Elsevier

Discrete Applied Mathematics

Volume 289, 31 January 2021, Pages 219-229
Discrete Applied Mathematics

Approximability of the dispersed p-neighbor k-supplier problem

https://doi.org/10.1016/j.dam.2020.10.007Get rights and content

Abstract

We consider the dispersed p-neighbor k-supplier problem. In the classical k-supplier problem, we have to select k suppliers in a metric space such that the maximum distance between a customer and its closest supplier is minimized. Here, we generalize this problem to the case where each customer possibly needs service from more than one supplier. Moreover, the selected suppliers should not be too close to each other, i.e., they need to be dispersed. For the classical k-supplier problem, and its special case the k-center problem, there is a 3- and a 2-approximation respectively. We show that these guarantees can also be given in the case when customers need service from multiple suppliers, without imposing dispersion constraints. If we generalize the problem to the dispersed case, without imposing neighboring constraints, we get inapproximability results depending on the measure of dispersion. We also show (almost) matching upper bounds. Finally, we show that adding both the neighbor requirement and the dispersion requirement leads to an inapproximable problem.

Introduction

In this paper, we discuss the approximability of a generalization of the k-supplier problem [10]. In this problem, we are given a metric space containing a set of customers and suppliers. We can choose k suppliers, and each customer will be served by its closest supplier. The goal is to find a set of suppliers that minimizes the maximum distance between any customer and its closest supplier. A special case of the k-supplier problem in which there is no distinction between suppliers and customers, i.e., every point in the metric space is a customer, and at each point a supplier can be selected, is the well-known k-center problem [9]. Here, the chosen suppliers are usually referred to as centers.

We generalize the k-supplier problem in two ways. We consider the generalization in which customer v needs service from multiple suppliers, say pv. These neighbor requirements may differ over the customers. The objective now becomes the maximum distance between any customer v and its pv-closest supplier. We also generalize the problem by adding a dispersion requirement, i.e., the distance between any two suppliers should be large enough. These two generalizations result in the dispersed p-neighbor k-supplier and k-center problem, which will be formally defined in the next section. In this paper, we study to what extend the approximability results for the k-supplier (and k-center) problem can be generalized when neighbor or dispersion requirements are added.

The problem originates from the following military application. Suppose we are going to deploy several units of special forces. There are some predefined locations at which a unit can be placed. There is also a set of potential targets, from which we will engage exactly one. To successfully engage a target, a predefined number of units is needed at the same time. The required number of units may differ over the targets. We want to place the units such that the time (assuming the units travel at unit speed) to successfully engage any target is minimized. However, if units are placed on locations close to each other, these units may become a target for the enemy. Therefore, the units should also be dispersed, i.e., the distance between any pair of units should be large enough. We can relate this application with the dispersed p-neighbor k-supplier problem, by interpreting the locations at which the units can be placed as suppliers, and the targets as customers.

An illustration of an instance and feasible solutions is given in Fig. 1, where dots represent suppliers, and squares represent customers. The number in a square represents the number of suppliers that is needed to serve a customer. In Fig. 1, we illustrate feasible solutions (in gray) that satisfy increasing dispersion requirements. Note that the time needed to serve a customer is determined by the supplier that is farthest (out of the chosen suppliers that will serve this customer) from the customer. Here, it is clear that increasing the dispersion requirement increases the objective value.

The novelty of the problems considered in this paper lies in two elements. Firstly, the dispersion requirement has not been considered. Secondly, we consider the case that the number of required suppliers differs per customer. In this paper, we investigate the consequences of these elements on the approximability of the k-supplier and the k-center problem. Here, an α-approximation algorithm will, for each instance, produce a feasible solution that has an objective value that is within a factor α of the optimal value for the instance. It also has to run in polynomial time. Before giving formal definitions (Section 2) and the main results, we give a brief overview of our results and relevant literature.

In Table 1, we show our results on approximation algorithms and inapproximability for problem variations considered in this paper. In Section 3, we show a 2-approximation for the p-neighbor k-center problem, and a 3-approximation for the p-neighbor k-supplier problem. This is done by generalizing the results in [2] for the p-neighbor k-center and k-supplier problem, in which the neighbor requirement is equal for all points and customers respectively. Here, we note that in the p-neighbor k-center problem the objective is the maximum distance, taken over all points without a center, from a point to a center. If the objective is the maximum distance, taken over all points, from a point to a center, we speak of the p-reliable k-center problem, for which we also give a 2-approximation (Theorem 2).

Then, in Section 4, we investigate the approximability of the dispersed k-center and k-supplier problem. We show lower bounds depending on Δ, the dispersion requirement, and also give approximation algorithms with (almost) matching upper bounds. Finally, in Section 5, we show that for the dispersed p-neighbor k-center problem, it is NP-complete to decide whether there exists a feasible solution. As a corollary, the problem is inapproximable. This also settles the approximability of the harder problem variations.

We also consider the dispersed k-median problem (Section 4.3). In this problem, the objective is the average distance of a point to its closest center. For the dispersed k-median problem, we show lower bounds on the approximability of 1.36, 2, and 78+316Δ (Theorem 9, Theorem 10). For simplicity, the results for the p-reliable k-center problem and the dispersed k-median problem are omitted in Table 1.

The k-center problem is a classical problem in combinatorial optimization. For clarity, we emphasize here that we consider the discrete version, and that we assume that the distances satisfy the triangle inequality. The problem is NP-hard, even on Euclidean instances [18]. For general metrics, there are several 2-approximations, for example the ones obtained by Feder and Greene [5], Gonzalez [6], and Hochbaum and Shmoys [9]. It was shown in [11] that there is no α-approximation algorithm for α<2, unless P = NP.

For the k-supplier problem, Hochbaum and Shmoys [10] gave a 3-approximation, and showed that this result is also tight. Recently, Nagarajan et al. [19] improved the approximation guarantee on Euclidean instances to 1+3. This was the first algorithm with a guarantee below 3. For the k-center problem in the Euclidean case the best approximation guarantee is still equal to 2.

The k-median problem is NP-hard [13], even for Euclidean instances [18]. The approximability of the k-median has not been settled yet. Currently, the best guarantee is given by Byrka et al. [1], who designed a (2.675+ϵ)-approximation, for any given ϵ>0. It was shown by Jain and Vazirani [12] that the k-median problem cannot be approximated within a factor 1+2e.

The p-neighbor k-center problem was considered first by Krumke [17]. He gave a 4-approximation by generalizing the ideas of Hochbaum and Shmoys [10]. Later, Chaudhuri et al. [2] improved this to a 2-approximation. They also gave a 2-approximation for the p-reliable k-center problem, and a 3-approximation for the p-neighbor k-supplier problem. Independently, Khuller et al. [16] gave a 2-approximation for the p-neighbor k-center problem, a 3-approximation for the p-reliable k-center problem, and a 3-approximation for the k-supplier problem.

A related problem we like to mention is the (discrete) k-dispersion problem. Here, one has to place k units in a metric space such that the dispersion, i.e., the distance between any two units, is maximized. The problem was shown to be NP-hard in both the general and the Euclidean case by Erkut [4]. Both Tamir [21] and Ravi et al. [20] gave an algorithm that produces a solution with a dispersion of at least half the optimal dispersion. Ravi et al. [20] also showed that this result is tight. Grigoriev et al. [7] consider the problem of maximizing the number of placed units given a certain dispersion requirement, where units can also be placed along an edge. They consider the computational complexity for small dispersion requirements.

Section snippets

Formal definitions

In this section, we will formally define our problem variations. We will also mention graph-theoretical notions that will be needed in this article.

The input of the dispersed p-neighbor k-supplier problem consists of a metric space containing a set of suppliers S and set of customers C. We denote by d(u,v) the distance between point u and v. We assume that the minimum non-zero distance is equal to 1, and that the distances satisfy the triangle inequality. Customer vC has a neighbor

The p-neighbor k-center and k-supplier problem

Here, we give a 2-approximation for the p-neighbor and the p-reliable k-center problem and a 3-approximation for the p-neighbor k-supplier problem by adjusting the algorithms presented in [2]. Recall that the objective of the p-neighbor k-center problem is the maximum distance, over all v, between any vertex v, that is not a center, and its pv-closest center. For the p-reliable k-center problem the objective is the maximum distance, over all v, between any vertex v and its pv-closest

The dispersed k-center and k-supplier problem

In this section, we look at the impact of the dispersion constraint on the approximability of the k-center, k-supplier, and k-median problem. For all problems, we show lower bounds depending on Δ. For the dispersed k-center and k-supplier problem, we also provide algorithms with (almost) matching upper bounds.

Dispersed p-neighbor k-center problem

In this section, we show that combining the neighbor and dispersion requirement leads to an inapproximable problem. That is, we show that the problem of deciding whether there exists a feasible solution for the dispersed p-neighbor (or p-reliable) k-center problem is NP-complete. Since any α-approximation algorithm should return a feasible solution, the problem is inapproximable. We give a reduction from the independent set problem. Here, we are given a graph G=(V,E) and an integer K. The

Conclusion

In this paper, we extended the results of [2] for the p-neighbor k-center, the p-reliable k-center and the p-neighbor k-supplier problem to the version in which the neighbor requirement differs over the vertices or customers. For the first two, this gives a 2-approximation, for the last one a 3-approximation. Then, we showed that adding a constraint on the dispersion of the centers or suppliers gives lower bounds on approximability that depend on Δ, the dispersion requirement. This holds for

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