Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks

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 ∈ S is a subset of X. The output is a coloring X of the sets in S that satisfies the following constraint: for every x ∈ X there exists a color i and a unique set S ∈ S, such that x ∈ S and χ(S) = i. The goal is to minimize the number of colors used by the coloring χ. 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 simple geometric regions in the plane. In particular, we study disks (both congruent and non-congruent), axis-parallel rectangles (with a constant ratio between the smallest and largest rectangle) regular hexagons (with a constant ratio between the smallest and largest hexagon), and general congruent centrally-symmetric convex regions in the plane. In all cases we have coloring algorithms that use O(log n) colors (where n is the number of regions). For rectangles and hexagons we obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also show that, even in the case of unit disks, Θ(log n) colors may be necessary.