A 2-Approximation Algorithm for the Soft-Capacitated Facility Location Problem

Transcription

1 A 2-Approximation Algorithm for the Soft-Capacitated Facility Location Problem Mohammad Mahdian Yinyu Ye Ý Jiawei Zhang Þ Abstract This paper is divided into two parts. In the first part of this paper, we present a 2-approximation algorithm for the soft-capacitated facility location problem. This achieves the integrality gap of the natural LP relaxation of the problem. The algorithm is based on an improved analysis of an algorithm for the linear facility location problem, and a bifactor approximate-reduction from this problem to the soft-capacitated facility location problem. We will define and use the concept of bifactor approximate reductions to improve the approximation factor of several other variants of the facility location problem. In the second part of the paper, we present an alternative analysis of the authors approximation algorithm for the uncapacitated facility location problem, using a single factor-revealing LP. This answers an open question of [18]. Furthermore, this analysis, combined with a recent result of Thorup [25] shows that our algorithm can be implemented in quasi-linear time, achieving the best known approximation factor in the best possible running time. 1 Introduction Variants of the facility location problem (FLP) have been studied extensively in the operations research and management science literatures and have received considerable attention in the area of approximation algorithms [21]. In the metric uncapacitated facility location problem (UFLP), which is the most basic facility location problem, we are given a set of facilities, a set of cities (a.k.a. clients), a cost for opening facility ¾, and a connection cost for connecting client to facility. The objective is to open a subset of the facilities in, and connect each city to an open facility so that the total cost is minimized. We assume that the connection costs are metric, meaning that they are symmetric and satisfy the triangle inequality. Since the first constant factor approximation algorithm due to Shmoys, Tardos and Aardal [22], a large number of approximation algorithm have been proposed for the UFLP [23, 12, 13, 17, 24, 14, 1, 3, 4, 5, 8, 14, 16], and the current best known approximation factor is 1.52 given by Mahdian, Ye and Zhang [18]. Guha and Khuller [8] proved that it is impossible to get an approximation guarantee of for the UFLP, unless NP ÌÁÅÒ Ç ÐÓ ÐÓ Òµ. Laboratory for Computer Science, MIT, Cambridge, MA 02139, USA. mahdian@theory.lcs.mit.edu. Ý Department of Management Science and Engineering, School of Engineering, Stanford University, Stanford, CA 94305, USA. yinyu-ye@stanford.edu. Research supported in part by NSF grant DMI Þ Department of Management Science and Engineering, School of Engineering, Stanford University, Stanford, CA 94305, USA. jiazhang@stanford.edu. Research supported in part by NSF grant DMI

2 The growing interests in the UFLP rely on not only its applications in a large number of settings [7], but also the fact that the UFLP is one of the most basic models among discrete location problems. The insights gained in dealing with the UFLP may also apply to more complicated location models, and in many cases the latter can be reduced directly to the UFLP. In this paper, we give a 2-approximation algorithm for the soft-capacitated facility location problem (SCFLP) by reducing it to the UFLP. The SCFLP is similar to the UFLP, except that there is a capacity Ù associated with each facility, which means that if we want this facility to serve Ü cities, we have to open it ÜÙ times at a cost of ÜÙ. This problem is also known as facility location problem with integer decision variables in the operations research literature (See [2]). Chudak and Shmoys [6] gave a -approximation algorithm for the SCFLP with uniform capacities (i.e., Ù Ù for all ¾ ) using LP rounding. For non-uniform capacities, Jain and Vazirani [14] showed how to reduce this problem to the UFLP, and by solving the UFLP through a primal-dual algorithm, they obtained a -approximation. A local search algorithm proposed by Arya et al [1] had an approximation ratio ¾. Following the approach of Jain and Vazirani [14], Jain, Mahdian, and Saberi [13] showed that the SCFLP can be solved within a factor of. This result was further improved by the authors [18] to a ¾-approximation for the SCFLP. This is the best previously known algorithm for this problem. We improve this factor to 2, achieving the integrality gap of the natural LP relaxation of the problem. The main idea of our algorithm is to consider algorithms and reductions that have separate (not necessarily equal) approximation factors for facility and connection costs. We will define the concept of bifactor approximate reduction in this paper, and show how it can be used to get an approximation factor of 2 for the SCFLP. We will also generalize our algorithm to a common generalization of the SCFLP and the concave-cost FLP. The idea of using bifactor approximation algorithms and reductions can be used to improve the approximation factor of several other problems in a straightforward manner. We show two examples of this in Appendix C. In the second part of this paper, we present an alternative analysis for the 1.52-approximation algorithm for the UFLP [18] using a single factor-revealing LP. This answers an open question of [18]. Furthermore, this analysis shows that the second phase of the 1.52 algorithm can be implemented in quasi-linear time. This, together with a recent result of Thorup [25] proves that our algorithm can be implemented in quasi-linear time, achieving the best known approximation factor in essentially the best possible running time. The rest of this paper is organized as follows: In Section 2 we present the necessary definitions and notations. In Section 3 we present a lemma on the approximability of the linear-cost facility location problem. In Section 4 we define the concept of bifactor approximate reductions between facility location problems, and present an algorithm for the SCFLP and a common generalization of the SCFLP and the concave-cost FLP using the lemma proved in Section 3 and a bifactor reduction from the SCFLP to the linear-cost FLP. Then, in Section 5, we present a new analysis on the 1.52 algorithm for the UFLP and show how it leads to an implementation in quasi-linear times. 2 Definitions and notations In this paper, we will define reductions between various facility location problems. Many such problems can be considered as special cases of the generalized facility location problem, as defined below. This problem was first defined and studied in [10]. Definition 1 In the metric generalized facility location problem, we are given a set of Ò cities, a set 2

3 of Ò facilities, a connection cost between city and facility for every ¾ ¾, and a facility cost function ¼ Ò Ê for every ¾. Connection costs are symmetric and obey the triangle inequality. The value of µ equals the cost of opening facility, if it is used to serve cities. A solution to the problem is a function assigning each city to a facility. The facility cost of the solution is defined as È ¾ µ µ, i.e., the total cost for opening facilities. The connection cost (a.k.a. service cost) of is È ¾ µ, i.e., the total cost of opening each city to its assigned facility. The objective is to find a solution that minimizes the sum. Now we can define uncapacitated and soft-capacitated facility location problems as special cases of the generalized FLP: Definition 2 The metric uncapacitated facility location problem (UFLP) is a special case of the generalized FLP in which all facility cost functions are of the following form: for each ¾, µ ¼ if ¼, and µ if ¼, where is a constant (which is called the facility cost of ). Definition 3 The metric soft-capacitated facility location problem (SCFLP) is a special case of the generalized FLP in which all facility cost functions are of the form µ Ù, where and Ù are constants for every ¾. Ù is called the capacity of facility. The 1.52-approximation algorithm of Mahdian, Ye, and Zhang [18] is built upon an earlier approximation algorithm of Jain, Mahdian, and Saberi [13]. We denote these two algorithms by the MYZ and the JMS algorithms, respectively. The analyses of both of these algorithms have the feature that allow the approximation factor for the facility cost to be different from the approximation factor for the connection cost, and give a way to compute the tradeoff between these two factors. The following definition captures this notion. Definition 4 An algorithm is called a µ-approximation algorithm for the generalized FLP, if for every instance Á of the generalized FLP, and for every solution ËÇÄ for Á with facility cost ËÇÄ and connection cost ËÇÄ, the cost of the solution found by the algorithm is at most ËÇÄ ËÇÄ. Recall the following theorem of Jain et al. [13] on the approximation factor of the JMS algorithm. Theorem A [13]. Let be fixed and ÙÔ Þ, where Þ is the solution of the following optimization program (which we call the factor-revealing LP). maximize È «È (1) subject to ««(2) Ö Ö (3) «Ö (4) ÑÜ Ö ¼µ ÑÜ «¼µ (5) «Ö ¼ (6) 3

4 Then the JMS algorithm is a µ-approximation algorithm for the UFLP. We will use the above theorem in this paper to give an alternative proof of the following theorem about the performance of the MYZ algorithm. Theorem B [18]. Let µ be a pair obtained from the above factor-revealing LP. Then for every, there is a ÐÒ µ µ-approximation algorithm for the UFLP. 3 The linear-cost facility location problem The linear-cost facility location problem is a special case of the generalized FLP in which the facility costs are of the form µ ¼ ¼ ¼ where and are nonnegative values for each ¾. and are called the setup and marginal (a.k.a. incremental) cost of facility, respectively. We denote an instance of the linear-cost FLP with marginal costs µ, setup costs µ, and connection costs µ by Ä ÄÈ µ. Clearly, the regular UFLP is a special case of the linear-cost FLP with ¼, i.e., Ä ÄÈ ¼ µ. Furthermore, it is straightforward to see that Ä ÄÈ µ is equivalent to an instance of the regular UFLP in which the marginal costs are added to the connection costs. More precisely, let for ¾ and ¾, and consider an instance of the UFLP with facility costs µ and connection costs µ. We denote this instance by Í ÄÈ µ. It is easy to see that Ä ÄÈ µ is equivalent to Í ÄÈ µ. Thus, the linear-cost FLP can be solved using any algorithm for the UFLP, and the overall approximation ratio will be the same. However, for applications in the next section, we need bifactor approximation factors of the algorithm (as defined in Definition 4). It is not necessarily true that applying a µ-approximation algorithm for the UFLP on the instance Í ÄÈ µ will give a µ-approximate solution for Ä ÄÈ µ. However, we will show that the JMS algorithm has this property. The following lemma, whose proof is given in Appendix A, generalizes Theorem A for the linear-cost FLP. Lemma 1 Let µ be a pair obtained from the factor-revealing LP in Theorem A. Then applying the JMS algorithm on the instance Í ÄÈ µ will give a µ-approximate solution for Ä ÄÈ µ. The above lemma and Theorem 9 in [13] give us the following corollary, which will be used in the next section. Corollary 2 There is a ¾µ-approximation algorithm for the linear-cost facility location problem. It is worth mentioning that the MYZ algorithm can also be generalized for the linear-cost FLP. The only trick is to scale up both and in the first phase by a factor of, and scale them both down in the second phase. The rest of the proof is almost the same as the proof of Lemma 1. 4

5 4 The soft-capacitated facility location problem In this section we will show how the soft-capacitated facility location problem can be reduced to the linearcost FLP. In Section 4.1 we define the concept of reduction between facility location problems. We will use this concept in Sections 4.2 and 4.3 to obtain approximation algorithms for the SCFLP and a generalization of the SCFLP and the concave-cost FLP. 4.1 Reduction between facility location problems Definition. A reduction from a facility location problem to another facility location problem is an efficient procedure Ê that maps every instance Á of to an instance Ê Áµ of. This procedure is called a µ-reduction if the following conditions hold. 1. For any instance Á of and any feasible solution for Á with facility cost and connection cost, there is a corresponding solution for the instance Ê Áµ with facility cost and connection cost. 2. For any feasible solution for the instance Ê Áµ, there is a corresponding feasible solution for Á whose total cost is at most as much as the total cost of the original solution for Ê Áµ. In other words, the FLP instance Ê Áµ is an over-estimate of the FLP instance Á. Theorem 3 If there is a µ-reduction from a facility location problem to another facility location problem, and a µ-approximation algorithm for, then there is a µ-approximation algorithm for. Proof. On an instance Á of the problem, we compute Ê Áµ, run the µ-approximation algorithm for on Ê Áµ, and output the corresponding solution for Á. In order to see why this is a µ- approximation algorithm for, let ËÇÄ denote an arbitrary solution for Á, Ä denote the solution that the above algorithm finds, and and È Ä È (È and Ä È, respectively) denote the facility and connection costs of ËÇÄ (Ä, respectively) when viewed as a solution for the problem È (È ). By the definition of µ-reductions and µ-approximation algorithms we have Ä Ä Ä Ä which completes the proof of the lemma. We will see examples of reductions in the rest of this paper. 4.2 The soft-capacitated facility location problem In this subsection, we give a ¾-approximation algorithm for the soft-capacitated FLP by reducing it to the linear-cost FLP. Theorem 4 There is a ¾-approximation algorithm for the soft-capacitated facility location problem. 5

6 Proof. We use the following reduction: Construct an instance of the linear-cost FLP, where we have the same sets of facilities and clients. The connection costs remain the same. However, the facility cost of the th facility is µ Ù if and ¼ if ¼. Note that, for every, Ù Ù ¾ Ù Therefore, it is easy to see that this reduction is a ¾ µ-reduction. By Lemma 2, there is a ¾µapproximation algorithm for the linear-cost FLP, which together with Theorem 3 completes the proof. Furthermore, we now illustrate that the following natural linear programming formulation of the SCFLP has an integrality gap of ¾. This means that we cannot obtain a better approximation ratio using this LP relaxation as the lower bound. minimize subject to Ý ¾ ¾ ¾ ¾ ¾ Ü Ý ¾ ¾ ¾ ¾ Ü (7) Ü Ù Ý Ü ¾ ¾ Ü ¾ ¼ (8) ¾ Ý is a nonnegative integer (9) In a natural linear program relaxation, we replace the constraints (8) and (9) by Ü ¼ and Ý ¼. Here we see that even if we only relax constraint (9), the integrality gap is ¾. Consider an instance of the SCFLP that consists of only one potential facility, and ¾ clients. Assume that the capacity of facility is, the facility cost is, and all connection costs are ¼. It is clear that the optimal integral solution has cost ¾. However, after relaxing constraint (9), the optimal fractional solution has cost. Therefore, the integrality gap between the integer program and its relaxation is ¾ which tends to ¾ as tends to infinity. 4.3 The concave soft-capacitated facility location problem In this subsection, we consider a common generalization of the soft-capacitated facility location problem and the concave-cost facility location problem. This problem, which we refer to as the concave softcapacitated FLP, is the same as the soft-capacitated FLP except that if Ö ¼ copies of facility are open, then the facility cost is Öµ where Öµ is a given concave function of Ö. In other words, the concave soft-capacitated FLP is a special case of the generalized FLP in which the facility cost functions are of the form ܵ ÜÙ µ for constants Ù and a concave function. It is also a special case of the so-called stair-case cost facility location problem [11]. On the other hand, it is a common generalization of the soft-capacitated FLP (when Öµ Ö) and the concave-cost FLP (when Ù for all ). The concave-cost FLP is a special case of the generalized FLP in which facility cost functions are required to be concave (See [10]). The main result of this subsection is the following. Theorem 5 The concave soft-capacitated FLP is ¾µ µ-reducible to the linear-cost FLP. µ 6

7 The proof of the above theorem is given in Appendix B. We first show that the concave soft-capacitated FLP is ¾µ µ reducible to the concave-cost FLP. Then we show that the latter is equivalent to the linearcost FLP. Therefore, by Theorem 5, a good approximation algorithm for linear-cost FLP would imply a µ good approximation for the concave soft-capacitated FLP. 5 The uncapacitated facility location problem In this section, we present a new analysis of the 1.52-approximation algorithm of Mahdian, Ye, and Zhang [18] for the UFLP. The analysis of the MYZ algorithm in [18] is based on combining a result of Jain et al. [13] (which is proved using factor-revealing LPs) with an analysis of a greedy augmentation procedure of Charikar et al. [3]. Here, we analyze the MYZ algorithm using a single factor-revealing LP. This gives us a new perspective on the MYZ algorithm. As a corollary, we use a recent result of Thorup [25] that the JMS algorithm can be implemented in quasi-linear time to improve the running time of the MYZ algorithm. We begin by sketching the MYZ algorithm. The algorithm consists of two phases. In the first phase, we scale up the facility costs in the instance by a factor of (which will be fixed later), and then run the JMS algorithm (see [13] for a description) on the modified instance. In addition to finding a solution for the scaled instance, the JMS algorithm outputs the share of each city of the total cost of the solution. Let «denote the share of city of the total cost (Therefore the total cost of the solution is È «). The main step in the analysis of the JMS algorithm is to prove that for any collection Ë of one facility Ë with opening cost ( in the original instance) and cities with connection costs to Ë and shares ««of the total cost, the values, s, «s and Ö s (whose definition is omitted here, since we don t need it) satisfy the inequalities (2)-(6) in Theorem A, except that the inequality (5) is replaced by ÑÜ Ö ¼µ ÑÜ «¼µ (10) In the second phase of the MYZ algorithm we reduce the scaling factor continuously, until it gets to 1. If at any point during this process a facility could be opened without increasing the total cost (i.e., if the opening cost of the facility equals the total amount that cities can save by switching their service provider to that facility), then we open the facility and connect each city to its closest open facility. The second phase of the MYZ algorithm is equivalent to a greedy augmentation procedure of [8, 3], and, in fact, a lemma from [3] is used in [18] in order to analyze the second phase. Here we analyze this phase differently. First, we modify the second phase as follows: Instead of decreasing the scaling factor continuously from to 1, we decrease it discretely in Ä steps where Ä is a constant. Let denote the value of the scaling factor in the th step. Therefore, ¾ Ä. We will fix the value of of s later. After decreasing the scaling factor from to, we consider facilities in an arbitrary order, and open those that can be opened without increasing the total cost. We denote this modified algorithm by MYZ Ä. Clearly, if Ä is sufficiently large (depending on the instance), the algorithm MYZ Ä computes the same solution as the MYZ algorithm. In order to analyze the above algorithm, we need to add extra variables and inequalities to the inequalities (2), (3), (4), (10) and (6). Let Ö denote the connection cost that city in Ë pays after we change the scaling factor to and process all facilities as described above (Thus, Ö is the connection cost of city 7

8 after the first phase). Therefore, by the description of the algorithm, we have Ä ÑÜ Ö ¼µ (11) since otherwise we could open Ë and decrease the total cost. Now, we compute the share of the city of the total cost of the solution that the MYZ Ä algorithm finds. In the first phase of the algorithm, the share of city of the total cost is «. Of this amount, Ö is spent on the connection cost, and «Ö is spent on the facility costs. However, since the facility costs are scaled up by a factor of in the first phase, therefore the share of city of facility costs in the original instance is equal to «Ö µ. After we reduce the scaling factor from to ( Ä ), the connection cost of city is reduced from Ö to Ö. Therefore, in this step, the share of city of the facility costs is Ö Ö with respect to the scaled instance, or Ö Ö µ with respect to the original instance. Thus, at the end of the algorithm, the total share of city of facility costs is Ä «Ö Ö Ö We also know that the final amount that city pays for the connection cost is Ö Ä. Therefore, the share of the facility of the total cost of the solution is: Ä «Ö Ö Ö Ö Ä «Ä Ö (12) This, together with a dual fitting argument similar to [13], imply the following. Theorem 6 Let µ be such that and is an upper bound on the solution of the following maximization program for every. maximize È «ÈÄ È Ö (13) subject to ¾µ µ µ ¼µ µ µ Then, the MYZ Ä algorithm is a µ-approximation algorithm for the UFLP. In the following theorem, we analyze the factor-revealing LP (13) and rederive the main result of [18]. In order to do this, we need to set the values of s. Here, for simplicity of computations, we set to Ä Ä ; however, it is easy to observe that any choice of s such that the limit of ÑÜ µ as Ä tends to infinity is zero, will also work. The proof is given in Appendix D. Theorem 7 Let µ be a pair given by the maximization program in Theorem A, and be an arbitrary number. Then for every, if Ä is a sufficiently large constant, the MYZ Ä algorithm is a ÐÒ µ µ-approximation algorithm for the UFLP. 8

9 The above analysis enables us to prove the following result. Corollary 8 For every ¼, there is a quasi-linear time ¾ µ-approximation algorithm for the UFLP, both in the distance oracle model (where the connection costs are given by a matrix) and in the sparse graph model (where the connection costs are distances in a given graph). Proof Sketch. We use the MYZ Ä algorithm for a large constant Ä. Thorup [25] shows that for every ¼, the JMS algorithm can be implemented in quasi-linear time (in both distance oracle and graph models) with an approximation factor of. It is straightforward to see that his argument actually implies the stronger conclusion that the quasi-linear algorithm is a µ-approximation, where µ are given by Theorem A. This shows that the first phase of the MYZ Ä algorithm can be implemented in quasi-linear time. The second phase consists of constantly many rounds. Therefore, we only need to show that each of these rounds can be implemented in quasi-linear time. This is easy to see in the distance oracle model. In the graph model, we can use the exact same argument as the one used by Thorup in the proof of Lemma 5.1 of [25]. Acknowledgements. We would like to thank Asaf Levin for pointing out that our analysis of the ¾- approximation algorithm for the soft-capacitated facility location problem is tight. References [1] V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, and V. Pandit. Local search heuristics for k-median and facility location problems. In Proceedings of 33rd ACM Symposium on Theory of Computing, [2] P. Bauer and R. Enders. A capacitated facility location problem with integer decision variables. In International Symposium on Mathematical Programming (ISMP), [3] M. Charikar and S. Guha. Improved combinatorial algorithms for facility location and k-median problems. In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, pages , October [4] F.A. Chudak. Improved approximation algorithms for uncapacited facility location. In R.E. Bixby, E.A. Boyd, and R.Z. Ríos-Mercado, editors, Integer Programming and Combinatorial Optimization, volume 1412 of Lecture Notes in Computer Science, pages Springer, Berlin, [5] F.A. Chudak and D. Shmoys. Improved approximation algorithms for the uncapacitated facility location problem. unpublished manuscript, [6] F.A. Chudak and D. Shmoys. Improved approximation algorithms for the capacitated facility location problem. In Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pages , [7] G. Cornuejols, G.L. Nemhauser, and L.A. Wolsey. The uncapacitated facility location problem. In P. Mirchandani and R. Francis, editors, Discrete Location Theory, pages John Wiley and Sons Inc.,

10 [8] S. Guha and S. Khuller. Greedy strikes back: Improved facility location algorithms. Journal of Algorithms, 31: , [9] S. Guha, A. Meyerson, and K. Munagala. Hierarchical placement and network design problems. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, [10] M. Hajiaghayi, M. Mahdian, and V.S. Mirrokni. The facility location problem with general cost functions. unpublished manuscript, [11] K. Holmberg. Solving the staircase cost facility location problem with decomposition and piecewise linearization. European Journal of Operational Research, 74:41 61, [12] K. Jain, M. Mahdian, E. Markakis, A. Saberi, and V.V. Vazirani. Approximation algorithms for facility location via dual fitting with factor-revealing LP. to appear in Journal of the ACM, [13] K. Jain, M. Mahdian, and A. Saberi. A new greedy approach for facility location problems. In Proceedings of the 34st Annual ACM Symposium on Theory of Computing, [14] K. Jain and V.V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM, 48: , [15] D. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, [16] M.R. Korupolu, C.G. Plaxton, and R. Rajaraman. Analysis of a local search heuristic for facility location problems. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1 10, January [17] M. Mahdian, E. Markakis, A. Saberi, and V.V. Vazirani. A greedy facility location algorithm analyzed using dual fitting. In Proceedings of 5th International Workshop on Randomization and Approximation Techniques in Computer Science, volume 2129 of Lecture Notes in Computer Science, pages Springer-Verlag, [18] M. Mahdian, Y. Ye, and J. Zhang. Improved approximation algorithms for metric facility location problems. In Proceedings of 5th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2002), [19] R. Ravi and A. Sinha. Integrated logistics: Approximation algorithms combining facility location and network design. In Proceedings of the 9th Conference on Integer Programming and Combinatorial Optimization, [20] G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the 10th ACM-SIAM symposium on Discrete Algorithms, pages , [21] D.B. Shmoys. Approximation algorithms for facility location problems. In K. Jansen and S. Khuller, editors, Approximation Algorithms for Combinatorial Optimization, volume 1913 of Lecture Notes in Computer Science, pages Springer, Berlin, [22] D.B. Shmoys, E. Tardos, and K.I. Aardal. Approximation algorithms for facility location problems. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages ,

11 [23] M. Sviridenko. An approximation algorithm for the metric uncapacitated facility location problem. In Proceedings of the 9th Conference on Integer Programming and Combinatorial Optimization, [24] M. Thorup. Quick -median, -center, and facility location for sparse graphs. In Automata, Languages and Programming, 28th International Colloquium, Crete, Greece, volume 2076 of Lecture Notes in Computer Science, pages , [25] M. Thorup. Quick and good facility location. In Proceedings of the 14th ACM-SIAM symposium on Discrete Algorithms, A Proof of Lemma 1 Proof. Let ËÇÄ be an arbitrary solution for Ä ÄÈ µ, which can also be viewed as a solution for Í ÄÈ µ for. Consider a facility that is open in ËÇÄ, and the set of clients connected to it in ËÇÄ. Let denote the number of these clients, µ (for ¼) be the facility cost function of, and denote the connection cost between client and the facility in the instance Í ÄÈ µ. Therefore, is the corresponding connection cost in the original instance Ä ÄÈ µ. Recall [13] the definition of «and Ö in the factor-revealing LP of Theorem A. It is proved [13] that «Ö. We strengthen this inequality as follows. Claim 9 «Ö Proof. It is true if ««since it happens only if Ö «. Otherwise, consider clients and µ at time Ø «. Let be the facility is assigned to at time Ø. By triangle inequality, we have Ö On the other hand «since otherwise could have connected to facility at a time earlier than Ø. It is also known [13] that ÑÜ Ö ¼µ ÑÜ «¼µ Notice that ÑÜ Ü ¼µ ÑÜ ¼µ Ü if Ü ¼. Therefore, we have ÑÜ Ö ¼µ ÑÜ «¼µ (14) Claim 9 and Inequality 14 show that the values «, Ö,,, and constitute a feasible solution of the following optimization program. 11

12 maximize subject to È «µ È ««Ö Ö «Ö ÑÜ Ö ¼µ «Ö ¼ ÑÜ «¼µ (15) However, it is clear that the above optimization program and the factor-revealing LP in Theorem A are equivalent. This completes the proof of this lemma. B Proof of Theorem 5 The theorem is established by the following lemmas which show the reductions between the concave soft-capacitated FLP, the concave-cost FLP and the linear-cost FLP. Lemma 10 The concave soft-capacitated FLP is ¾µ µ reducible to the concave-cost FLP. µ Proof. Given an instance Á of the concave soft-capacitated FLP, where the facility cost function of the facility is µ Ù µ, we construct an instance Ê Áµ of the concave-cost FLP as follows: We have the same sets of facilities and clients and the same connection costs as in Á. The facility cost function of the th facility is given by ¼ µ Öµ Ö µ Öµµ ¼ Ù Ö µ if ¼ Ö Ù if ¼ Concavity of implies that the above function is also concave, and therefore Ê Áµ is an instance of concave-cost FLP. Also, it is easy to see from the above definition that By the concavity of the function, we have Ö µ Öµ and number, This completes the proof of the lemma. Ù µ ¼ µ Ù µ ¾µ for every Ö. Therefore, for every facility µ ¼ ¾µ µ µ µ µ Moreover, we will show a simple µ-reduction from the concave-cost FLP to the linear-cost FLP. This, together with the above lemma, reduce the concave soft-capacitated facility location problem to the linearcost FLP. 12

13 Lemma 11 There is a µ-reduction from the concave-cost FLP to the linear-cost FLP. Proof. Given an instance Á of concave-cost FLP, we construct an instance Ê Áµ of linear-cost FLP as follows: Corresponding to each facility in Á with facility cost function µ, we put Ò copies of this facility in Ê Áµ (where Ò is the number of clients), and let the facility cost function of the Ð th copy be е ¼ if ¼ µ е е Ð µµ е if ¼ In other words, the facility cost function is the line that passes through the points Ð Ð µµ and Реµ. The set of clients, and the connection costs between clients and facilities are unchanged. We prove that this reduction is a µ-reduction. For any feasible solution ËÇÄ for Á, we can construct a feasible solution ËÇÄ ¼ for Ê Áµ as follows: If a facility is open and clients are connected to it in ËÇÄ, we open the th copy of the corresponding facility in Ê Áµ, and connect the clients to it. Since µ µ µ, the facility and connection costs of ËÇÄ ¼ is the same as those of ËÇÄ. Conversely, consider an arbitrary feasible solution ËÇÄ for Ê Áµ. We construct a solution ËÇÄ ¼ for Á as follows. For any facility, if at least one of the copies of is open in ËÇÄ, we open and connect all clients that were served by a copy of in ËÇÄ to it. We show that this does not increase the total cost of the solution: Assume the Ð th, Ð ¾ th,..., and Ð th copies of were open in ËÇÄ, serving ¾, and clients, respectively. By concavity of, and the fact that е µ µ µ µ for every Ð, we have µ µ µ Ð µ µ Ð µ µ This shows that the facility cost of ËÇÄ ¼ is at most the facility cost of ËÇÄ. C Other examples The main trick we used in Section 4 was to differentiate between the parts of the approximation ratio that correspond to the facility costs and connection costs. This trick can be used to improve the approximation factor of other problems that are solved by reductions to a facility location problem. In this appendix, we present two such examples. C.1 The capacitated cable facility location problem with non-uniform demands The capacitated cable facility location (CCFLP), each client has a demand that needs to be routed to an open facility. This connection can be done by installing cables of capacity Ù (i.e., capable of routing units of demand in total) on the edges of the graph. More than one cable can be installed on each edge, at a cost equal to the number of cables times the cost of the edge. The objective is to open some facilities and install some cables to minimize the total cost of facilities and cables. This problem was introduced by Ravi and Sinha [19]. If all the clients have uniform demands, Ravi and Sinha [19] show that the CCFLP can be approximated by a factor of Í ÄÈ ËÌ where Í ÄÈ and ËÌ denote the approximation ratio 13

14 for the UFLP and the Steiner Tree problem. For non-uniform demands, they show that the CCFLP can be approximated by a factor of ¾ Í ÄÈ ËÌ (using Í ÄÈ ¾ [18] and ËÌ [20]). As noted by Ravi and Sinha [19], the coefficient ¾ in front of Í ÄÈ in the above expression is only because of doubling connection costs (and not facility costs). Therefore, we can get the following result. Observation 12 The capacitated cable facility location problem with non-uniform demands can be approximated within a factor of. Proof. It is implicitly proved in [19] that a Ê Ê µ-approximation algorithm for the UFLP and a ËÌ - approximation algorithm for the Steiner tree problem imply a ÑÜ Ê ¾Ê µ ËÌ µ-approximation algorithm for the CCFLP with non-uniform demands. Using ËÌ [20] and Theorem B, we get an algorithm with the approximation ratio at most ÑÜ ÐÒ ¾ µµ for the CCFLP. This expression is minimized at µ µ and, yielding a factor of. C.2 The load-balanced facility location In the load-balanced facility location (LBFLP), each open facility is required to serve at least Ä clients. This problem was introduced independently by Guha, Meyerson and Munagala [9] and by Karger and Minkoff [15]. They reduce the LBFLP to the UFLP, and show that if the UFLP can be approximated within a factor of, there is a bicriteria µ-approximation for the LBFLP, i.e, the algorithm produce a solution in which each facility serves at least Ä clients for some, and the total cost is at most times the optimal cost of a solution satisfying the lower bounds. Their result can be used to obtain constant factor approximation algorithms for the connected facility location problem and some network design problems, in particular the access network design problem. This, together with ¾, lead to bicriteria approximation ratios ¼µ and ¼ µ for the LBFLP. This result is based on a reduction that transforms an LBFLP instance of total cost into an instance of the UFLP of total cost at most ¾ µ µ, such that any solution for the reduced UFLP instance gives a solution to the original LBFLP instance violating the lower bounds by at most a factor of. A close look at this result [15, 9] shows that for the same reduction the reduced UFLP instance has a solution with facility cost at most ¾ and connection cost at most. Therefore, we have the following Observation 13 If the UFLP can be approximated within a factor of µ, then there is a bicriteria ¾ µ-approximation for the load-balanced facility location problem. In particular, using the pairs µ ¾µ and µ µ, we get bicriteria approximation algorithms for the LBFLP with factors ¾µ and ¾ µ. D Proof of Theorem 2 Proof. Since inequalities of the factor-revealing LP (13) are a superset of the inequalities of the factorrevealing LP (1), by Theorem A and the definition of µ, we have «(16) 14

15 By inequality (11), for every Ä, we have Therefore, «Ö «µ Ä Ä ¼ ÑÜ Ö ¼µ µ Ö µ Ä ¼ Ö µ Ä µ Ä Ä µ Ä µ µ µ µ µ (17) This, together with Theorem 6 shows that the MYZ Ä is a Ä µ Ä µ µ µ- approximation algorithm for the UFLP. The fact that the limit of Ä µ Ä µ µ as Ä tends to infinity is ÐÒ µ completes the proof. 15