Coloring and Maximum Weight Independent Set of Rectangles

TL;DR

This paper improves the coloring bound for intersection graphs of axis-parallel rectangles from quadratic to near-linear in the clique size, and provides a better approximation algorithm for the maximum weight independent set problem.

Contribution

It proves a new $O( ext{omega} \, ext{log} \, ext{omega})$ coloring bound and offers a polynomial-time algorithm for it, enhancing previous results.

Findings

Improved coloring bound from $O( ext{omega}^2)$ to $O( ext{omega} \, ext{log} \, ext{omega})$.

Developed a polynomial-time algorithm for the new coloring bound.

Achieved an $O( ext{log} \, ext{log} \, n)$ approximation for maximum weight independent set.

Abstract

In 1960, Asplund and Gr\"unbaum proved that every intersection graph of axis-parallel rectangles in the plane admits an $O(\omega^2)$-coloring, where $\omega$ is the maximum size of a clique. We present the first asymptotic improvement over this six-decade-old bound, proving that every such graph is $O(\omega\log\omega)$-colorable and presenting a polynomial-time algorithm that finds such a coloring. This improvement leads to a polynomial-time $O(\log\log n)$-approximation algorithm for the maximum weight independent set problem in axis-parallel rectangles, which improves on the previous approximation ratio of $O(\frac{\log n}{\log\log n})$.