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

TL;DR

This paper establishes computational hardness results for detecting multiple disjoint planted structures in random graphs, extending known single-plant thresholds to multi-plant scenarios using SoS and SQ frameworks.

Contribution

It provides the first Sum-of-Squares integrality gap and SQ lower bounds for multi-plant planted structures, revealing fundamental computational limitations.

Findings

Degree-$d$ SoS cannot certify upper bounds below $kt$ in certain regimes.

No polynomial-time SQ algorithm can distinguish planted bicliques when $kt = O(n^{1/2- ext{delta}})$.

Extends single-plant planted clique bounds to multi-plant settings with explicit disjointness constraints.

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 $kt = O(n^{1/2-\delta})$ for any fixed $\delta>0$, no polynomial-time SQ algorithm can distinguish the planted and null distributions with constant advantage.