Elsevier

Computational Geometry

Volume 89, August 2020, 101591
Computational Geometry

Approximation algorithms for geometric conflict free covering problems

https://doi.org/10.1016/j.comgeo.2019.101591Get rights and content

Abstract

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 CGO with O as vertex set. An edge (Oi,Oj) in CGO denotes conflict between Oi and Oj 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 CGO. 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, GC with vertex set C and there is an edge (Ci,Cj) in GC, if and only if there exist two vertices in G, vaCi and vbCj such that there is an edge (va,vb) 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 4γ approximation algorithm for conflict graphs having Clique Partition Chromatic Number γ. We show that unit interval graphs and unit disk graphs have constant Clique Partition Chromatic Number while for chordal graphs, it is bounded by logn. 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.

Introduction

Geometric Set Cover problem is one of the most extensively studied problems in computational geometry. In Geometric Set Cover, we are given a set of points P={p1,p2,,pn} and a set of closed geometric objects, O={O1,O2,,Om} in the plane. A geometric object O is said to cover a point p if and only if p lies inside O. The objective is to cover the set of points P with a minimum number of objects from the set O. The Geometric Set Cover problem has a wide range of applications including VLSI floor planning, wireless sensor networks, facility location, etc. For example in a wireless sensor network, a fundamental problem is to find suitable locations to place mobile towers. Let U be the given set of users modeled as a point in the plane and T be the set of possible locations in the plane where a tower can be placed. There is a range of radius r within which a mobile tower can communicate. Thus transmission zone of each tower can be modeled as a disk of fixed radius. Given P and T our objective is to find the minimum subset of locations TT such that each user in U can be served by at least one tower placed in T (see Fig. 1). This is a well-studied problem and a PTAS is known for the problem given by Mustafa and Ray [11].

Interestingly with the advancement of all such application areas, there are other constraints that have been introduced along with the basic Geometric Set Cover. For example, consider the problem of mobile tower placement. In view of the adverse effect of radio waves, it might not be preferred to place two mobile towers in close proximity. That is, the placement of a tower at one point should preclude the placement of towers in all nearby points. We call such kind of restrictions as conflicts. In a recent paper Banik et al. [6] provided a unified model to capture such constraints. The authors denote such problems as the Geometric Conflict Free Set Cover (GCFcov ). The input of the GCFcov consists of a set of points P and a set O of closed geometric objects in Rd and a graph CGO termed as conflict graph. The vertices of CGO are the geometric objects O. If there is an edge between two vertices representing geometric objects Oi and Oj in CGO then we denote that there is a conflict between the objects Oi,OjO and at most one among Oi among Oj can be present in any valid solution. An independent set in CGO is called a conflict-free set. Given P, O and CGO, the objective of GCFcov is to find a minimum cardinality conflict free subset OO such that the union of objects in O covers P. We formally define the geometric conflict-free set cover problem as follows, Arkin et al. [2] considered a special case of Geometric Conflict Free Set Cover which they refer as Rainbow Covering. The input to the rainbow covering problem is P, I and CGI. Here P is a set of points on the real line and I is a set of intervals. The conflict graph CGI is a matching i.e. a set of edges such that no pair of edges share a common endpoint. The authors show that Rainbow Covering is NP -complete.

Banik et al. considered some special cases of Geometric Conflict Free Set Cover from the perspective of parameterized complexity in [6]. They give the following results for the case when the conflict graph has bounded arboricity.

  • If the Geometric Coverage problem is fixed parameter tractable (FPT), then so is its conflict free version.

  • If the Geometric Coverage problem admits a factor α-approximation, then the conflict free version admits a factor α-approximation algorithm running in FPT time.

The next natural question is whether there exists an approximation algorithm for the maximization version of the problem, which can be defined as follows. Note that it is possible to consider the minimization version of Geometric Conflict Free Set Cover where the objective is to find a minimum cardinality conflict free subset of objects that cover all the points. It is NP -hard to find out whether any feasible solution for GCFcov exists or not, which immediately implies that the minimization version is Poly-APX -hard even when CGO is a matching.

Our goal here is to find efficient approximation algorithms for Max-GCFcov. But to illustrate the inherent difficulty of the Max-GCFcov we start with the following result. We show that Max-GCFcov is APX -hard when the covering objects are intervals on the real line and the conflict graph is bipartite. We show a reduction from MAX-3-SAT. In MAX-3-SAT, we are given a 3-SAT formula F with variable set X={x1,x2xn} and clause set C={C1,C2,Cm}. The objective is to find an assignment for X so as to maximize the number of clauses in C that evaluate to True. It is known that if there is an r-approximate algorithm for MAX-3-SAT, where r<8/7, then P=NP [10].

Lemma 1.1

There is no approximation algorithm with approximation factor less that 8/7 for Max-GCFcov unless P=NP, when the conflict graph is bipartite.

Proof

Given an instance of MAX-3-SAT, with 3-SAT formula F on variable set X={x1,x2xn} and clause set C={C1,C2,Cm}, we create an equivalent instance of Max-GCFcov as follows. For each clause Ci we create a point pi=(i,0). Assume no variable appears twice in a clause and each variable appears more than once in F. We number the appearance of a literal according to its order of appearance in the sequence C1,C2,Cm. For each literal xj (or xj) that appears in the clause Ci we create an interval Ijl=[i0.1,i+0.1] (or Ijl) where this is lth appearance of the literal (see Fig. 2). We create the conflict graph CG(U,Eu) as follows. We take U={Ijl,Ijk:1jn}. For all 1jn we create an edge between (Ijl,Ijk) for all values of l and k. Observe that the conflict graph is bipartite. Next, we prove that there is an assignment satisfying k clauses if and only if there is a conflict free covering of k points. Suppose there exists an assignment which satisfies k clauses. Observe that in each such clause there is a literal which is set to True. By choosing the intervals corresponding to those literals we can cover k points. On the other hand, suppose there exist k points that can be covered by conflict free choice of intervals. By choosing the literals corresponding to these conflict free intervals, it is possible to satisfy k clauses. Thus, the result holds. 

In this paper, we consider the following problem. Initially, our objective was to present an approximation algorithm when the conflict graph is a unit interval graph. After obtaining an approximation algorithm for this problem, we further considered generalizations and were able to extend our result for a special graph class which also includes unit interval graphs. We define this graph class in the next section. Note that most of the work on the CFI-covhas been done assuming the conflict graph is a matching [1], [2], [3], [5], [7]. First time in this paper we are considering approximability for a broader class of graphs. We hope that this will create a new direction of research for such problems.

Section snippets

A generalized (geometric) graph class

In this paper, we consider that CGI belongs to a special graph class. In order to define the graph class let us recall certain graph-theoretic definitions. A clique of a simple graph G=(V,E), is a subset WV such that between every pair of vertices in W, there is an edge in E. The chromatic number of a graph G is the smallest number of colors needed to color the vertices of G, so that no two adjacent vertices share the same color. It is denoted by χ(G).

A Clique Partition of the graph G is a

Approximation algorithm for CFI-cov(I,P,CGI,γ)

In this section we provide a framework to design approximation algorithms for CFI-cov(I,P,CGI,γ). We start our discussion with the following result. Recall that P is a set of points on the real line and I={I1,,Im} is a set of unit intervals. CGI is the conflict graph with Clique Partition Chromatic Number γ. For any set of intervals IjI, let P(Ij) be the set of points covered by Ij. Let all intervals in I lie between 0 and a on real line. Also, let every point pP be covered by at least one

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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This work was partially supported by the Science & Engineering Research Board (SERB) (ECR/2016/000769).

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