Non-negative submodular stochastic probing via stochastic contention resolution schemes

In a stochastic probing problem we are given a universe E, where each element e ∈ E isactive independently with probability pe ∈ [0, 1], and only a probe of e can tell us whether itis active or not. On this universe we execute a process that one by one probes elements —if a probed element is active, then we have to include it in the solution, which we graduallyconstruct. Throughout the process we need to obey inner constraints on the set of elementstaken into the solution, and outer constraints on the set of all probed elements. The objectiveis to maximize a function of successfully probed elements.This abstract model was presented by Gupta and Nagarajan (IPCO’13), and providesa unified view of a number of problems. Adamczyk, Sviridenko, Ward (STACS’14) gavebetter approximation for matroid environments and linear objectives. At the same timethis method was easily extendable to settings, where the objective function was monotonesubmodular. However, the case of non-negative submodular function could not be handledby previous techniques.In this paper we address this problem, and our results are twofold. First, we adaptthe notion of contention resolution schemes of Chekuri, Vondrák, Zenklusen (SICOMP’14)to show that we can optimize non-negative submodular functions in this setting with aconstant factor loss with respect to the deterministic setting. Second, we show a new con-tention resolution scheme for transversal matroids, which yields better approximations in thestochastic probing setting than the previously known tools. The rounding procedure under-lying the scheme can be of independent interest — Bansal, Gupta, Li, Mestre, Nagarajan,Rudra (Algorithmica’12) gave two seemingly different algorithms for stochastic matchingand stochastic k-set packing problems with two different analyses, but we show that oursingle technique can be used to analyze both their algorithms. ∗Supported by the ERC StG project PAAl no. 259515.1