[Paper Review] Approximability and proof complexity

Ryan O’Donnell, Yuan Zhou|arXiv (Cornell University)|Nov 8, 2012

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper establishes that bounded-degree Sum of Squares (SOS) proofs can efficiently certify near-optimal solutions for two prominent hard optimization problems: Balanced-Separator and Max-Cut instances derived from the Khot–Vishnoi Unique-Games construction. It proves that a degree-4 SOS proof refutes the existence of good balanced cuts in Devanur et al.'s integrality gap instances, and a constant-degree SOS proof certifies within 95.2% of the true Max-Cut optimum for Khot–Vishnoi instances, leveraging a novel SOS proof of the KKL Theorem.

ABSTRACT

This work is concerned with the proof-complexity of certifying that optimization problems do \emph{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 system is automatizable using semidefinite programming (SDP), meaning that any $n$-variable degree-$d$ proof can be found in time $n^{O(d)}$. Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "$d/2$-round Lasserre value" 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 instances" for the Unique-Games problem are in fact solved close to optimally 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?

Motivation & Objective

• To investigate the power of bounded-degree Sum of Squares (SOS) proofs in certifying that optimization problems do not have good solutions.
• To determine whether known hard instances for Balanced-Separator and Max-Cut admit efficient SOS refutations.
• To establish that the KKL Theorem admits a degree-4 SOS proof, enabling certification of integrality gaps.
• To analyze the extent to which Khot–Vishnoi Max-Cut instances can be certified via constant-degree SOS proofs.
• To explore the broader mathematical question of whether a constant-degree SOS proof exists for the Central Limit Theorem.

Proposed method

• Leverages the duality between the Lasserre SDP hierarchy and bounded-degree SOS proofs to analyze proof strength.
• Applies a novel SOS proof of the KKL Theorem to certify the integrality gap of Balanced-Separator instances using degree-4 SOS.
• Uses the KKMO reduction to transform Unique-Games instances into Max-Cut instances and analyzes their SOS refutability.
• Employs a local consistency argument via expected label agreement to derive a degree-d SOS refutation from a fractional assignment of value at least ε.
• Combines polynomial identities and degree-bounded SOS reasoning to bound error terms in the SDP relaxation.
• Uses duality and polynomial inequalities to derive a degree-Õ(1/δ²) SOS refutation for Khot–Vishnoi instances with high confidence.

Experimental results

Research questions

• RQ1 Can the Balanced-Separator integrality gap instances of Devanur et al. be certified as having no good solution using a low-degree SOS proof?
• RQ2 Does the KKL Theorem admit a bounded-degree SOS proof, and can it be used to certify integrality gaps in optimization problems?
• RQ3 To what extent can the Khot–Vishnoi Max-Cut integrality gap instances be certified via constant-degree SOS proofs?
• RQ4 Is there a constant-degree SOS proof of the Central Limit Theorem, and what would that imply for proof complexity?
• RQ5 What is the minimal SOS degree required to certify that the optimum value of a given Max-Cut instance is bounded away from the true optimum?

Key findings

• A degree-4 SOS proof can certify that the optimal value of Balanced-Separator instances from Devanur et al. exceeds the integrality gap, resolving a key open question.
• The KKL Theorem admits a degree-4 SOS proof, which is instrumental in certifying the hardness of Balanced-Separator instances.
• For Khot–Vishnoi Max-Cut instances with ρ₀ ≈ −0.689, a constant-degree SOS SDP certifies the optimum value to within 95.2% of the true value.
• For general ρ ∈ (−1, 0), the degree-O(1) SOS SDP certifies a value within 93.1% of the optimum, with the factor derived from the arccos(ρ)/π expression.
• The paper shows that a degree-Õ(1/δ²) SOS refutation exists for Khot–Vishnoi instances when the fractional assignment value is at least ε, with error terms controlled via polynomial identities.
• The results suggest that constant-degree SOS proofs may be powerful enough to capture central limit phenomena, raising the open question of whether a constant-degree SOS proof of the Central Limit Theorem exists.