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Cost sharing methods for makespan and completion time scheduling
 IN IN PROCEEDINGS OF THE 24TH INTERNATIONAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS), LECTURE NOTES IN COMPUTER SCIENCE
, 2007
"... Roughgarden and Sundararajan recently introduced an alternative measure of efficiency for cost sharing mechanisms. We study cost sharing methods for combinatorial optimization problems using this novel efficiency measure, with a particular focus on scheduling problems. While we prove a lower bound o ..."
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Roughgarden and Sundararajan recently introduced an alternative measure of efficiency for cost sharing mechanisms. We study cost sharing methods for combinatorial optimization problems using this novel efficiency measure, with a particular focus on scheduling problems. While we prove a lower bound of Ω(log n) for a very general class of problems, we give a best possible cost sharing method for minimum makespan scheduling. Finally, we show that no budget balanced cost sharing methods for completion or flow time objectives exist.
Optimal efficiency guarantees for network design mechanisms
 In Proceedings of the 12th Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 4513 of Lecture Notes in Computer Science
, 2007
"... Abstract. A costsharing problem is defined by a set of players vying to receive some good or service, and a cost function describing the cost incurred by the auctioneer as a function of the set of winners. A costsharing mechanism is a protocol that decides which players win the auction and at what ..."
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Abstract. A costsharing problem is defined by a set of players vying to receive some good or service, and a cost function describing the cost incurred by the auctioneer as a function of the set of winners. A costsharing mechanism is a protocol that decides which players win the auction and at what prices. Three desirable but provably mutually incompatible properties of a costsharing mechanism are: incentivecompatibility, meaning that players are motivated to bid their true private value for receiving the good; budgetbalance, meaning that the mechanism recovers its incurred cost with the prices charged; and efficiency, meaning that the cost incurred and the value to the players served are traded off in an optimal way. Our work is motivated by the following fundamental question: for which costsharing problems are incentivecompatible mechanisms with good approximate budgetbalance and efficiency possible? We focus on cost functions defined implicitly by NPhard combinatorial optimization problems, including the metric uncapacitated facility location problem, the Steiner tree problem, and rentorbuy network design problems. For facility location and rentorbuy network design, we establish for the first time that approximate budgetbalance and efficiency are simultaneously possible. For the Steiner tree problem, where such a guarantee was previously known, we prove a new, optimal lower bound on the approximate efficiency achievable by the wide and natural class of “Moulin mechanisms”. This lower bound exposes a latent approximation hierarchy among different costsharing problems. 1
Prizecollecting Steiner networks via iterative rounding
 in Proceedings of The 9th Latin American Theoretical Informatics Symposium (LATIN
, 2010
"... Abstract. In this paper we design an iterative rounding approach for the classic prizecollecting Steiner forest problem and more generally the prizecollecting survivable Steiner network design problem. We show as an structural result that in each iteration of our algorithm there is an LP variable ..."
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Abstract. In this paper we design an iterative rounding approach for the classic prizecollecting Steiner forest problem and more generally the prizecollecting survivable Steiner network design problem. We show as an structural result that in each iteration of our algorithm there is an LP variable in a basic feasible solution which is at least onethirdintegral resulting a 3approximation algorithm for this problem. In addition, we show this factor 3 in our structural result is indeed tight for prizecollecting Steiner forest and thus prizecollecting survivable Steiner network design. This especially answers negatively the previous belief that one might be able to obtain an approximation factor better than 3 for these problems using a natural iterative rounding approach. Our structural result is extending the celebrated iterative rounding approach of Jain [13] by using several new ideas some from more complicated linear algebra. The approach of this paper can be also applied to get a constant factor (bicriteria)approximation algorithm for degree constrained prizecollecting network design problems. We emphasize that though in theory we can prove existence of only an LP variable of at least onethirdintegral, in practice very often in each iteration there exists a variable of integral or almost integral which results in a much better approximation factor than provable factor 3 in this paper (see patent application [11]). This is indeed the advantage of our algorithm in this paper over previous approximation algorithms for prizecollecting Steiner forest with the same or slightly better provable approximation factors. 1
PrizeCollecting Steiner Network Problems
"... In the Steiner Network problem we are given a graph G with edgecosts and connectivity requirements ruv between node pairs u, v. The goal is to find a minimumcost subgraph H of G that contains ruv edgedisjoint paths for all u, v ∈ V. In PrizeCollecting Steiner Network problems we do not need to ..."
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In the Steiner Network problem we are given a graph G with edgecosts and connectivity requirements ruv between node pairs u, v. The goal is to find a minimumcost subgraph H of G that contains ruv edgedisjoint paths for all u, v ∈ V. In PrizeCollecting Steiner Network problems we do not need to satisfy all requirements, but are given a penalty function for violating the connectivity requirements, and the goal is to find a subgraph H that minimizes the cost plus the penalty. The case when ruv ∈ {0, 1} is the classic PrizeCollecting Steiner Forest problem. In this paper we present a novel linear programming relaxation for the PrizeCollecting Steiner Network problem, and by rounding it, obtain the first constantfactor approximation algorithm for submodular and monotone nondecreasing penalty functions. In particular, our setting includes allornothing penalty functions, which charge the penalty even if the connectivity requirement is slightly violated; this resolves an open question posed in [SSW07]. We further generalize our results for elementconnectivity and nodeconnectivity.
Quantifying Inefficiency in CostSharing Mechanisms
, 2009
"... In a costsharing problem, several participants with unknown preferences vie to receive some good or service, and each possible outcome has a known cost. A costsharing mechanism is a protocol that decides which participants are allocated a good and at what prices. Three desirable properties of a co ..."
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In a costsharing problem, several participants with unknown preferences vie to receive some good or service, and each possible outcome has a known cost. A costsharing mechanism is a protocol that decides which participants are allocated a good and at what prices. Three desirable properties of a costsharing mechanism are: incentivecompatibility, meaning that participants are motivated to bid their true private value for receiving the good; budgetbalance, meaning that the mechanism recovers its incurred cost with the prices charged; and economic efficiency, meaning that the cost incurred and the value to the participants are traded off in an optimal way. These three goals have been known to be mutually incompatible for thirty years. Nearly all the work on costsharing mechanism design by the economics and computer science communities has focused on achieving two of these goals while completely ignoring the third. We introduce novel measures for quantifying efficiency loss in costsharing mechanisms and prove simultaneous approximate budgetbalance and approximate efficiency guarantees for mechanisms for a wide range of costsharing problems, including all submodular and Steiner tree problems. Our key technical tool is an exact characterization of worstcase efficiency loss in Moulin mechanisms, the dominant paradigm in costsharing mechanism design.
Prizecollecting Network Design on Planar Graphs
, 2010
"... In this paper, we reduce PrizeCollecting Steiner TSP (PCTSP), PrizeCollecting Stroll (PCS), PrizeCollecting Steiner Tree (PCST), PrizeCollecting Steiner Forest (PCSF) and more generally Submodular PrizeCollecting Steiner Forest (SPCSF) on planar graphs (and more generally boundedgenus graphs) ..."
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In this paper, we reduce PrizeCollecting Steiner TSP (PCTSP), PrizeCollecting Stroll (PCS), PrizeCollecting Steiner Tree (PCST), PrizeCollecting Steiner Forest (PCSF) and more generally Submodular PrizeCollecting Steiner Forest (SPCSF) on planar graphs (and more generally boundedgenus graphs) to the same problems on graphs of bounded treewidth. More precisely, we show any αapproximation algorithm for these problems on graphs of bounded treewidth gives an (α + ɛ)approximation algorithm for these problems on planar graphs (and more generally boundedgenus graphs), for any constant ɛ> 0. Since PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth using dynamic programming, we obtain PTASs for these problems on planar graphs and boundedgenus graphs. In contrast, we show PCSF is APXhard to approximate on seriesparallel graphs, which are planar graphs of treewidth at most 2. This result is interesting on its own because it gives the first provable hardness separation between prizecollecting and nonprizecollecting (regular) versions of the problems: regular Steiner Forest is known to be polynomially solvable on seriesparallel graphs and admits a PTAS on graphs of bounded treewidth. An analogous hardness result can be shown for Euclidian PCSF. This ends the common belief that prizecollecting variants should not add any new hardness to the problems.
acyclic mechanisms and their applications to scheduling problems
"... Abstract. Mehta, Roughgarden, and Sundararajan recently introduced a new class of cost sharing mechanisms called acyclic mechanisms. These mechanisms achieve a slightly weaker notion of truthfulness than the wellknown Moulin mechanisms, but provide additional freedom to improve budget balance and s ..."
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Abstract. Mehta, Roughgarden, and Sundararajan recently introduced a new class of cost sharing mechanisms called acyclic mechanisms. These mechanisms achieve a slightly weaker notion of truthfulness than the wellknown Moulin mechanisms, but provide additional freedom to improve budget balance and social cost approximation guarantees. In this paper, we investigate the potential of acyclic mechanisms for combinatorial optimization problems. In particular, we study a subclass of acyclic mechanisms which we term singleton acyclic mechanisms. We show that every ρapproximate algorithm that is partially increasing can be turned into a singleton acyclic mechanism that is weakly groupstrategyproof and ρbudget balanced. Based on this result, we develop singleton acyclic mechanisms for parallel machine scheduling problems with completion time objectives, which perform extremely well both with respect to budget balance and social cost. 1
Elementary approximation algorithms for prize collecting Steiner tree problems
 Information Processing Letters
"... Abstract. This paper deals with approximation algorithms for the prize collecting generalized Steiner forest problem, defined as follows. The input is an undirected graph G =(V,E), a collection T = {T1,...,Tk}, each a subset of V of size at least 2, a weight function w: E → R +, and a penalty functi ..."
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Abstract. This paper deals with approximation algorithms for the prize collecting generalized Steiner forest problem, defined as follows. The input is an undirected graph G =(V,E), a collection T = {T1,...,Tk}, each a subset of V of size at least 2, a weight function w: E → R +, and a penalty function p: T → R +. The goal is to find a forest F that minimizes the cost of the edges of F plus the penalties paid for subsets Ti whose vertices are not all connected by F.)approximation for the prize collecting generalized Steiner forest problem, where n ≥ 2isthenumber of vertices in the graph. This obviously implies the same approximation for the special case called the prize collecting Steiner forest problem (all subsets Ti are of size 2). The approximation ratio we achieve is better than that of the best known combinatorial algorithm for this problem, which is the 3approximation of Sharma, Swamy, and Williamson [13]. Furthermore, our algorithm is obtained using an elegant application of the local ratio method and is much simpler and practical, since unlike the algorithm of Sharma et al., it does not use submodular function minimization. Our main result is a combinatorial (3 − 4 n)approximation for the prize collecting Steiner tree problem (all subsets Ti are of size 2 and there is some root vertex r that belongs to all of them). This latter algorithm is in fact the local ratio version of the primaldual algorithm of Goemans and Williamson [7]. Another special case of our main algorithm is BarYehuda’s local ratio Our approach gives a (2 − 1 n−1)approximation for the generalized Steiner forest problem (all the penalties are infinity) [3]. Thus, an important contribution of this paper is in providing a natural generalization of the framework presented by Goemans and Williamson, and later by BarYehuda.
On budgetbalanced groupstrategyproof cost sharing mechanisms
, 2012
"... A costsharing mechanism defines how to share the cost of a service among serviced customers. It solicits bids from potential customers and selects a subset of customers to serve and a price to charge each of them. The mechanism is groupstrategyproof if no subset of customers can gain by lying abou ..."
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A costsharing mechanism defines how to share the cost of a service among serviced customers. It solicits bids from potential customers and selects a subset of customers to serve and a price to charge each of them. The mechanism is groupstrategyproof if no subset of customers can gain by lying about their values. There is a rich literature that designs groupstrategyproof costsharing mechanisms using schemes that satisfy a property called crossmonotonicity. Unfortunately, Immorlica et al showed that for many services, crossmonotonic schemes are provably not budgetbalanced, i.e., they can recover only a fraction of the cost. While crossmonotonicity is a sufficient condition for designing groupstrategyproof mechanisms, it is not necessary. Pountourakis and Vidali recently provided a complete characterization of groupstrategyproof mechanisms. Using their characterization, we construct a fully budgetbalanced groupstrategyproof mechanism for the edgecover problem. This improves upon the crossmonotonic approach which can recover only half the cost, and provides a proofofconcept as to the usefullness of the complete characterization. This raises the question of whether all “natural ” problems have budgetbalanced groupstrategyproof mechanisms. We answer this question in the negative by
GroupStrategyproof Cost Sharing Mechanisms for Makespan and Other Scheduling Problems
, 2007
"... Classical results in economics show that no truthful mechanism can achieve budget balance and efficiency simultaneously. Roughgarden and Sundararajan recently proposed an alternative efficiency measure, which was subsequently used to exhibit that many previously known cost sharing mechanisms approxi ..."
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Classical results in economics show that no truthful mechanism can achieve budget balance and efficiency simultaneously. Roughgarden and Sundararajan recently proposed an alternative efficiency measure, which was subsequently used to exhibit that many previously known cost sharing mechanisms approximate both budget balance and efficiency. In this work, we investigate cost sharing mechanisms for combinatorial optimization problems using this novel efficiency measure, with a particular focus on scheduling problems. Our contribution is threefold: First, for a large class of optimization problems that satisfy a certain coststability property, we prove that no budget balanced Moulin mechanism can approximate efficiency better than Ω(log n), where n denotes the number of players in the universe. Second, we present a groupstrategyproof cost sharing mechanism for the minimum makespan scheduling problem that is tight with respect to budget balance and efficiency. Finally, we show a general lower bound on the budget balance factor for cost sharing methods, which can be used to prove a lower bound of Ω(n) on the budget balance factor for completion and flow time scheduling objectives.