[Paper Review] Sum-of-squares proofs and the quest toward optimal algorithms

Boaz Barak, David Steurer|arXiv (Cornell University)|Apr 21, 2014

Computational Geometry and Mesh Generation47 references96 citations

TL;DR

This paper investigates the Sum-of-Squares (SOS) hierarchy as a unified framework for designing optimal approximation algorithms, linking it to the Unique Games Conjecture (UGC) and Small-Set Expansion Hypothesis (SSEH). It shows that constant-degree SOS proofs can certify key inequalities underlying hard instances for UGC/SSEH, suggesting these problems may not be as hard as believed and potentially refuting the conjectures if stronger degree bounds are achieved.

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.

Motivation & Objective

• To investigate whether the Sum-of-Squares (SOS) method can provide optimal approximation guarantees for a wide range of computational problems.
• To examine the connection between the SOS method and the Unique Games Conjecture (UGC), particularly in terms of algorithmic performance and hardness of approximation.
• To determine whether constant-degree SOS proofs can refute the UGC or SSEH by solving known hard instances.
• To develop tools for bounding the degree of SOS proofs required to certify polynomial inequalities arising in optimization.
• To assess whether the SOS method can achieve better approximation guarantees than existing algorithms for problems like graph expansion and independent set.

Proposed method

• Uses the SOS method to systematically analyze and certify polynomial inequalities that arise in optimization problems such as graph expansion and independent set.
• Applies degree- and dimension-based bounds on the SOS proof degree, leveraging the fact that low-degree SOS proofs exist for subspaces without sparse vectors.
• Employs the Bonami-Beckner-Gross hypercontractivity theorem (in the form of Eq. 6) as a core component to certify that certain subspaces do not contain sparse vectors.
• Leverages the fact that if a subspace W satisfies E[x_i^4] ≤ 9^k (E[x_i^2])^2 for all x ∈ W_k, then it admits a constant-degree SOS proof, implying robustness against sparse vectors.
• Uses the SOS degree bound to derive running time guarantees: for distinguishing small-set expansion, SOS achieves time exp(O(n^τ)) for τ → 0 as ε → 0.
• Analyzes the implications of SOS proof degree on the complexity of problems like Unique Games and Small-Set Expansion, showing that constant-degree SOS can solve known hard instances.

Experimental results

Research questions

• RQ1 Can the Sum-of-Squares method provide optimal approximation guarantees across a broad class of problems, as predicted by the Unique Games Conjecture?
• RQ2 Are the known hard instances for the Unique Games Conjecture actually solvable by constant-degree SOS proofs?
• RQ3 What is the minimal SOS degree required to certify that a subspace does not contain a sparse vector, and how does this relate to computational complexity?
• RQ4 Can the SOS method refute the Small-Set Expansion Hypothesis by solving its known hard instances with low-degree proofs?
• RQ5 To what extent do existing hardness results for problems like independent set and expansion rely on assumptions that may be invalidated by SOS-based algorithms?

Key findings

• Constant-degree SOS proofs can certify the Bonami-Beckner-Gross hypercontractivity inequality, which underlies many known hard instances for the UGC and SSEH.
• The existence of constant-degree SOS proofs for key inequalities implies that these hard instances may not be as difficult as previously thought, challenging the validity of the UGC and SSEH.
• For the small-set expansion problem, the SOS method can distinguish between graphs with expansion ≤ ε and those with expansion ≥ 1−ε in time exp(O(n^τ)) for τ → 0 as ε → 0, significantly improving over brute-force enumeration.
• If the SOS degree bound in Corollary 5.3 could be improved to eliminate dependence on d (dimension), this would refute the Small-Set Expansion Hypothesis.
• The paper shows that all known hardness results for UGC/SSEH based on specific hard instances can be refuted by constant-degree SOS proofs, suggesting no known hard instances remain for the SOS algorithm.
• The SOS method may unify a wide range of approximation algorithms, potentially achieving optimal guarantees without tailoring to individual problems, as suggested by the UGC.