Approximation algorithms for geometric conflict free covering problems

In the GEOMETRIC CONFLICT FREE COVERING, we are given a set of points P, a set of closed geometric objects O and a conflict graph CG(O) with O as vertex set. An edge (O-i,O-j) in CG(O) denotes conflict between O-i and O-j and at most one among these can be part of any feasible solution. A set of objects is conflict free if they form an independent set in CG(O). The objective is to find a conflict free set of objects that maximizes the number of points covered. We consider the UNIT INTERVAL COVERING where P is a set of points on the real line, and O is a set of closed unit-length intervals. The objective is to find a smallest subset of given intervals that covers P. We prove that for an arbitrary conflict graph the problem is Poly-APX-hard. We present an approximation algorithm for a special class of conflict graphs with a bounded graph parameter Clique Partition. A Clique Partition of the graph G is a set of cliques such that every vertex in the graph is part of exactly one clique. For any Clique Partition C, we define the Clique Partition Graph, G(C) with vertex set C and there is an edge (C-i,C-j) in G(C), if and only if there exist two vertices in G, v(a) is an element of C-i and v(b) is an element of C-j such that there is an edge (v(a),v(b)) in G. For a graph G, Clique Partition Chromatic Number is defined as the minimum chromatic number among all possible Clique Partitions of the Clique Partition Graph. In this paper, we consider those graph classes for which Clique Partition Chromatic Number can be computed in polynomial time. We present a 4 gamma approximation algorithm for conflict graphs having Clique Partition Chromatic Number gamma. We show that unit interval graphs and unit disk graphs have constant Clique Partition Chromatic Number while for chordal graphs, it is bounded by log n. Note that, Clique Partition Chromatic Number is less than or equal to the chromatic number. Thus our algorithm achieves a constant approximation factor for graphs with constant chromatic number (e.g. planar graphs ). This is the first result regarding the approximability in Geometric CONFLICT FREE COVERING. (C) 2019 Elsevier B.V. All rights reserved.