Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

Abstract

Given a set of points P ⊆ R 2 , a conflict-free coloring of P is an assignment of colors to points of P , such that each non-empty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in R 2 can be conflict-free colored with˜Owith˜ with˜O(n .382+) colors in expected polynomial time, for any arbitrarily small > 0. This improves upon the previously known bound of O(p n log log n/ log n).

Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent set

“…In the opposite direction, Chen et al [10] proved that dg-indep(n, B 2 ) = O(n log 2 log n/ log n). In Section 2, we refine Ajwani et al's proof approach [1] to obtain the following result: Theorem 1.5 dg-indep(n, B 2 ) = Ω(n α ) for some α > 0.632.…”

“…One proof [19] uses the Erdős-Szekeres theorem to find a chain of Ω( √ n) points which is monotone in x and y simultaneously; then an independent set can be obtained by taking, say, all the even-indexed points in the chain. Another proof is divided into two cases: every graph with maximum degree at most √ n has independence number Ω( √ n); on the other hand, if some vertex in the Delaunay graph has degree exceeding √ n, then its neighborhood would contain a monotone chain 1 All rectangles and boxes are axis-parallel in this paper. 2 The O or Ω notation hides log O(1) n and log O(1) m factors in this paper.…”

“…The square-root barrier was finally broken in a more substantial way by Ajwani et al [1], by building on ideas from a previous paper of Elbassioni and Mustafa [13]. Ajwani et al showed that cf-color(n, B 2 ) = O(n 2−φ+ε ) = O(n 0.382 ), where φ is the golden ratio (1 + √ 5)/2 and ε is an arbitrarily small constant.…”

An Optimal Algorithm for Conflict-Free Coloring for Tree of Rings

“…In [7] the conflict-free coloring of n points with respect to axis-parallel rectangles were studied. Various other conflict-free coloring problems have been considered in very recent papers [8,12,13,14,15,16,17,18].…”

On variants of conflict-free-coloring for hypergraphs

“…Such coloring have attracted many researchers from the computer science and mathematics community. As to CF-coloring of hypergraphs that arise in geometry, refer to Smorodinsky [20], Even et al [12], Har-Peled and Smorodinsky [15], Smorodinsky [21], Pach and Tardos [19], Ajwani et al [2], Chen et al [10], Alon and Smorodinsky [3], Lev-Tov and Peleg [17] and etc. As to CF-coloring of arbitrary hypergraphs, refer to Pach and Tardos [18].…”