Probabilistic Partial Set Covering with an Oracle for Chance Constraints

Abstract

We consider a class of chance-constrained combinatorial optimization problems, which we refer to as probabilistic partial set-covering (PPSC) problems. Given a prespecified risk level $\epsilon\in[0,1]$, the PPSC problem aims to find the minimum cost selection of subsets of items such that a target number of items is covered with probability at least $1-\epsilon$. We show that PPSC admits an efficient probability oracle that computes the coverage probability exactly, under certain distributions of the random variables representing the coverage relation. Using this oracle, we give a compact mixed-integer program that solves the PPSC problem for a special case. For large-scale instances for which an exact oracle-based method exhibits slow convergence, we propose a sampling-based approach that exploits the special structure of PPSC. The oracle can be used as a tool for checking and fixing the feasibility of the solution given by this approach. In particular, we introduce a new class of facet-defining inequalities for a submodular substructure of PPSC, and show that a sampling-based algorithm coupled with the probability oracle effectively provides high-quality feasible solutions to the large-scale test instances.