Product theorems via semidefinite programming

TL;DR

This paper extends the theory of semidefinite programming product theorems to encompass all known examples, explaining when and why these programs behave perfectly under product operations in complexity theory.

Contribution

It generalizes Mittal and Szegedy's theory to include previously unexplained cases, unifying all known semidefinite product theorems.

Findings

Unified theory captures all known semidefinite product theorems

Explains exceptions like Feige-Lovasz parallel repetition and discrepancy direct product

Provides a comprehensive framework for semidefinite program composition

Abstract

The tendency of semidefinite programs to compose perfectly under product has been exploited many times in complexity theory: for example, by Lovasz to determine the Shannon capacity of the pentagon; to show a direct sum theorem for non-deterministic communication complexity and direct product theorems for discrepancy; and in interactive proof systems to show parallel repetition theorems for restricted classes of games. Despite all these examples of product theorems--some going back nearly thirty years--it was only recently that Mittal and Szegedy began to develop a general theory to explain when and why semidefinite programs behave perfectly under product. This theory captured many examples in the literature, but there were also some notable exceptions which it could not explain--namely, an early parallel repetition result of Feige and Lovasz, and a direct product theorem for the discrepancy method of communication complexity by Lee, Shraibman, and Spalek. We extend the theory of Mittal and Szegedy to explain these cases as well. Indeed, to the best of our knowledge, our theory captures all examples of semidefinite product theorems in the literature.