Submodular Stochastic Probing with Prices

Abstract

We introduce Stochastic Probing with Prices (SPP), a variant of the Stochastic Probing (SP) model in which we must pay a price to probe an element. A SPP problem involves two set systems (N , Iin) and (N , Iout) where each e ∈ N is active with probability pe. To discover whether an element e is active, it must be probed by paying the price ∆e. If an element is probed and is active, then it is irrevocably added to the solution. Moreover, at all times, the set of probed elements must lie in Iout, and the solution (the set of probed and active elements) must lie in Iin. The goal is to maximize a submodular set function f minus the cost of the probes. We give a bi-criteria approximation algorithm to the online version of this problem, in which the elements are shown to the algorithm in a possibly adversarial order. Our results translate to state-of-the-art approximations for the traditional (online) stochastic probing problem. * benjamin.chugg@maths.ox.ac.uk. Supported by a research internship at RIKEN. † takanori.maehara@riken.jp 1 We leave the question of whether a pair of socks is the optimal gift for another paper. 2 I ⊂ 2 N is downward-closed if A ∈ I and B ⊂ A implies B ∈ I.