Conflict-free coloring of unit disks

Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call Minimum Conflict-Free Coloring (Min-CF-Coloring). In its general form, the input of the Min-CF-coloring problem is a set system (X;S), where each S 2 S is a subset of X . The output is a coloring of the sets in S that satisfies the following constraint: for every x 2 X there exists a color i and a unique set S 2 S, such that x 2 S and (S) = i. The goal is to minimize the number of colors used by the coloring . Min-CF-coloring of general set systems is not easier than the classic graph coloring problem. However, in view of our motivation, we consider set systems induced by unit disks in the plane. Our main result is a poly-time algorithm that finds a CF-coloring of any set of unit disks, using a number of colors that is only logarithmic in the maximum number of centers of disks that reside in a square of diameter 1. Based on this result we obtain two bi-criteria algorithms. Both algorithms find colorings that use a number of colors that is essentially as small as desired. The first comes at a cost of not serving an exponentially small fraction of the area covered by the disks. The second serves all the area but increases the radius of the disks by an exponentially small amount. In “exponentially small” we mean as a function of the number of colors used. Dept. of Electrical Engineering Systems, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail:[email protected]. yDept. of Electrical Engineering Systems, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail:[email protected]. zDept. of Electrical Engineering Systems, Tel-Aviv University, Tel-Aviv 69978, Israel. E-mail:[email protected].