LP-based Approximations for Disjoint Bilinear and Two-Stage Adjustable Robust Optimization

Abstract

We consider the class of disjoint bilinear programs max {x T y | x ∈ X , y ∈ Y} where X and Y are packing polytopes. We present an O( log log m1 log m1 log log m2 log m2 )-approximation algorithm for this problem where m 1 and m 2 are the number of packing constraints in X and Y respectively. In particular, we show that there exists a near-optimal solution (x, y) such that x and y are "nearintegral". We give an LP relaxation of this problem from which we obtain the near-optimal near-integral solution via randomized rounding. As an application of our techniques, we present a tight approximation for the two-stage adjustable robust problem with covering constraints and right-hand side uncertainty where the separation problem is a bilinear optimization problem. In particular, based on the ideas above, we give an LP restriction of the two-stage problem that is an O( log n log log n log L log log L )-approximation where L is the number of constraints in the uncertainty set. This significantly improves over state-of-the-art approximation bounds known for this problem.