Aurko Roy

The matching problem has no small symmetric SDP

Yannakakis (Proceedings of the STOC, pp 223–228, 1988; J Comput Syst Sci 43(3):441–466, 1991. doi:10.1016/0022-0000(91)90024-Y) showed that the matching problem does not have a small symmetric linear program. Rothvoß (Proceedings of the STOC, pp 263–272, 2014) recently proved that any, not necessarily symmetric, linear program also has exponential size. In light of this, it is natural to ask whether the matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization. We answer this question negatively for symmetric SDPs: any symmetric SDP for the matching problem has exponential size. We also show that an O(k)-round Lasserre SDP relaxation for the asymmetric metric traveling salesperson problem yields at least as good an approximation as any symmetric SDP relaxation of size

Strong reductions for extended formulations

n^{k}

Adversarial Patch.

nk. The key technical ingredient underlying both these results is an upper bound on the degree needed to derive polynomial identities that hold over the space of matchings or traveling salesperson tours.

Hierarchical Clustering via Spreading Metrics

We generalize the reduction mechanism between linear programming problems from [1] in two ways 1 relaxing the requirement of affineness, and 2 extending to fractional optimization problems. As applications we provide several new LP-hardness and SDP-hardness results, e.g., for the SparsestCut problem, the BalancedSeparator problem, the MaxCut problem and the Matching problem on 3-regular graphs. We also provide a new, very strong Lasserre integrality gap for the IndependentSet problem, which is strictly greater than the best known LP approximation, showing that the Lasserre hierarchy does not always provide the tightest SDP relaxation.

Reinforcement Learning under Model Mismatch

AbstractWe generalize the reduction mechanism for linear programming problems and semidefinite programming problems from Braun et al. (Inapproximability of combinatorial problems via small LPs and SDPs, 2015) in two ways (1) relaxing the requirement of affineness, and (2) extending to fractional optimization problems. As applications we provide several new LP-hardness and SDP-hardness results, e.g., for the problem, the problem, and the problem and show how to extend ad-hoc reductions between Sherali–Adams relaxations to reductions between LPs.