Minimizing Movement in Mobile Facility Location Problems

Zachary Friggstad, Mohammad R. Salavatipour
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
In the mobile facility location problem, which is a variant of the classical facility location and k-median problems, each facility and client is assigned to a start location in a metric graph and our goal is to find a destination node for each client and facility such that every client is sent to a node which is the destination of some facility. The quality of a solution can be measured either by the total distance clients and facilities travel or by the maximum distance traveled by any client
more » ... veled by any client or facility. As we show in this paper (by an approximation preserving reduction), the problem of minimizing the total movement of facilities and clients generalizes the classical k-median problem. The class of movement problems was introduced by Demaine et al. in SODA 2007 [8], where it was observed q simple 2-approximation for the minimum maximum movement mobile facility location while an approximation for the minimum total movement variant and hardness results for both were left as open problems. Our main result here is an 8-approximation algorithm for the minimum total movement mobile facility location problem. Our algorithm is obtained by rounding an LP relaxation in five phases. For the minimum maximum movement mobile facility location problem, we show that we cannot have a better than a 2-approximation for the problem, unless P = N P ; so the simple algorithm observed in [8] is essentially best possible. * Supported by NSERC. † Supported by NSERC and a University Start-up fund minimizing the total cost of movements of facilities and clients. Formally, we want to assign a destination v j for each facility j to minimize j∈F d jv j + i∈C d iv i D i where v i is the nearest facility destination to client i. This is called the minimum total movement mobile facility location problem, or TM-MFL. If we wish to minimize the maximum distance a client or facility travels then we obtain minimum maximum movement mobile facility location, or MM-MFL. Total movement can be thought of as the total amount of resources (e.g. gasoline) consumed by all facilities and clients in reaching a valid solution while maximum movement can be viewed as the time it takes to simultaneously move all units to a valid configuration (e.g. response time). Note that the demand (number of individual clients) per node is irrelevant in MM-MFL since we are only concerned with the distance. These problems fall into a natural class of problems, called movement problems, which were introduced by Demaine et al. [8] . In these types of problems, we are typically given an instance which contains a weighted graph G together with some pebbles on the vertices (and/or edges) and a desired property P ; some examples of this property P can be connectivity, independent set, or facility location. We are looking to obtain a movement of pebbles so that the final configuration of pebbles in the graph satisfies the desired property P while minimizing some objective cost function. Some of the natural objective cost functions considered are the total movements of pebbles or the maximum distance a pebble has to move. Many problems of this type arise naturally in other areas, such as operation research, robotics, and design of systems of wireless networks. For instance, suppose each pebble corresponds to a wireless sensor and our goal is to move these sensors around so that they form a connected network. This corresponds to the movement problem with property P being the subgraph induced by the final pebbles' locations being connected. (see e.g. [12, 3] and the references in [8] for more applications). The movement problems can be defined for different properties P . Demaine et al. [8] considered some specific movement problems, including the problem of connectivity (in which our desired property P is that the induced subgraph by the final pebbles' locations is connected), s, t-connectivity (in which we want the induced subgraph by the final pebbles' locations contains both s and t in the same connected component), and independent set (pebbles should form an independent set) and gave approximation algorithms and hardness results for each (for different objective functions). They also raised the question of movement mobile facility location problem. For the minimum maximum movement mobile facility location (MM-MFL), they [8] observed that there is a simple 2-approximation and asked whether this can be improved. They also left the problem of finding a good approximation algorithm for the minimum total movement mobile facility location (TM-MFL) as an open question. In this paper, we answer both these questions. For MM-MFL, we show that it is NP-hard to obtain better than a 2-approximation. The main contribution of this paper is to present a constant factor approximation algorithm for the TM-MFL defined earlier. As we will see, this problem in fact generalizes the classical kmedian problem. We show that there is an approximation preserving reduction from k-median to minimum total movement facility location. Related Works: In the classical (uncapacitated) facility location problem UFL, we are given a graph G(V, E) with metric costs d ij on the edges, a set of clients C ⊆ V , and a set of facilities F ⊆ V with each i ∈ F having an opening cost f i . The goal is to open some of the facilities and assign each client to an open facility such that the total cost of opening facilities plus the costs of clients traveling to open facilities is minimized. The first approximation algorithm for facility location had ratio O(log n) and is due to Hochbaum [11] . Shmoys, Tardos, Aardal [20] were the first to give a constant ratio approximation for this problem; their algorithm had ratio 3.16. Later, in a series of papers several constant approximation algorithm were obtained for this problem with better ratios (see [10, 7, 13, 15, 1, 21, 18, 4] and references in [4]). The best known algorithm has ratio 1.5 [4] . Guha and Khuller [10] showed that, unless N P ⊆ DTIME(n polylog(n) ), there is no better than a 1.463-approximation for UFL. Several variations of the facility location problem have been studied such as capacitated facility location, in which there is a capacity on the number of clients that can be served at each facility f i (e.g. see [17] and the references there). Another well-studied related problem is the classical k-median. In the k-median problem there is no opening
doi:10.1109/focs.2008.12 dblp:conf/focs/FriggstadS08 fatcat:bob35nhc4rf4jim3lrnhxit4qe