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 $O(omegalogomega)$-colorable and presenting a polynomial-time algorithm that finds such a coloring. This improvement leads to a polynomial-time $O(loglog n)$-approximation algorithm for the maximum weight independent set problem in axis-parallel rectangles, which improves on the previous approximation ratio of $O(frac{log n}{loglog n})$.