Sum-of-squares proofs and the quest toward optimal algorithms

TL;DR

This paper explores the connection between the Sum-of-Squares method and the Unique Games Conjecture, highlighting how SOS could lead to optimal algorithms and potentially resolve longstanding complexity questions.

Contribution

It uncovers new links between SOS and UGC, offering tools to bound SOS's runtime and improve guarantees for solving complex optimization problems.

Findings

New bounds on SOS runtime for approximate solutions

Potential for SOS to refute the Unique Games Conjecture

Enhanced understanding of SOS's role in computational complexity

Abstract

In order to obtain the best-known guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible guarantees for a wide range of different problems. The Unique Games Conjecture (UGC) is a tantalizing conjecture in computational complexity, which, if true, will shed light on the complexity of a great many problems. In particular this conjecture predicts that a single concrete algorithm provides optimal guarantees among all efficient algorithms for a large class of computational problems. The Sum-of-Squares (SOS) method is a general approach for solving systems of polynomial constraints. This approach is studied in several scientific disciplines, including real algebraic geometry, proof complexity, control theory, and mathematical programming, and has found applications in fields as diverse as quantum information theory, formal verification, game theory and many others. We survey some connections that were recently uncovered between the Unique Games Conjecture and the Sum-of-Squares method. In particular, we discuss new tools to rigorously bound the running time of the SOS method for obtaining approximate solutions to hard optimization problems, and how these tools give the potential for the sum-of-squares method to provide new guarantees for many problems of interest, and possibly to even refute the UGC.