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New Approximation Algorithms for the Unsplittable Capacitated Facility Location Problem∗

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New Approximation Algorithms for the Unsplittable Capacitated Facility Location Problem∗ New Approximation Algorithms for the Unsplittable Capacitated Facility Location Problem∗ Babak Behsazy Mohammad R. Salavatipourz Zoya Svitkinax May 24, 2015 Abstract In this paper, we consider the Unsplittable (hard) Capacitated Facility Location Problem (UCFLP) with uniform capacities and present new approximation algorithms for it. This prob- lem is a generalization of the classical facility location problem where each facility can serve at most u units of demand and each client must be served by exactly one facility. This problem is motivated by its applications in many practical problems including supply chain problems of indivisible goods [36] and the assignment problem in the content distribution networks [9]. While there are several approximation algorithms for the soft capacitated version of this problem (in which one can open multiple copies of each facility) or the splittable version (in which the demand of each client can be divided to be served by multiple open facilities), there are very few results for the UCFLP. It is known that it is NP-hard to approximate this problem within any factor without violating the capacities. So we consider bicriteria (α; β)-approximations where the algorithm returns a solution whose cost is within factor α of the optimum and violates the capacity constraints within factor β. Shmoys, Tardos, and Aardal [35] were the first to consider this problem and gave a (9; 4)-approximation. Later results imply (O(1); 2)-approximations, however, no constant factor approximation is known with capacity violation of less than 2. We present a framework for designing bicriteria approximation algorithms for this problem and show two new approximation algorithms with factors (9; 3=2) and (29:315; 4=3). These are the first al- gorithms with constant approximation in which the violation of capacities is below 2. The heart of our algorithm is a reduction from the UCFLP to a restricted version of the problem. One feature of this reduction is that any (O(1); 1+)-approximation for the restricted version implies an (O(1); 1 + )-approximation for the UCFLP and we believe our techniques might be useful towards finding such approximations or perhaps (f(); 1 + )-approximation for the UCFLP for some function f. In addition, we present a quasi-polynomial time (1 + , 1 + )-approximation for the (uniform) UCFLP in Euclidean metrics, for any constant > 0. ∗A preliminary version of this paper has appeared in the Proceedings of 13th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), Pages 237-248, 2012. yDepartment of Computing Science, University of Alberta, Edmonton, Alberta T6G 2E8, Canada. e-mail: [email protected]. Supported in part by Alberta Innovates Graduate Student Scholarship. zDepartment of Computing Science, University of Alberta, Edmonton, Alberta T6G 2E8, Canada. e-mail: [email protected]. Supported by NSERC and an Alberta Ingenuity New Faculty Award. xGoogle Inc., 1600 Amphitheatre Parkway, Mountain View, CA 94043, USA, e-mail: [email protected]. Sup- ported in part by Alberta Ingenuity. 1 1 Introduction We consider the Unsplittable Capacitated Facility Location Problem (UCFLP) with uniform ca- pacities. In this problem, we are given a set of clients C and a set of facilities F where each client j has demand dj and each facility i has opening cost fi and capacity u. There is a metric cost cij which denotes the cost of serving one unit of demand of client j at facility i. The goal is to open a subset of facilities I F and assign each client j to exactly one open facility φ(j) ⊆ to serve its entire demand dj so that the total amount of demand assigned to each open facility is no more than u, while minimizing the total cost of opening facilities and connecting (serving) P P clients, i.e., minimizing i I fi + j C djcφ(j)j. This problem generalizes the bin packing, the minimum makespan, and some2 facility2 location problems. If the demands of clients can be served by multiple open facilities, then we have the splittable capacitated version of the problem. If each facility can be opened multiple times then we have the so-called soft capacitated version. Each of these relaxations (i.e., allowing splitting the demands of clients and/or having multiple copies of each facility) makes the problem significantly easier as discussed below. When each facility i has a given capacity ui, the problem is called the non-uniform version. Facility location problems have been studied extensively in operations research and manage- ment sciences and even a few books are devoted to these problems (e.g., see [17, 29]). They are also well studied in theoretical computer science and various approximation algorithms are designed. While these problems arise in a wide range of practical applications, the most common context of their employment has been supply chain. In a supply chain that consists of suppliers, distribution centres, or warehouses and customers, these problems emerge in making location decisions [36]. When we deal with indivisible goods, the unsplittable demand assumption is a necessity. In par- ticular, the UCFLP has been studied in operations research literature from the eighties, where it is called the capacitated facility location problem with single sourcing or the capacitated concentrator location problem [17]. The latter name comes from the problem of assigning a set of terminals or workstations to some concentrator devices in telecommunications networks. Here, each terminal has a demand that must be served by exactly one concentrator and each concentrator has a capacity that shows the amount of traffic that it can manage. As Bateni and Hajiaghayi [9] pointed out, solving the UCFLP without relaxation of capacities is NP-hard even in very special cases. This is done via a simple reduction from the special case of minimum makespan when the size of each client j is in a set p ; . In fact, it can be shown that f j 1g unless P= NP, any approximation algorithm with a bounded approximation ratio for the UCFLP F violates the capacities of at least j j facilities in some instances [10]. Thus, research has focused b 2 c on the design of bicriteria approximation algorithms. An (α; β)-approximation algorithm for the UCFLP returns a solution whose cost is within factor α of the optimum and violates the capacity constraints within factor β. It should be noted that if we violate capacity of a facility within factor β, we must pay β times its opening cost. In the context of approximation algorithms, Shmoys, Tardos, and Aardal [35] were the first to consider this problem and presented a (9; 4)-approximation algorithm. They used a filtering and rounding technique to get an approximation algorithm for the splittable version and used a rounding for the generalized assignment problem (GAP) [34] to obtain their algorithm for the unsplittable version. This technique of reducing the unsplittable version using the rounding for the GAP to the splittable version was a cornerstone of the subsequent approximation algorithms. In addition, in the same place, by a randomized version of the filtering technique, they got a new algorithm with the ratio (7:62; 4:29). Korupolu, Plaxton, and Rajaraman [25] gave the first constant factor approximation algorithm for the splittable hard capacitated version, and applied the GAP rounding technique of [35] to get a (O(1); 2)-approximation algorithm for the UCFLP. Applying the current best approximation algorithms for the splittable capacitated 2 version with non-uniform capacities [37] and uniform capacities [1], it is straightforward to get factor (9; 2) and (5; 2) approximation algorithms for the UCFLP with non-uniform and uniform capacities, respectively [10]. Bateni and Hajiaghayi [9] designed the first approximation algorithms for the UCFLP having violation ratio less than 2. They modelled an assignment problem in content distribution networks by the UCFLP. This assignment problem has been first considered by Alzoubi et al. [2] and is basically the assignment of downloadable objects, such as media files or softwares, to some servers. We cannot split a downloadable object and we need to store it in a single server. As Alzoubi et al. mention, the server capacities is very crucial in practice and a high overload amount on a server can disrupt a large numbers of connections. Motivated by this strict requirement on capacities, the authors of [9] designed a (1 + , 1 + )-approximation algorithm for tree metrics (for any constant > 0) using a dynamic programming approach. They also presented a quasi-polynomial time (1 + , 1 + )-approximation algorithm (again for trees) for the non-uniform capacity case. Using Fakcharoenphol et al.'s [18] improvement of Bartal's machinery [8], this implies a polynomial time (O(log n); 1 + )-approximation algorithm for almost uniform capacities and a quasi-polynomial time (O(log n); 1 + )-approximation algorithm for non-uniform version for an arbitrary constant > 0. 1.1 Related Work Perhaps the most well-studied facility location problem is the uncapacitated facility location problem (UFLP). In this problem, we do not have the capacity constraints and we only need to decide which facilities to open, as each client will be assigned to its closest open facility. The first constant approximation for the UFLP was a 3:16-approximation algorithm by Shmoys, Tardos, and Aardal [35]. This algorithm was based on a filtering method due to Lin and Vitter [28] and rounding a linear programming (LP) formulation of the problem. There is a long series of works that improve this constant approximation ratio for the UFLP. Guha and Khuller [20] improved the ratio to 2:41 by combining a simple greedy heuristic with the algorithm of [35]. This greedy heuristic adds unopened facilities one by one greedily based on some measure of effectiveness for each facility.
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