The k-level facility location game

The k-level facility location game Dachuan Xua,∗,†and Donglei Dub a College of Applied Sciences, Beijing University of Technology, P.R. China b Faculty of Administration, University of New Brunswick, Canada Abstract We propose a cost-sharing scheme for the k-level facility location game that is cross-monotonic, competitive, and 6-approximate cost recovery. This extends the recent result for the 1-level facility location game of Pál and Tardos. Keywords: k-level facility location game, cross-monotonic, competitive, approximate cost recovery. 1 Introduction In the traditional (single level) facility location problem (FLP), agents from a given universal set U jointly pay to open and access facilities from a given set F . Opening a facility i ∈ F causes a fixed nonnegative open cost fi and accessing this facility by any agent j ∈ U imposes a nonnegative connection cost cij . The problem seeks to open facilities and connect each agent to a facility such that the total open and connection cost is minimized. In the corresponding facility location game (FLG), each agent is a player. For any given subset of agents J ⊂ U , there is a minimum total cost c∗ (J) that assigns all agents in J. The goal is to design a cost-sharing scheme, or pricing policy that allocates the total cost to the different agents. More precisely, a cost-sharing method is to compute the cost shares α(J, j) for each j ∈ J. ∗ Corresponding author. College of Applied Sciences, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing 100022, P.R. China. E-mail addresses: xudc@bjut.edu.cn (D. Xu), ddu@unb.ca (D. Du). † The earliest manuscript was done while the author was a Postdoctoral Scholar at Department of Management Science and Engineering, Stanford University. 1 In many practical situations, the potential facilities F are organized into k-levels F1 , · · · , Fk hierarchically. Each agent j ∈ J must be served by a path p = (i1 , · · · , ik ) ∈ F1 × · · · × Fk formed by k different facilities, one from each level. This leads to the k-level facility location problem k-FLP and its corresponding game k-FLG. In this paper we are interested in the k-FLG. We also impose the assumption that c is a metric on U S F from now on, i.e., we consider the metric facility location games. k-FLG is a special cooperative game. In cooperative game theory, the central problem is to develop cost-sharing methods that fulfill certain desirable properties, such as fairness, group strategyproof, competitiveness, exact cost recovery or approximate cost recovery, and cross-monotonicity. A cost share is fair if no subset of agents I is charged more than the optimum cost, i.e., P j∈I α(I, j) ≤ c∗ (I) for every I ⊂ U . It is group strategyproof if no agent or coalition of agents can benefit by deviating from the truthful strategy. It is competitive if the agents are charged no more than the true cost, i.e., P j∈J α(J, j) ≤ c∗ (J). It satisfies r-approximate cost recovery if the agents are required to recover an 1/r fraction of the total cost, i.e., P j∈J α(J, j) ≥ c∗ (J)/r, where r ≥ 1. It is exact cost recovery if r = 1. It is cross-monotonic if the price charged to any individual in a group never goes up as the group expands, i.e., α(J, j) ≥ α(J ′ , j) for all J ⊆ J ′ . Thus agents have an economic incentive to cooperate. However, these aforementioned properties are not unrelated. Pál and Tardos [9] observe that competitiveness and cross-monotonicity together imply fairness. Moulin and Shenker [8] show that cross-monotonic cost-sharing leads to group strategyproof mechanisms. Therefore to obtain fair and group strategyproof mechanisms, it is sufficient to focus on cost-sharing that is crossmonotonic, competitive, and exact or approximate cost recovery. Unfortunately, no cross-monotonic and competitive cost-sharing can exactly recover the cost for even the single level facility location game 1-FLG (cf. [9]). Actually Immorlica et al [4] recently prove a stronger result for 1-FLG that no such method can recover more than 1/3rd of the cost. This together with the recent scheme of Pál and Tardos [9] completely settles the 1-FLG. The main contribution of this paper is to develop a cost-sharing method for the k-FLG that is cross-monotonic, competitive, and 6-approximate cost recovery. Moreover, we actually give a primal-dual algorithm for the k-FLP with a performance guarantee of 6. Although an approximation algorithm with the same performance was already known in the literature [2], the algorithm in this paper has certain nice properties that might be helpful in solving stochastic k-FLP (A similar 2 property has helped Gupta et al [3] solve a stochastic 1-FLP). Since the focus of this paper is on cooperative games, we only briefly review some approximation results for the underlying NP-hard facility location problems before leaving this section. For 1-FLP, the first constant approximation algorithm is offered by Shmoys et al [10], and the best currently known ratio is 1.52 due to Mahdian et al [6]. For k-FLP, Aardal et al [1] propose an 3-approximation algorithm using linear programming relaxation, and Zhang [11] recently improves the ratio to 1.77 for the special 2-FLP via a quasi-greedy approach. The rest of the paper is organized as follows. After describing the k-FLP and k-FLG formally in section 2, we proceed to the main algorithm and analysis in section 3. Some final remarks are given in section 4. 2 The k-level facility location game Given a set of potential agents U , and k level sets of pairwise disjoint facilities F1 , · · · , Fk , we denote F = F1 S ··· S Fk . There is a facility cost fil for opening facility il ∈ Fl (1 ≤ l ≤ k), and a connection cost c(i, j) for any i, j ∈ U U S S F . Throughout this paper we assume c is a metric on F. Denote P = F1 × F2 × · · · × Fk . A path p ∈ P is open if and only if every facility on the path is open. For j ∈ U and p = (i1 , i2 , · · · , ik ) ∈ P , we let jp denote the path (j, i1 , i2 , · · · , ik ). Let c(p) = Pk l=2 c(il−1 , il ) and c(jp) = c(j, i1 ) + c(p). For an union of some paths H ⊆ P , denote by f (H) the sum of open costs of the facilities in H. Given a subset J ⊆ U of agents, the k-FLP is to open a subset of facilities such that each agent j ∈ J is assigned to some open path pj ∈ P and the cost of the solution X c(jpj ) + f ( j∈J [ pj ) j∈J is minimized. In the corresponding game k-FLG, the players are the agents. The objective is construct a solution such that each agent pay certain amount to recover a constant fraction of the cost of the solution constructed while respecting competitiveness and cross-monotonicity. Consider the integer program for the k-FLP ([2]). yi (i ∈ F ) is an indicator variable denoting whether facility i is open, and xjp (j ∈ J, p ∈ P ) is an indicator variable denoting whether agent j 3 is served by p. The first constraint ensures that each agent j gets connected to one path, and the second constraint ensures that the paths only use open facilities. Instead of studying the integer program, we consider the LP-relaxation and its dual program as follows min P j∈J, p∈P s. t. i∈F fi yi xjp = 1, ∀j ∈ J P xjp ≤ yi , ∀i ∈ F, j ∈ J p:i∈p 3 P P p∈P (LP) c(jp)xjp + max P j∈J s. t. (DLP) αj αj − P j∈J P i∈p βij ≤ c(jp), ∀p ∈ P, j ∈ J βij ≤ fi , ∀i ∈ F xjp ≥ 0, ∀p ∈ P, j ∈ J βij ≥ 0, ∀i ∈ F, j ∈ J yi ≥ 0, ∀i ∈ F. αj ≥ 0, ∀j ∈ J. The cross-monotonic cost-sharing method In this section, we present our cross-monotonic cost-sharing method for the metric k-FLG by adopting a similar ghost-process as suggested in [9] for 1-FLG. First we define the ghost-process tailored for our problem. Then we produce the cost-share to generate a dual feasible solution for (DLP). Finally, we specify the rules respectively for opening facilities and assigning agents to construct a primal feasible solution for (LP). The purpose of the ghost-process is to guarantee crossmonotonicity. The competitiveness is a direct consequence of dual feasibility. The approximate ratio of cost recovery will be guaranteed by comparing the values of these two feasible solutions due to the weak duality of linear programming. We describe the ghost-process and introduce some necessary notations in the following. We introduce the notion of time advancing from 0 to ∞. For every agent j ∈ J, the ghost of j is a ball centered at j and radius equal to the current time t. The ghost-process grows a ghost for every agent over time. The contribution of the ghost j towards filling facility i is βij which will be updated over time. Set βij = 0 initially at time 0. We need further concepts before introducing the updating rule for βij . A facility i ∈ F is said to be fully paid at some time t(i) if P j∈J βij = fi . A path is fully paid if and only if every facility on the path is fully paid. The ghost of agent j ∈ J is said to reach facility iℓ ∈ Fℓ (1 ≤ ℓ ≤ k) at time t, if for some fully paid path p = (i1 , · · · , iℓ ) ∈ F1 × · · · × Fℓ , c(jp) + P i∈p βij = t. Whenever the ghost of some j ∈ J reaches a facility i ∈ F that is not fully paid, we start increasing βij with unit speed until i is fully paid. Let Si = {j ∈ J|βij > 0} be the set of agents that contributed towards i ∈ F . The predecessor of i ∈ Fℓ is the facility in level ℓ − 1 through which i was reached by a ghost for 4 the first time, i.e., pred(i) := argmini′ ∈Fℓ−1 t(i′ ) + c(i′ , i). The predecessor of a facility i ∈ F1 is its closest agent. For any facility i ∈ Fk , we say that pi = (i1 , · · · , ik = i) is the associated path of i if iℓ = pred(iℓ+1 ) (ℓ = 1, · · · , k − 1). The neighborhood of i ∈ Fk is the set of agents contributing to the associated path pi , denoted as Ni := {j ∈ J|βi′ j > 0 for some i′ ∈ pi }. We now are ready to give the cost share αj of each agent j ∈ J. Increase αj with unit speed until j’s ghost reaches some fully paid facility i ∈ Fk or some facility i ∈ Fk that it is reaching becomes fully paid, i.e., αj = min{ min min [c(jp) + i:j ∈S / i ,i∈Fk p=(i1 ,···,ik−1 ,ik )∈P :ik =i X βij ], i∈p min i:j∈Si ,i∈Fk t(i)}. (1) These αj together with the βij generated in the ghost process constitute a dual feasible solution for (DLP) as can be easily verified. Hence the competitiveness follows. Furthermore, note that by adding more agents, each facility i ∈ Fk will be fully paid more quickly and each βij (i ∈ J, j ∈ F ) will not increase. This implies cross-monotonicity. To prove the cost recovery ratio, we construct a primal feasible solution. This involves specifying the rules on how to construct the open facilities and assign agents to open facilities, respectively. First we sort all the k th -level facilities in a nondecreasing order of the fully paid time t(i) (i ∈ Fk ). According to this order, we open the facility i ∈ Fk and the associated path pi if and only if there is no already open facility i′ ∈ Fk such that c(i, i′ ) ≤ 3t(i). The aim of this construction is to keep open paths sufficiently far away from each other so that their neighborhoods are pairwise disjoint (proved in Fact 1), and hence no agent will contribute towards filling more than one open facility. This guarantees the validity of the following agent-assigning policy. Suppose that i ∈ Fk is open, we assign the agents in Ni to the associated path. The remaining agents will be assigned to the closest open path. Fact 1 If facilities i ∈ Fk and i′ ∈ Fk are both open, then Ni Proof : Assume on the contrary that Ni T T Ni′ = ∅. Ni′ 6= ∅. Let i be opened later than i′ , then c(i, i′ ) > 3t(i) > 2t(i) ≥ t(i) + t(i′ ) (2) On the other hand, since j ∈ Ni , there exists ĩ ∈ pi such that βĩj > 0. This implies that there exists a fully paid path p(ĩ) from F1 to ĩ such that c(jp(ĩ)) ≤ c(jp(ĩ))+ 5 P i′′ ∈p(ĩ) βij ≤ t(ĩ). Consider the concatenated path p(i) = (p(ĩ), pi ) which travels first from F1 to ĩ along path p(ĩ), and then from ĩ to Fk along i’s associated path pi . By the definition of predecessor, we have c(jp(i)) ≤ t(i). Similarly, there exists a path p(i′ ) from F1 to i′ such that c(jp(i′ )) ≤ t(i′ ). Combining (2), we get c(j, i) + c(j, i′ ) ≤ c(jp(i)) + c(jp(i′ )) ≤ t(i) + t(i′ ) < c(i, i′ ). This contradicts the triangle inequality. ✷ Now we study the performance of the proposed algorithm. Lemma 2 Let i ∈ Fk be an open facility, and let j ∈ Ni be an agent. Then 2αj ≥ t(i). Proof : Assume on the contrary that 2αj < t(i). Since j ∈ Ni , from the proof of Lemma 1, there exists a path p(i) from F1 to i such that c(jp(i)) ≤ t(i). Let i′ ∈ Fk be the first fully paid facility that j reached. Then we have t(i′ ) ≤ αj and there exists a path p(i′ ) such that c(jp(i′ )) ≤ αj . Now we consider two possibilities, each of which leads to a contradiction: (1) If i′ is open, then c(i, i′ ) ≤ c(jp(i)) + c(jp(i′ )) ≤ t(i) + αj < 3t(i). (2) If i′ is not open, there exists an open facility i′′ ∈ Fk such that c(i′ , i′′ ) ≤ 3t(i′ ). Note that i 6= i′′ , since t(i′′ ) ≤ t(i′ ) ≤ αj < t(i). We have c(i, i′′ ) ≤ c(jp(i)) + c(jp(i′ )) + c(i′ , i′′ ) ≤ t(i) + αj + 3t(i′ ) ≤ t(i) + 4αj < 3t(i). ✷ Lemma 3 Let i ∈ Fk be an open facility and pi = (i1 , · · · , ik = i) be the associated path. Then f (pi ) + P j∈Ni c(jpi ) ≤ 6 P j∈Ni αj . Proof : First note that the sets Siℓ are nested, i.e., Siℓ ⊆ Siℓ+1 for ℓ = 1, ..., k − 1. Indeed, if a client j happens to contribute to the facility il , which, by the definition of the associated path pi , is the first ℓ-level facility that reaches iℓ+1 , then j must have a positive contribution to iℓ+1 . Therefore we can partition the set Ni into k pairwise disjoint sets as follows: S i1 = Si1 , and S iℓ = Siℓ \ Siℓ−1 , ℓ = 2, · · · , k. Consider j ∈ S iℓ which implies βiℓ j > 0. Then we have a path p(jiℓ ) from F1 to iℓ 6 such that 0 < βiℓ j ≤ t(iℓ ) − c(jp(jiℓ )). From the triangle inequality, we have c(jpi ) = c(j, i1 ) + k P ℓ′ =2 ≤ c(jp(jiℓ )) + c(iℓ′ −1 , iℓ′ ) ℓ P ℓ′ =2 c(iℓ′ −1 , iℓ′ ) + k P ℓ′ =2 c(iℓ′ −1 , iℓ′ ) ≤ c(jp(jiℓ )) + 2t(i) ≤ t(iℓ ) − βiℓ j + 2t(i). Combining Lemma 2 and the above inequality, we have f (pi ) + P j∈Ni c(jpi ) = = k P P j∈Ni ℓ′ =1 k P P ≤ k P P ℓ=1 j∈S i ≤ k P ( ℓ=1 j∈S i k P P ℓ=1 j∈S i ≤ 6 P j∈Ni ℓ ℓ′ =ℓ j∈Ni c(jpi ) βiℓ′ j + c(jpi )) k P ( ℓ P βiℓ′ j + ℓ′ =ℓ+1 βiℓ′ j + t(iℓ ) + 2t(i)) 3t(i) ℓ αj . ✷ Lemma 4 Let j be an agent that does not belong to the neighborhood Ni of any open facility i ∈ Fk . Suppose j is assigned to the closest open path pi , where i ∈ Fk is an open facility. Then 6αj ≥ c(jpi ). Proof : Let i′ ∈ Fk be the first fully paid facility that j reached. Then we have t(i′ ) ≤ αj and there exists a path p(i′ ) such that c(jp(i′ )) ≤ αj . Now we consider two possibilities: (1) If i′ is open, let pi′ = (i′1 , · · · , i′k−1 , i′k = i′ ). We have c(jpi ) ≤ c(jpi′ ) = c(j, i′1 ) + ≤ c(jp(i′ )) + 2 Pk Pk ′ ′ ℓ=2 c(iℓ−1 , iℓ ) ′ ′ ℓ=2 c(iℓ−1 , iℓ ) ≤ c(jp(i′ )) + 2t(i′ ) ≤ 3αj . (2) If i′ is not open, there exists an open facility i′′ ∈ Fk such that c(i′ , i′′ ) ≤ λt(i′ ) and 7 t(i′′ ) ≤ t(i′ ). Let pi′′ = (i′′1 , · · · , i′′k−1 , i′′k = i′′ ). We have c(jpi ) ≤ c(jpi′′ ) = c(j, i′′1 ) + Pk ′′ ′′ ℓ=2 c(iℓ−1 , iℓ ) ≤ c(jp(i′ )) + c(i′ , i′′ ) + 2 Pk ′′ ′′ ℓ=2 c(iℓ−1 , iℓ ) ≤ c(jp(i′ )) + c(i′ , i′′ ) + 2t(i′′ ) ≤ αj + 5t(i′ ) ≤ 6αj . ✷ Lemma 3 and 4 together prove the following theorem. Theorem 5 The cost of the solution constructed is at most 6 P j∈J αj . Finally we conclude that the proposed algorithm is a 6-approximate cross-monotonic costsharing method for the metric k-FLG. 4 Final remarks In this paper, we design a cost-sharing strategy for the k-FLG that is cross-monotonic, competitive, and 6-approximate cost recovery. Since k-FLG contains 1-FLG as a special case, therefore the lower bound result for 1-FLG in [4] also holds here. The obvious open question is to close the gap. Acknowledgements The first author is grateful to Xiaodong Hu, Yinyu Ye and Jiawei Zhang for helpful discussions and comments. The authors also thank the anonymous referees for pointing out an error in the proof of Lemma 3 on an earlier version of this paper. The first author’s research was supported by NSF of China (Grant Nos. 10401038, 10171108 and 70271014) and Startup grant for doctoral research of Beijing University of Technology. The second author’s research was supported by NSERC grant 10004901. References [1] K. I. Aardal, F. A. Chudak and D. B. Shmoys, A 3-approximation algorithm for the k-level uncapacitated facility location problem, Information Processing Letters, 72 (1999) 161-167. 8 [2] A. Bumb and W. Kern, A simple dual ascent algorithm for the multilevel facility location problem, Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), LNCS 2129, pages 55-63, 2001. [3] A. Gupta, M. Pál, R. Ravi and A. Sinha, Boosted sampling: Approximation algorithms for stochastic optimization, Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), 2004. [4] N. Immorlica, M. Mahdian and V. S. Mirrokni, Lower bounds for cost sharing and groupstrategyproof mechanisms, submitted. [5] K. Jain and Vijay V. Vazirani, Primal-dual approximation algorithms for metric facility location and k-median problems, Journal of the ACM, 48 (2001) 274-296. [6] M. Mahdian, Y. Ye and J. Zhang, Improved approximation algorithms for metric facility location problems, 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, (APPROX), LNCS 2462, pages 229-242, 2002. [7] R. R. Mettu and C. G. Plaxton, The online median problem, Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 339-348, 2000. [8] H. Moulin and S. Shenker, Strategyproof sharing of submodular costs: budget balance versus efficiency, Economic Theory, 18 (2001) 511-533. [9] M. Pál and E. Tardos, Group strategyproof mechanisms via primal-dual algorithms, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 584-593, 2003. [10] D. B. Shmoys, E. Tardos and K. I. Aardal, Approximation algorithms for facility location problems, Proceedings of the 29th Annual ACM Symposium on Theory of Coputing (STOC), pages 265-274, 1997. [11] J. Zhang, Approximating the two-level facility location problem via a quasi-greedy approach, Proceedings of The 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 801-810, 2004. 9