Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams

Abstract

We study a general family of facility location problems defined on planar graphs and on the 2-dimensional plane. In these problems, a subset of k objects has to be selected, satisfying certain packing (disjointness) and covering constraints. Our main result is showing that, for each of these problems, the n O(k) time brute force algorithm of selecting k objects can be improved to n O( √ k) time. The algorithm is based on an idea that was introduced recently in the design of geometric QPTASs, but was not yet used for exact algorithms and for planar graphs. We focus on the Voronoi diagram of a hypothetical solution of k objects, guess a balanced separator cycle of this Voronoi diagram to obtain a set that separates the solution in a balanced way, and then recurse on the resulting subproblems.The following list is an exemplary selection of concrete consequences of our main result. We can solve each of the following problems in time n O( √ k) , where n is the total size of the input:• d-Scattered Set: find k vertices in an edge-weighted planar graph that pairwise are at distance at least d from each other (d is part of the input).• d-Dominating Set (or (k, d)-Center): find k vertices in an edge-weighted planar graph such that every vertex of the graph is at distance at most d from at least one selected vertex (d is part of the input).• Given a set D of connected vertex sets in a planar graph G, find k disjoint vertex sets in D.• Given a set D of disks in the plane (of possibly different radii), find k disjoint disks in D.• Given a set D of simple polygons in the plane, find k disjoint polygons in D.• Given a set D of disks in the plane (of possibly different radii) and a set P of points, find k disks in D that together cover the maximum number of points in P.• Given a set D of axis-parallel squares in the plane (of possibly different sizes) and a set P of points, find k squares in D that together cover the maximum number of points in P.It is known from previous work that, assuming the Exponential Time Hypothesis (ETH), there is no f (k)n o( √ k) time algorithm for any computable function f for any of these problems.Furthermore, we give evidence that packing problems have n O( √ k) time algorithms for a much more general class of objects than covering problems have. For example, we show that, assuming ETH, the problem where a set D of axis-parallel rectangles and a set P of points are given, and the task is to select k rectangles that together cover the entire point set, does not admit an f (k)n o(k) time algorithm for any computable function f .

On the Minimum Consistent Subset Problem

“…We consider the decision version of this problem in which we are given a set P of n colored points in the plane and an integer k, and we want to decide whether or not P has a consistent subset of size k. Moreover, if the answer is positive, then we want to find such a subset. This problem can be solved in time n O(k) by checking all possible subsets of size k. We show how to solve this problem in time n O( √ k) ; we use a recursive separator-based technique that was introduced in 1993 by Hwang et al [7] for the Euclidean k-center problem, and then extended by Marx and Pilipczuk [10] for planar facility location problems. Although this technique is known before, its application in our setting is not straightforward and requires technical details which we give in this section.…”

“…We introduce a new vertex at infinity and connect these three rays to that vertex. To this end we obtain a 2-connected 3-regular planar graph, namely G. Marx and Pilipczuk [10] showed that such a graph has a polygonal separator δ of size O( √ k) (going through O( √ k) faces and vertices) that is face balanced, in the sense that there are at most 2k/3 faces of G strictly inside δ and at most 2k/3 faces of G strictly outside δ. The vertices of δ alternate between points of S and the vertices of G as depicted in Figure 1(a).…”

Subexponential algorithms for variants of the homomorphism problem in string graphs

“…For example, in unit disk graphs, i.e., intersection graphs of unit-radius disks in the plane, I S , H C , V C can be solved in time 2Õ ( √ n) [1,39,16], and k C can be solved in time 2Õ ( √ nk) for every k [29,3]. All these bounds are essentially tight under the ETH, up to polylogarithmic factors in the exponent.…”

“…The algorithm is a simple win-win strategy: either we have a vertex of large degree and we branch on choosing it to the solution or not, or all degrees are small and thus there exists a small balanced separator, which allows us for one step of divide & conquer. Recently, Marx and Pilipczuk [39] used a di erent approach to obtain a 2 O( √ n) p O (1) algorithm for I S in string graphs, where p is the number of geometric vertices in the representation.…”

Near-Linear Time Approximations Schemes for Clustering in Doubling Metrics

“…1{ε q [37]. Therefore, to get an efficient approximation scheme one need to approximate both the number of centers and the cost.…”