Max Weight Independent Set in sparse graphs with no long claws

TL;DR

This paper improves algorithms for the maximum weight independent set problem in certain sparse graphs by providing a faster method and extending polynomial-time solvability to broader graph classes.

Contribution

The authors present a simpler, faster algorithm for MWIS in graphs excluding subdivided claws and extend results to graphs excluding a fixed subdivided claw and biclique.

Findings

Faster algorithm with running time $n^{O(\Delta^2)}$ for MWIS

Polynomial-time algorithm for graphs excluding subdivided claws and bicliques

Simplified approach compared to previous methods

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, Rz\k{a}\.zewski, SODA 2022]. First, we show that with an arguably simpler approach we can obtain a faster algorithm with running time $n^{\mathcal{O}(\Delta^2)}$, where $n$ is the number of vertices of the instance and $\Delta$ 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.