TY - JOUR

T1 - Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks

AU - Even, Guy

AU - Lotker, Zvi

AU - Ron, Dana

AU - Smorodinsky, Shakhar

PY - 2003/11

Y1 - 2003/11

N2 - 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 χ 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 noncongruent), 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). Tightness is demonstrated by showing that even in the case of unit disks, Θ(log n) colors may be necessary. For rectangles and hexagons we also obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also consider a dual problem of CF-coloring points with respect to sets. Given a set system (X, S), the goal in the dual problem is to color the elements in X with a minimum number of colors so that every set S ∈ S contains a point whose color appears only once in S. We show that O(log |X|) colors suffice for set systems in which X is a set of points in the plane and the sets are intersections of X with scaled translations of a convex region. This result is used in proving that O(log n) colors suffice in the primal version.

AB - 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 χ 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 noncongruent), 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). Tightness is demonstrated by showing that even in the case of unit disks, Θ(log n) colors may be necessary. For rectangles and hexagons we also obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also consider a dual problem of CF-coloring points with respect to sets. Given a set system (X, S), the goal in the dual problem is to color the elements in X with a minimum number of colors so that every set S ∈ S contains a point whose color appears only once in S. We show that O(log |X|) colors suffice for set systems in which X is a set of points in the plane and the sets are intersections of X with scaled translations of a convex region. This result is used in proving that O(log n) colors suffice in the primal version.

KW - Approximation algorithms

KW - Computational geometry

KW - Conflict-free coloring

KW - Frequency assignment

UR - http://www.scopus.com/inward/record.url?scp=1842459431&partnerID=8YFLogxK

U2 - 10.1137/s0097539702431840

DO - 10.1137/s0097539702431840

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AN - SCOPUS:1842459431

SN - 0097-5397

VL - 33

SP - 94

EP - 136

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 1

ER -