Coloring and Maximum Weight Independent Set of Rectangles
Parinya Chalermsook, Bartosz Walczak

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 -coloring, where 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 -colorable and presenting a polynomial-time algorithm that finds such a coloring. This improvement leads to a polynomial-time -approximation algorithm for the maximum weight independent set problem in axis-parallel rectangles, which improves on the previous approximation ratio of .
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Optimization and Packing Problems
