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arXiv · 2203.04836

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

Konrad Majewski·Tomáš Masařík·Jana Novotná·Karolina Okrasa·Marcin Pilipczuk·Paweł Rzążewski·Marek Sokołowski
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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.

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Konrad Majewski, Tomáš Masařík, Jana Novotná, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, Marek Sokołowski. 2022-03-09. Max Weight Independent Set in graphs with no long claws: An analog of the Gy\'arf\'as' path argument. https://doi.org/10.1145/3636422%2010.4230%2Flipics.icalp.2022.93

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