Multiple Planted Structures Below $\sqrt{n}$: An SoS Integrality Gap and an SQ Lower Bound

2026-04-08

AI Summary

This work investigates the computational hardness of detecting multiple disjoint hidden structures in the average case, focusing on the multi-hidden-clique problem where the total planted size is below $\sqrt{n}$. By constructing Sum-of-Squares (SoS) pseudo-expectations with explicit mutual exclusivity constraints and leveraging the statistical query (SQ) complexity framework, the study extends existing SoS and SQ lower bounds—previously established only for a single hidden structure—to the multi-structure setting. It establishes that degree-$d$ SoS fails to certify an upper bound on the total clique size when $kt \leq n^{1/2 - c\sqrt{d/\log n}}$, and that no SQ algorithm can distinguish the planted model from the null with constant advantage when $kt = O(n^{1/2-\delta})$. These results reveal a fundamental computational barrier at this size threshold.

Abstract

We study computational limitations in \emph{multi-plant} average-case inference problems, in which $t$ disjoint planted structures of size $k$ are embedded in a random background on $n$ elements. A natural parameter in this setting is the total planted size $K := kt$. For several classic planted-subgraph problems, including planted clique, existing algorithmic and lower-bound evidence suggests a characteristic computational threshold near $\sqrt{n}$ in the single-plant setting. Our main result is a Sum-of-Squares (SoS) integrality gap for refuting the presence of multiple planted cliques. Specifically, for $G \sim G(n,1/2)$, we construct a degree-$d$ SoS pseudoexpectation for the natural relaxation that maximizes the total size of up to $t$ disjoint cliques. Throughout the regime $kt \le n^{1/2 - c\sqrt{d/\log n}},$ for a universal constant $c>0$, this relaxation achieves objective value $kt(1-o(1))$, and therefore degree-$d$ SoS cannot certify an upper bound below $kt$. This extends the planted-clique SoS lower bounds of~\cite{BarakHKKMP19} to a multi-plant setting with explicit disjointness constraints. As complementary evidence from a different computational model, we prove a lower bound in the statistical query (SQ) framework, extending the results of~\cite{FeldmanGRVX17}. We show that for detecting $t$ disjoint planted $k \times k$ bicliques (equivalently, a row-mixture distribution), when $kt = O(n^{1/2-δ})$ for any fixed $δ>0$, no polynomial-time SQ algorithm can distinguish the planted and null distributions with constant advantage.

Problem

• multi-plant
• computational hardness
• planted clique
• integrality gap
• statistical query

Innovation

• Sum-of-Squares integrality gap
• statistical query lower bound
• multi-plant inference
• disjoint planted cliques
• computational hardness threshold