A Stress-Free Sum-of-Squares Lower Bound for Coloring

TL;DR

This paper establishes a fundamental lower bound on the effectiveness of sum-of-squares semidefinite programs in refuting colorability of random graphs, showing that even high-degree relaxations cannot significantly improve over basic SDP methods.

Contribution

It introduces the first lower bound for coloring random graphs using any single-round sum-of-squares hierarchy, demonstrating limitations of SDP-based refutation methods.

Findings

Sum-of-squares SDP cannot refute k-colorability for k = n^{1/2 + ε} in random graphs.

The lower bound applies to high-degree, polynomial-time SDPs.

The proof uses a novel reduction from coloring to independent set detection.

Abstract

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, a natural $n^{O(\varepsilon^2 \log n)}$-time, degree $O(\varepsilon^2 \log n)$ sum-of-squares semidefinite program cannot refute the existence of a valid $k$-coloring of $G$ for $k = n^{1/2 +\varepsilon}$. Our result implies that the refutation guarantee of the basic semidefinite program (a close variant of the Lov\'asz theta function) cannot be appreciably improved by a natural $o(\log n)$-degree sum-of-squares strengthening, and this is tight up to a $n^{o(1)}$ slack in $k$. To the best of our knowledge, this is the first lower bound for coloring $G(n,1/2)$ for even a single round strengthening of the basic SDP in any SDP hierarchy. Our proof relies on a new variant of instance-preserving non-pointwise complete reduction within SoS from coloring a graph to finding large independent sets in it. Our proof is (perhaps surprisingly) short, simple and does not require complicated spectral norm bounds on random matrices with dependent entries that have been otherwise necessary in the proofs of many similar results [BHK+16, HKP+17, KB19, GJJ+20, MRX20]. Our result formally holds for a constraint system where vertices are allowed to belong to multiple color classes; we leave the extension to the formally stronger formulation of coloring, where vertices must belong to unique colors classes, as an outstanding open problem.