the recent polynomial-time algorithm for the MAX WEIGHT INDEPENDENT SET (MWIS) problem in bounded-degree graphs that do not contain a fixed graph whose every component is a subdivided claw as an induced subgraph [Abrishami, Dibek, Chudnovsky, Rzążewski, SODA 2022]. First, we show that with an arguably simpler approach we can obtain a faster algorithm with running time , where is the number of vertices of the instance and is the maximum degree. Then we combine our technique with known results concerning tree decompositions and provide a polynomial-time algorithm for MWIS in graphs excluding a fixed graph whose every component is a subdivided claw as an induced subgraph, and a fixed biclique as a subgraph.
Paper Structure (12 sections, 12 theorems, 12 equations)

This paper contains 12 sections, 12 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tree decompositions.
  4. Extended strip decompositions.
  5. Bounded-degree graphs: Proof of Theorem \ref{['thm:main-maxdeg']}
  6. Graphs with no large bicliques: Proof of Theorem \ref{['thm:main-biclique']}
  7. Conclusion
  8. Acknowledgements
  9. Appendix: Proof of Lemma \ref{['lem:lozinrazgon']}
  10. Case 1 (large $Q$): $|Q| \geqslant q$.
  11. Case 2 (small $Q$): $Q < q$.
  12. Appendix: Proof of Lemma \ref{['lem:esd-border-mwis']}

Key Result

Theorem 1

There exists an algorithm that, given a vertex-weighted graph $(G,\mathbf{w})$ on $n$ vertices with maximum degree $\Delta$ and integers $d,t$ in time $2^{\mathcal{O}(dt \Delta ^2)}n^{\mathcal{O}(t\Delta^2)}$ either finds an induced $dS_{t,t,t}$ or the maximum possible weight of an independent set i

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3: ICALP-qptas in a semi-weighted setting
  • Theorem 4: Weißauer DBLP:journals/siamdm/Weissauer19
  • Lemma 5
  • Corollary 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Conjecture 10
  • ...and 9 more