We study a general stochastic probing problem defined on a universe V, where each element e in V is "active" independently with probability p_e. Elements have weights w_e and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the p_e values-- to determine whether or not an element e is active, our algorithm must probe e. If element e is probed and happens to be active, then e must irrevocably be added to the chosen set S; if e is not active then it is not included in S. Moreover, the following conditions must hold in every random instantiation: (1) the set Q of probed elements satisfy an "outer" packing constraint, and (2) the set S of chosen elements satisfy an "inner" packing constraint. The kinds of packing constraints we consider are intersections of matroids and knapsacks. Our results provide a simple and unified view of results in stochastic matching and Bayesian mechanism design, and can also handle more general constraints. As an application, we obtain the first polynomial-time @math -approximate "Sequential Posted Price Mechanism" under k-matroid intersection feasibility constraints.
The papers of @cite_3 , Yan @cite_15 , and Kleinberg and Weinberg @cite_12 study the performance of Sequential Posted Price Mechanisms (SPMs) for Bayesian single-parameter auctions, and relate the revenue obtained by SPMs to the optimal (non-posted-price) mechanism given by Myerson @cite_19 . Our algorithm for stochastic probing also yields SPMs for Bayesian auctions where the feasible sets of buyers are specified by, e.g., @math -matroid intersection and unsplittable flow on trees. Our proof relates an LP relaxation of the optimal mechanism to the LP used for stochastic probing. Linear programs have been used to model optimal auctions in a number of settings; e.g., see Vohra @cite_13 . @cite_14 also used LP relaxations to obtain approximately optimal mechanisms in a Bayesian setting with multiple items and budget constrained buyers.
@cite_14 @cite_3 @cite_19 @cite_15 @cite_13 @cite_12
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For revenue and welfare maximization in single-dimensional Bayesian settings, (STOC10) recently showed that sequential posted-price mechanisms (SPMs), though simple in form, can perform surprisingly well compared to the optimal mechanisms. In this paper, we give a theoretical explanation of this fact, based on a connection to the notion of correlation gap. Loosely speaking, for auction environments with matroid constraints, we can relate the performance of a mechanism to the expectation of a monotone submodular function over a random set. This random set corresponds to the winner set for the optimal mechanism, which is highly correlated, and corresponds to certain demand set for SPMs, which is independent. The notion of correlation gap of (SODA10) quantifies how much we "lose" in the expectation of the function by ignoring correlation in the random set, and hence bounds our loss in using certain SPM instead of the optimal mechanism. Furthermore, the correlation gap of a monotone and submodular function is known to be small, and it follows that certain SPM can approximate the optimal mechanism by a good constant factor. Exploiting this connection, we give tight analysis of a greedy-based SPM of for several environments. In particular, we show that it gives an @math -approximation for matroid environments, gives asymptotically a @math -approximation for the important sub-case of @math -unit auctions, and gives a @math -approximation for environments with @math -independent set system constraints. For revenue and welfare maximization in single-dimensional Bayesian settings, (STOC10) recently showed that sequential posted-price mechanisms (SPMs), though simple in form, can perform surprisingly well compared to the optimal mechanisms. In this paper, we give a theoretical explanation of this fact, based on a connection to the notion of correlation gap. Loosely speaking, for auction environments with matroid constraints, we can relate the performance of a mechanism to the expectation of a monotone submodular function over a random set. This random set corresponds to the winner set for the optimal mechanism, which is highly correlated, and corresponds to certain demand set for SPMs, which is independent. The notion of correlation gap of (SODA10) quantifies how much we "lose" in the expectation of the function by ignoring correlation in the random set, and hence bounds our loss in using certain SPM instead of the optimal mechanism. Furthermore, the correlation gap of a monotone and submodular function is known to be small, and it follows that certain SPM can approximate the optimal mechanism by a good constant factor. Exploiting this connection, we give tight analysis of a greedy-based SPM of for several environments. In particular, we show that it gives an e (e − 1)-approximation for matroid environments, gives asymptotically a 1 (1--1 √2πk)-approximation for the important sub-case of k-unit auctions, and gives a (p + 1)-approximation for environments with p-independent set system constraints. We give a general technique to obtain approximation mechanisms that are truthful in expectation. We show that for packing domains, any spl alpha -approximation algorithm that also bounds the integrality gap of the IF relaxation of the problem by a can be used to construct an spl alpha -approximation mechanism that is truthful in expectation. This immediately yields a variety of new and significantly improved results for various problem domains and furthermore, yields truthful (in expectation) mechanisms with guarantees that match the best known approximation guarantees when truthfulness is not required. In particular, we obtain the first truthful mechanisms with approximation guarantees for a variety of multi-parameter domains. We obtain truthful (in expectation) mechanisms achieving approximation guarantees of O( spl radic m) for combinatorial auctions (CAs), (1 + spl epsi ) for multiunit CAs with B = spl Omega (log m) copies of each item, and 2 for multiparameter knapsack problems (multiunit auctions). Our construction is based on considering an LP relaxation of the problem and using the classic VCG mechanism by W. Vickrey (1961), E. Clarke (1971) and T. Groves (1973) to obtain a truthful mechanism in this fractional domain. We argue that the (fractional) optimal solution scaled down by a, where a is the integrality gap of the problem, can be represented as a convex combination of integer solutions, and by viewing this convex combination as specifying a probability distribution over integer solutions, we get a randomized, truthful in expectation mechanism. Our construction can be seen as a way of exploiting VCG in a computational tractable way even when the underlying social-welfare maximization problem is NP-hard. We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity.