A Stochastic Probing Problem with Applications

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Abstract

We study a general stochastic probing problem defined on a universe V, where each element e ∈ V is “active” independently with probability p e . Elements have weights {w e :e ∈ V} 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:

  • the set Q of probed elements satisfy an “outer” packing constraint,

  • 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 [1, 2] and Bayesian mechanism design [3], and can also handle more general constraints. As an application, we obtain the first polynomial-time Ω(1/k)-approximate “Sequential Posted Price Mechanism” under k-matroid intersection feasibility constraints, improving on prior work [3-5].