Max Weight Independent Set in graphs with no long claws: An analog of the Gy\'arf\'as' path argument

TL;DR

This paper develops improved algorithms for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw, using a novel analog of Gyárfás' path argument for such graphs.

Contribution

It introduces a polynomial-time method to find a small vertex set and a decomposition in $S_{t,t,t}$-free graphs, extending Gyárfás' path argument to this class.

Findings

Subexponential-time algorithm with $2^{O(\sqrt{n}\log n)}$ running time.

Quasipolynomial-time approximation scheme with $2^{O(\varepsilon^{-1} \log^{5} n)}$ running time.

New structural decomposition for $S_{t,t,t}$-free graphs.

Abstract

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw $S_{t,t,t}$ as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomass\'{e}, SODA 2020] and provide a subexponential-time algorithm with improved running time $2^{\mathcal{O}(\sqrt{n}\log n)}$ and a quasipolynomial-time approximation scheme with improved running time $2^{\mathcal{O}(\varepsilon^{-1} \log^{5} n)}$. The Gy\'arf\'as' path argument, a powerful tool that is the main building block for many algorithms in $P_t$-free graphs, ensures that given an $n$-vertex $P_t$-free graph, in polynomial time we can find a set $P$ of at most $t-1$ vertices, such that every connected component of $G-N[P]$ has at most $n/2$ vertices. Our main technical contribution is an analog of this result for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, in polynomial time we can find a set $P$ of $\mathcal{O}(t \log n)$ vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of $G-N[P]$ such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most $n/2$ vertices.