On the Lasserre/Sum-of-Squares Hierarchy with Knapsack Covering Inequalities

The Lasserre/Sum-of-Squares hierarchy is a systematic procedure to strengthen LP relaxations by constructing a sequence of increasingly tight formulations. For a wide variety of optimization problems, this approach captures the convex relaxations used in the best available approximation algorithms. The capacitated covering IP is an integer program of the form min{cx : Ux ≥ d, 0 ≤ x ≤ b, x ∈ Z+}, where all the entries of c, U and d are nonnegative. One difficulty in approximating the capacitated covering problems lies in the fact that the ratio between the optimal IP solution to the optimal LP solution can be as large as ||d||∞, even when U consists of a single row (i.e. the Min Knapsack problem). Currently the strongest linear program relaxation is obtained by adding (exponentially many) valid knapsack cover (KC) inequalities introduced by Wolsey [44], which yields a very powerful way to cope with these problems. For the Min Knapsack problem we prove that even after non-constant number of levels the Lasserre/Sum-of-Squares hierarchy does not improve the integrality gap of 2 implied by the starting (KC) inequalities. Furthermore, we show that the integrality gap of the relaxation stays M for the special case of ∑ i xi ≥ 1/M when starting with the standard Min Knapsack polytope. We note that Min Knapsack admits an FPTAS and our results quantify a fundamental weakness of the Lasserre/Sum-of-Squares hierarchy for this basic problem.