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

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 $c&gt;0$, 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 $kt = O(n<sup>{1/2-δ})$ for any fixed $δ&gt;0$, no polynomial-time SQ algorithm can distinguish the planted and null distributions with constant advantage.

• The paper shows that distributing signal across t planted cliques does not lower computational limits below the sqrt(n) threshold.
• It constructs a low-degree SoS pseudoexpectation that satisfies stringent disjointness constraints, achieving nearly kt objective value in G(n,1/2).
• The analysis establishes SQ lower bounds, demonstrating that when kt = O(n^(1/2-δ)), any efficient algorithm must use polynomial query complexity.

Multiple Planted Structures Below $\sqrt{n}$: Integrality Gaps and Statistical Query Lower Bounds

Problem Formulation and Context

The paper investigates the computational complexity of average-case inference problems involving multiple disjoint planted structures in random graphs and related models. Specifically, the focus is on the detection and refutation of $t$ disjoint planted cliques, each of size $k$, embedded in a random background (typically $G(n,1/2)$). The parameter of interest is the total planted size $K=kt$, and the paper explores whether distributing the signal across multiple small planted objects changes the established computational threshold in the single-plant setting—most prominently, the $\sqrt{n}$ barrier for planted clique detection.

Historically, the single planted clique problem exhibits a pronounced statistical–computational gap: while detection becomes information-theoretically possible for $k = 2\log_2 n$, no known polynomial-time algorithm effectively finds cliques below $k = \Theta(\sqrt{n})$, and several lower bounds (notably for Sum-of-Squares (SoS) and Statistical Query (SQ) models) support the conjectured hardness in this regime. The extension to multiple plants, particularly when the individual structures are small but collectively sum to a large $K$, is nontrivial as naive reductions degrade the thresholds.

Sum-of-Squares Integrality Gaps for Multi-Plant Planted Clique

The principal technical contribution is the construction of a low-degree SoS pseudoexpectation for the relaxation optimizing the sum of at most $t$ disjoint cliques in the random graph $t$0. The constructed pseudoexpectation is shown to satisfy all constraints of the multi-plant SoS program (Booleanity, clique constraints on non-edges, and stringent variable-disjointness), and, critically, achieves an objective value $t$1 throughout the regime:

$t$2

where $t$3 is the SoS degree and $t$4 is a universal constant.

Strongly, this result matches the single-plant planted clique SoS lower bound, demonstrating that even with $t$5 planted cliques, no degree-$t$6 SoS algorithm can refute $t$7 in the above regime. The technical novelty lies in extending the moment-matrix/ribbon-based analysis of low-degree pseudoexpectations to handle explicit global disjointness constraints, where each vertex is allowed to participate in at most one planted structure. The analysis leverages a local calibration and truncation scheme on the Fourier basis for $t$8 and iterates the separator-based factorization for moment matrix PSDness, following and refining the approaches of [Barak et al., SIAM J. Comput., 2019] and [Meka, Potechin, Wigderson, STOC 2015]. The main ribbon and separator combinatorics remain robust under the multi-plant generalization, owing to the sparsity of relevant Fourier coefficients and the control provided by the planted size $t$9.

A key implication is that even in the multi-plant scenario—where the naive embedding strategy is susceptible to threshold degradation due to vertex partitioning—the SoS lower bound is sharp with respect to $k$0, and any potential computational advantage from distributing the planted signal is tightly limited.

Statistical Query Lower Bound for Multi-Plant Planted Biclique Detection

In parallel, the paper establishes rigorous lower bounds for the complexity of detecting $k$1 disjoint planted $k$2 bicliques (row-mixture distributions) in the Statistical Query (SQ) model. By constructing a family of “challenging” distributions corresponding to all orderings of $k$3 disjoint $k$4-subsets (the set of plantings), the authors show that, for

$k$5

for any fixed $k$6, no randomized polynomial-time SQ algorithm can distinguish a planted multi-biclique instance from the null model with constant statistical advantage.

The proof follows the statistical dimension framework of [Feldman et al., JACM 2017], quantifying the average correlation among the constructed planted distributions and leveraging hypergeometric and Poisson tail bounds to show that small overlaps dominate the average correlation within sufficiently large subfamilies. Notably, the analysis targets the equal-size, fully disjoint case, making stronger demands than reductions from single-plant instances would permit.

Crucially, the established computational threshold in the SQ model is again governed by the total planted size $k$7, not $k$8 or $k$9 individually. The construction excludes the existence of efficient algorithms even when detection is straightforward in the information-theoretic sense for larger plants.

Numerical and Structural Claims

• SoS integrality gap holds for all $G(n,1/2)$0, matching the best known lower bounds for single planted cliques.
• SQ lower bound: for $G(n,1/2)$1, polynomial query complexity becomes necessary for distinguishing plantings, and the statistical dimension grows as $G(n,1/2)$2 in appropriate regimes.
• The paper directly confirms that multi-plant refutation is at least as hard as single-plant, even under explicit disjointness, and demonstrates the inadequacy of partition/tensor reductions for achieving tight thresholds in the multi-plant regime.

Implications and Future Directions

From a theoretical standpoint, the results decisively characterize the computational barriers for multiple disjoint planted structures in random graphs and mixture models, showing that distributing the planted signal amongst many structures does not lower the fundamental SoS and SQ–based algorithmic barriers below $G(n,1/2)$3, modulo small-degree and logarithmic corrections. This sharply constrains the possibilities for algorithmic improvement exploiting signal distribution across many weak structures.

Practically, these findings reinforce the generality and limitations of SoS and SQ paradigms in average-case detection—any future attempts to break the $G(n,1/2)$4 barrier for planted clique–type problems, whether with multiple plants or otherwise, must fundamentally escape these frameworks or violate their underlying assumptions.

Potential future research directions include:

• Tightening the characterization of detection thresholds in transitional regimes where $G(n,1/2)$5 by information-theoretic and algorithmic analyses.
• Extending these SoS and SQ integrality gap/lower bound techniques to more complex combinatorial structures (beyond cliques and bicliques), such as motifs, hypergraphs, or heterogeneously sized plants.
• Investigating whether alternative algorithmic paradigms (e.g., sublinear-time algorithms, non-SQ learning) could break the demonstrated barriers in special cases.

Conclusion

This paper establishes tight computational lower bounds for the detection and refutation of multiple disjoint planted structures below the $G(n,1/2)$6 regime, both in the Sum-of-Squares and Statistical Query models (2604.07278). The results demonstrate that, even with $G(n,1/2)$7 structures, the total planted size $G(n,1/2)$8 remains the governing parameter for computational hardness, and neither the SoS relaxation nor statistical query approaches can overcome the classic planted clique thresholds in this richer, multi-plant setting. These advances clarify the limits of algorithmic approaches to planted structure detection and provide a conceptual framework for understanding adversarial planting and composite inference tasks.