Vanishing Ideals and the Computational Tractability of Sum-of-Squares over Boolean Domains

2026-09-29

AI Summary

This study addresses the computational tractability and exact rational solving challenges of Sum-of-Squares optimization over the Boolean hypercube. To overcome these limitations, this work constructs enhanced moment semidefinite programs by exploiting truncated vanishing identities, and introduces geometric realization and transfer theorems that integrate bit-complexity frameworks, Gröbner bases, the ellipsoid method, and spectral coefficient bounds. The main contributions demonstrate that polynomial systems closed under Min/Max operations are solvable in polynomial time with constructible certificates. Furthermore, it establishes a polynomial-time algorithm for searching rational proofs at fixed relaxation degrees, achieving exact rational feasibility and arbitrary-precision optimization. These results effectively break through the longstanding semidefinite programming bottleneck for Boolean Sum-of-Squares hierarchies.

Abstract

Building on the bit-complexity framework of Raghavendra-Weitz and the moment-SOS criteria of Gribling-Polak-Slot, we study the effective use of truncated vanishing identities over Boolean polynomial systems. A complete, constructible identity space gives an augmented moment SDP that can be optimized with exact rational feasibility and arbitrary additive accuracy. We give a self-contained geometric implementation: an explicit affine reduction and a simplex of Boolean evaluations supply the radius bounds required by rational ellipsoids. The identity space can be constructed directly or extracted from a supplied graded Groebner basis, including one supplied at a higher truncation degree. An explicit transfer theorem connects this augmented formulation to the original system. Two-sided SoS derivations eliminate the added equality axioms from certificates, with controlled degree and coefficient growth, and imply containment of a projected higher-level moment relaxation in the augmented body. Together with spectral coefficient bounds, this gives polynomial-time search for rational proofs with an additive perturbation. For Min-closed linear systems, propagation constructs the identity space in polynomial time at fixed degree and gives degree-(4t+4) certificates for both signs of each degree-t basis element. Consequently, a rational degree-(8d+4) proof of f + epsilon >= 0 can be found in polynomial time for fixed d whenever f >= 0 has a degree-2d proof. The augmented degree-2d moment SDP can be optimized in polynomial time with exact rational feasibility and a comparison to the original degree-(8d+4) relaxation. Boolean complementation gives the same results for Max-closed systems, including generalized packing and covering. Min-closed systems thus provide a concrete application of the general criteria. All complexity bounds are in the Turing model.

Problem

• Vanishing Ideals
• Sum-of-Squares
• Boolean Domains
• Computational Tractability
• Moment SDP

Innovation

• Vanishing Ideals
• Sum-of-Squares
• Boolean Domains
• Moment SDP
• Computational Tractability