Online submodular welfare maximization: Greedy is optimal

We design and analyze deterministic truthful approximation mechanisms for multiunit Combinatorial Auctions involving only a constant number of distinct goods, each in arbitrary limited supply. Prospective buyers (bidders) have preferences over multisets of items, i.e., for more than one unit per distinct good. Our objective is to determine allocations of multisets that maximize the Social Welfare. Our main results are for multiminded and submodular bidders. In the first setting each bidder has a positive value for being allocated one multiset from a prespecified demand set of alternatives. In the second setting each bidder is associated to a submodular valuation function that defines his value for the multiset he is allocated. For multi-minded bidders, we design a truthful Fptas that fully optimizes the Social Welfare, while violating the supply constraints on goods within factor (1 + ), for any fixed > 0 (i.e., the approximation applies to the constraints and not to the Social Welfare). This result is best possible, in that full optimization is impossible without violating the supply constraints. For submodular bidders, we obtain a Ptas that approximates the optimum Social Welfare within factor (1 + ), for any fixed > 0, without violating the supply constraints. This result is best possible as well. Our allocation algorithms are Maximal-in-Range and yield truthful mechanisms, when paired with Vickrey-Clarke-Groves payments.

show abstract

“…We consider submodular valuation functions over multisets in U, as defined by Kapralov, Post, and Vondrák (2013):…”

Section: Submodular Valuation Functionsmentioning

confidence: 99%

Mechanisms for Multi-unit Combinatorial Auctions with a Few Distinct Goods

Krysta

Telelis

Ventre

2015

jair

View full text Add to dashboard Cite

show abstract

“…model [7,8], and a similar ratio can be obtained for the more general online SWM problem [18]. However, all results based on such techniques seem to only apply to the i.i.d.…”

Section: Related Workmentioning

confidence: 60%

“…However, all results based on such techniques seem to only apply to the i.i.d. model [7,18] and not the random order model. Generalizing such results to the random order model remains an interesting open problem in the area.…”

Section: Related Workmentioning

confidence: 99%

“…For example, in the adversarial model, 1 − 1/e-competitive algorithms have been achieved for the special case of the online matching problem, as well as the the budgeted allocation and online weighted matching problems under the so-called large capacity assumption [20,24,11]. However, such a result is not possible for the general online SWM problem in the adversarial setting, where the simple greedy algorithm is the best possible: A recent result by Kapralov, Post, and Vondrak [18] shows that this 1/2-approximation is tight for online SWM unless NP=RP. This hardness result, however, does not rule out improving the approximation factor of 1/2 for the random order model.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Online Submodular Welfare Maximization

Korula

Mirrokni

Zadimoghaddam

2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

View full text Add to dashboard Cite

In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents. Each agent has a valuation function v , where v (S) denotes the welfare obtained by this agent if she receives the set of items S. The functions v are all submodular; as is standard, we assume that they are monotone and v (∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, . . . Sm in order to maximize the social welfare, defined as m =1 v (S ). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondrak's recent (1 − 1/e)-approximation algorithm [34].In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary / worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n [9, 10], barely better than the ratio for the adversarial order. Obtaining a competitive ratio of 1/2 + Ω(1) for the random order model has been an important open problem for several years. We solve this open problem by

show abstract

“…The greedy strategy is already half competitive. Kapralov et al [36] showed that for adversarial arrival greedy is the best possible in general (competitive ratio of 1/2), but under a "large capacities" assumption, a primal-dual algorithm can obtain 1 − 1/e-competitive ratio [18]. For random arrival Korula et al [41] showed that greedy can beat half; obtaining 1 − 1/e in this settings remains open.…”

Section: Further Related Workmentioning

confidence: 99%

Combinatorial Prophet Inequalities

Rubinstein¹,

Singla²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

We introduce a novel framework of Prophet Inequalities for combinatorial valuation functions. For a (non-monotone) submodular objective function over an arbitrary matroid feasibility constraint, we give an O(1)-competitive algorithm. For a monotone subadditive objective function over an arbitrary downwardclosed feasibility constraint, we give an O(log n log 2 r)-competitive algorithm (where r is the cardinality of the largest feasible subset).Inspired by the proof of our subadditive prophet inequality, we also obtain an O(log n · log 2 r)-competitive algorithm for the Secretary Problem with a monotone subadditive objective function subject to an arbitrary downward-closed feasibility constraint. Even for the special case of a cardinality feasibility constraint, our algorithm circumvents an Ω( √ n) lower bound by Bateni, Hajiaghayi, and Zadimoghaddam [10] in a restricted query model.En route to our submodular prophet inequality, we prove a technical result of independent interest: we show a variant of the Correlation Gap Lemma [14, 1] for nonmonotone submodular functions.

show abstract

Online submodular welfare maximization: Greedy is optimal

Cited by 65 publications

References 28 publications

Mechanisms for Multi-unit Combinatorial Auctions with a Few Distinct Goods

Mechanisms for Multi-unit Combinatorial Auctions with a Few Distinct Goods

Online Submodular Welfare Maximization

Combinatorial Prophet Inequalities

Contact Info

Product

Resources

About