Random Sampling with Removal

Random sampling is a classical tool in constrained optimization. Under favorable conditions, the optimal solution subject to a small subset of randomly chosen constraints violates only a small subset of the remaining constraints. Here we study the following variant that we call random sampling with removal: suppose that after sampling the subset, we remove a fixed number of constraints from the sample, according to an arbitrary rule. Is it still true that the optimal solution of the reduced sample violates only a small subset of the constraints? The question naturally comes up in situations where the solution subject to the sampled constraints is used as an approximate solution to the original problem. We study random sampling with removal in a generalized, completely abstract setting where we assign to each subset R of the constraints an arbitrary set V (R) of constraints disjoint from R; in applications, V (R) corresponds to the constraints violated by the optimal solution subject to only the constraints in R. Furthermore, our results are parametrized by the dimension δ, i.e., we assume that every set R has a subset B of size at most δ with the same set of violated constraints. This is the first time this generalized setting is studied. In this setting, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of k elements. For a large range of values of k, the new upper bounds improve the previously best bounds for LPtype problems, which moreover had only been known in special cases. We show that this bound on special LP-type problems can be derived in the much more general setting of violator spaces, and with very elementary proofs.