Coloring and Maximum Independent Set of Rectangles

In this paper, we consider two geometric optimization problems: Rectangle Coloring problem (RCOL) and Maximum Independent Set of Rectangles (MISR). In RCOL, we are given a collection of n rectangles in the plane where overlapping rectangles need to be colored differently, and the goal is to find a coloring that minimizes the number of colors. Let q be the maximum clique size of the instance, i.e. the maximum number of rectangles containing the same point. We are interested in bounding the ratio σ(q) between the total number of colors used and the clique size. This problem was first raised by graph theory community in 1960 when the ratio of σ(q) ≤ O(q) was proved. Over decades, except for special cases, only the constant in front of q has been improved. In this paper, we present a new bound for σ(q) that significantly improves the known bounds for a broad class of instances. The bound σ(q) has a strong connection with the integrality gap of natural LP relaxation for MISR, in which the input is a collection of rectangles where each rectangle is additionally associated with non-negative weight, and our objective is to find a maximum-weight independent set of rectangles. MISR was studied extensively and has applications in various areas of computer science. Our new bounds for RCOL imply new approximation algorithms for a broad class of MISR, including (i) O(log log n) approximation algorithm for unweighted MISR, matching the result by Chalermsook and Chuzhoy, and (ii) O(log log n) approximation algorithm for the MISR instances arising in the Unsplittable Flow Problem on paths. Our technique builds on and generalizes past works.