Approximability and proof complexity
Ryan W. O’Donnell, Yuan Zhou · Symposium on Discrete Algorithms · 2013
This work is concerned with the proof-complexity of certifying that optimization problems do not have good solutions. Specifically we consider bounded-degree Sum of Squares (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof is automatizable using semidefinite programming (SDP), meaning that any n-variable degree-d proof can be found in time nO(d). Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the d/2-round Lasserre of an optimization problem is equal to the best bound provable using a degree-d SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known hard for the Unique-Games problem are in fact optimally solved by a constant level of the Lasserre SDP hierarchy.We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 (> .878) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?