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Integrality Gaps for Bounded Color Matchings

We investigate the performance of two common lift-and-project techniques, the Sherali-Adams (SA) and the Balas, Ceria and Cornu\'ejols (BCC) hierarchies, on the natural linear relaxation of the Bounded Color Matching polyhedron and generalizations. We prove the following unconditional inapproximability results: even a large family of linear programs, generated by an asymptotically linear number of rounds of the Sherali-Adams hierarchy or a linear number of rounds of the BCC hierarchy, is not enough to improve the integrality gap of the natural LP relaxation even in bipartite graphs. We complement these results by showing in a non-algorithmic way that if we exclude certain sub-structures from our instance graphs, then the integrality gap of the natural linear formulation improves.