Approximation algorithms for geometric conflict free covering problems☆
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 and a set of closed geometric objects, 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 . 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 such that each user in U can be served by at least one tower placed in (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 of closed geometric objects in and a graph termed as conflict graph. The vertices of are the geometric objects . If there is an edge between two vertices representing geometric objects and in then we denote that there is a conflict between the objects and at most one among among can be present in any valid solution. An independent set in is called a conflict-free set. Given P, and , the objective of GCFcov is to find a minimum cardinality conflict free subset such that the union of objects in 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, and . Here P is a set of points on the real line and is a set of intervals. The conflict graph 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.
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If the Geometric Coverage problem is fixed parameter tractable (FPT), then so is its conflict free version.
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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 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 and clause set . The objective is to find an assignment for X so as to maximize the number of clauses in that evaluate to True. It is known that if there is an r-approximate algorithm for MAX-3-SAT, where , 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 and clause set , we create an equivalent instance of Max-GCFcov as follows. For each clause we create a point . 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 . For each literal (or ) that appears in the clause we create an interval (or ) where this is lth appearance of the literal (see Fig. 2). We create the conflict graph as follows. We take . For all we create an edge between 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. □
Section snippets
A generalized (geometric) graph class
In this paper, we consider that 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 , is a subset 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 .
A Clique Partition of the graph G is a
Approximation algorithm for CFI-cov()
In this section we provide a framework to design approximation algorithms for CFI-cov(). We start our discussion with the following result. Recall that P is a set of points on the real line and is a set of unit intervals. is the conflict graph with Clique Partition Chromatic Number γ. For any set of intervals , let be the set of points covered by . Let all intervals in lie between 0 and a on real line. Also, let every point 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).