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.

Abstract. A cost-sharing 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 cost-sharing mechanism are: incentive-compatibility, meaning that players are motivated to bid their true private value for receiving the good; budget-balance, 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 cost-sharing problems are incentive-compatible mechanisms with good approximate budget-balance and efficiency possible? We focus on cost functions defined implicitly by NP-hard combinatorial optimization problems, including the metric uncapacitated facility location problem, the Steiner tree problem, and rent-or-buy network design problems. For facility location and rent-or-buy network design, we establish for the first time that approximate budget-balance 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 cost-sharing problems. 1

In the Steiner Network problem we are given a graph G with edge-costs and connectivity requirements ruv between node pairs u, v. The goal is to find a minimum-cost subgraph H of G that contains ruv edge-disjoint paths for all u, v ∈ V. In Prize-Collecting 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 Prize-Collecting Steiner Forest problem. In this paper we present a novel linear programming relaxation for the Prize-Collecting Steiner Network problem, and by rounding it, obtain the first constant-factor approximation algorithm for submodular and monotone non-decreasing penalty functions. In particular, our setting includes all-or-nothing 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 element-connectivity and node-connectivity.

In a cost-sharing problem, several participants with unknown preferences vie to receive some good or service, and each possible outcome has a known cost. A cost-sharing mechanism is a protocol that decides which participants are allocated a good and at what prices. Three desirable properties of a cost-sharing mechanism are: incentive-compatibility, meaning that participants are motivated to bid their true private value for receiving the good; budget-balance, 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 cost-sharing 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 cost-sharing mechanisms and prove simultaneous approximate budget-balance and approximate efficiency guarantees for mechanisms for a wide range of cost-sharing problems, including all submodular and Steiner tree problems. Our key technical tool is an exact characterization of worst-case efficiency loss in Moulin mechanisms, the dominant paradigm in cost-sharing mechanism design.

Abstract. In this paper we design an iterative rounding approach for the classic prize-collecting Steiner forest problem and more generally the prize-collecting 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 one-third-integral resulting a 3-approximation algorithm for this problem. In addition, we show this factor 3 in our structural result is indeed tight for prize-collecting Steiner forest and thus prize-collecting 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 prize-collecting network design problems. We emphasize that though in theory we can prove existence of only an LP variable of at least one-third-integral, 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 prize-collecting Steiner forest with the same or slightly better provable approximation factors. 1

In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF) and more generally Submodular Prize-Collecting Steiner Forest (SPCSF) on planar graphs (and more generally bounded-genus graphs) to the same problems on graphs of bounded treewidth. More precisely, we show any α-approximation algorithm for these problems on graphs of bounded treewidth gives an (α + ɛ)-approximation algorithm for these problems on planar graphs (and more generally bounded-genus 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 bounded-genus graphs. In contrast, we show PCSF is APX-hard to approximate on series-parallel 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 prize-collecting and non-prize-collecting (regular) versions of the problems: regular Steiner Forest is known to be polynomially solvable on series-parallel 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 prize-collecting variants should not add any new hardness to the problems.

the primal-dual framework for approximation algorithms and applied it to a class of network design optimization problems. Since then literally hundreds of results appeared that extended, modified and applied the technique to a wide range of optimization problems. In this paper we define a class of cost-sharing games arising from Goemans and Williamson’s original network design problems. We then show how to derive a groupstrategyproof (i.e., collusion resistant) mechanism for such a game, using an existing primal-dual algorithm for the underlying optimization problem as a black box. The budgetbalance factor of this mechanism is proportional to the performance ratio of the primal-dual algorithm if the optimization problem satisfies an additional technical condition. Most existing collusion-resistant cost-sharing mechanisms are obtained through skillful adaptation of existing primal-dual algorithms for the associated optimization problems. This paper shows that, at least for a large class of games arising from network design problems, no such adaptation is necessary. 1

An instance of the k-generalized connectivity problem consists of an undirected graph G = (V, E), whose edges are associated with non-negative costs, and a collection D = {(S1, T1),..., (Sd, Td)} of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph H ⊆ G connects a demand (Si, Ti) when it contains a path with one endpoint in Si and the other in Ti. Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in D. Alon, Awerbuch, Azar, Buchbinder and Naor (SODA ’04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiplechoice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log 2 n log 2 d) approximation due to Chekuri, Even, Gupta and Segev (SODA ’08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA ’06) related its inapproximability to that of dense k-subgraph. In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n 2/3 · polylog(n, k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.

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. Our main result is a (3 − 4 n)-approximation for the prize collecting generalized Steiner forest problem, where n � 2 is the number 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 areofsize2).The approximation algorithm is obtained by applying the local ratio method, and is much simpler than the best known combinatorial algorithm for this problem. Our approach gives a (2 − 1 n−1)-approximation for the prize collecting Steiner tree problem (all subsets Ti areofsize2andthere is some root vertex r that belongs to all of them). This latter algorithm is in fact the local ratio version of the primal-dual algorithm of Goemans and Williamson [M.X. Goemans, D.P. Williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing 24 (2) (April 1995) 296–317]. Another special case of our main algorithm is Bar-Yehuda’s local ratio (2 − 2 n)-approximation for the generalized Steiner forest problem (all the penalties are infinity) [R. Bar-Yehuda, One for the price of two: a unified approach for approximating covering problems, Algorithmica 27 (2) (June 2000) 131–144]. Thus, an important contribution of this paper is in providing a natural generalization of the framework presented by Goemans and Williamson, and later by Bar-Yehuda.

We consider the design of strategyproof cost-sharing mechanisms. We give two simple, but extremely versatile, black-box reductions, that in combination reduce the cost-sharing mechanism-design problem to the algorithmic problem of finding a minimum-cost solution for a set of players. Our first reduction shows that any truthful, α-approximation mechanism for the socialcost minimization (SCM) problem satisfying a technical no-bossiness condition can be morphed into a truthful mechanism that achieves an O(α log n)-approximation where the prices recover the cost incurred. Thus, we decouple the task of truthfully computing an outcome with near-optimal social cost from the cost-sharing problem. This is fruitful since truthful mechanism-design, especially for single-dimensional problems, is a relatively well-understood and manageable task. Our second reduction nicely complements the first one by showing that any LP-based ρ-approximation for the problem of finding a min-cost solution for a set of players yields a truthful, no-bossy, (ρ + 1)-approximation for the SCM problem (and hence, a truthful (ρ + 1) log n-approximation cost-sharing mechanism). These reductions find a slew of applications, yielding, as corollaries, the first or improved polytime costsharing mechanisms for a variety of problems. For example, our first reduction coupled with the celebrated VCG mechanism shows that for any costsharing problem (with a monotone cost function) one can obtain a truthful mechanism that achieves an O(log n)-approximation where the prices recover the cost incurred. Other applications include O(log n)approximation mechanisms for: survivable network design problems, facility location (FL) problems including capacitated and connected FL problems, and minimummakespan scheduling on unrelated machines. Our results demonstrate that in contrast with our current understanding of group-strategyproof and acyclic mechanisms, strategyproofness allows for ample flexibility in cost-sharing mechanism design enabling one to effectively leverage various algorithmic results.