A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem (1604.03084v2)

Abstract: We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\log n)^{1/2}}$ for some constant $c>0$. This yields a nearly tight $n^{1/2 - o(1)}$ bound on the value of this program for any degree $d = o(\log n)$. Moreover we introduce a new framework that we call \emph{pseudo-calibration} to construct Sum of Squares lower bounds. This framework is inspired by taking a computational analog of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.

• The paper establishes nearly optimal lower bounds for the Sum-of-Squares algorithm applied to the planted clique problem.
• It introduces a pseudo-calibration framework that constructs pseudo-distributions as dual certificates by leveraging Bayesian probability.
• The results highlight that, under low-degree constraints, surpassing the detection threshold of n^(1/2 - o(1)) is computationally intractable.

Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

The paper addresses foundational concerns in average-case complexity, specifically focusing on the planted clique problem. This problem, originally conceptualized by Karp and formulated in seminal works by Jerrum and Kucera, is pivotal for testing algorithms under the average-case scenario. It involves detecting a randomly planted clique of size $\omega$ within a graph generated from the Erdős–Rényi model $G(n,1/2)$. The challenge is finding sophisticated polynomial-time algorithms that can surpass the currently effective size bound of $\epsilon \sqrt{n}$ for the clique, especially as $\omega$ approaches $n^{0.5 - o(1)}$.

The paper establishes a nearly optimal lower bound on the performance of the Sum-of-Squares (SoS) algorithm applied to the planted clique problem, setting a limit at $n^{1/2 - o(1)}$ when the algorithm's degree is $d = o(\log n)$. This result is significant especially in light of previous limitations demonstrated with weaker relaxation methodologies like Lovász-Schrijver hierarchies. The research introduces a methodological shift through a pseudo-calibration framework inspired by Bayesian probability theory, adding robustness to the lower bound proof. The pseudo-calibration strategy offers a recipe for deriving pseudo-distributions, acting as dual certificates in the SoS semidefinite programming relaxation, thus enhancing insights into the comparative strength of the SoS hierarchy.

This pseudo-calibration allows the SoS algorithm to be positioned as a proxy for Bayesian computational probabilities within certain constraints. The framework demands that low-degree polynomial approximations of these pseudo-expectations satisfy a particular consistency—rooted in expectations from the planted model $G(n, 1/2, \omega)$. This refinement in handling weak global effects offers a more nuanced view for problems like planted cliques, where these subtle correlations cannot be ignored, contrary to more straightforward scenarios in constraint satisfaction problems seen in prior works.

The results imply computational intractability at the suggested level, aligning with evidence posited by previous research suggesting the hardness of the SoS framework in approximating complex structural configurations like cliques. Thus, these results conjecture strong implications for related problems in network analysis, cryptographic assumptions in planted-based cryptographic constructs, and beyond. They also foresee no polynomially quicker discovery of cliques beyond the current threshold under the SoS constraints.

In sum, this paper not only cements the understanding of the limits of SoS for planted cliques but also provides a blueprint that could inspire future research into constraints and relaxations for other combinatorial optimization problems within artificial intelligence and theoretical computer science. The framework may inspire more generalized adaptions for other average-case scenario problems, efficiently delineating boundaries and potentials for viable algorithmic strategies. The forward-looking inference suggests broader implications and potential refinements through pseudo-distribution strategies, hinting at new realms of analysis within SoS applications.