Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, May 17, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Aaron Potechin (KTH Stockholm)

Sum of Squares Lower Bounds from Symmetry and a Good Story

The sum of squares hierarchy is a hierarchy of semidefinite programs which has the three advantages of being broadly applicable (it can be applied whenever the problem can be phrased in terms of polynomial equations over R), powerful (it captures the best known algorithms for several problems including max cut, sparsest cut, and unique games), and in some sense, simple (all it is really using is the fact that squares are non-negative over R). The sum of squares hierarchy can also be viewed as the Positivstellensatz proof system. However, the sum of squares does have a few known weaknesses. One such weakness, as shown by Grigoriev's sum of squares lower bound on the knapsack problem, is that the sum of squares hierarchy is poor at capturing integer arguments. In this talk, we further explore this weakness for symmetric problems. In particular, we describe an analogous sum of squares lower bound for the following Turan type problem: Minimize the number of triangles a graph of a given edge density contains. We then describe the core of the lower bound, which is a simple prover/adversary game, and sketch how this game implies sum of squares lower bounds.

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