A stochastic probing problem with applications

We study a general stochastic probing problem defined on a universe V, where each element e ∈ V is “active ” independently with probability pe. Elements have weights {we: e ∈ V} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the pe 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 [12, 3] and Bayesian mechanism design [9], and can also handle more general constraints. As an application, we ob-tain the first polynomial-time Ω(1/k)-approximate “Sequential Posted Price Mechanism ” under k-matroid intersection feasibility constraints, improving on prior work [9, 25, 19].