Coloring and Maximum Weight Independent Set of Rectangles

In 1960, Asplund and Grunbaum 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 $O(\frac{\log n}{\log\log n})$.