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1. Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs

   Afonso S Bandeira, Jess Banks (October 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   Abstract We study the problem of efficiently refuting the k-colorability of a graph, or equivalently, certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, suggesting that there is no polynomial-time algorithm that improves upon a classical spectral algorithm. Our evidence takes the form of a "computationally-quiet planting": we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model. 

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2. Random Subgraph Detection Using Queries

   Huleihel, W; Mazumdar, A; Pal, S (April 2024, Journal of machine learning research)

   The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on n vertices. Under the null hypothesis, the graph is a realization of an Erdös-R{\'e}nyi graph with edge probability (or, density) q. Under the alternative, there is a subgraph on k vertices with edge probability p>q. The statistical as well as the computational barriers of this problem are well-understood for a wide range of the edge parameters p and q. In this paper, we consider a natural variant of the above problem, where one can only observe a relatively small part of the graph using adaptive edge queries. For this model, we determine the number of queries necessary and sufficient (accompanied with a quasi-polynomial optimal algorithm) for detecting the presence of the planted subgraph. We also propose a polynomial-time algorithm which is able to detect the planted subgraph, albeit with more queries compared to the above lower bound. We conjecture that in the leftover regime, no polynomial-time algorithms exist. Our results resolve two open questions posed in the past literature. 

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3. Detection-Recovery and Detection-Refutation Gaps via Reductions from Planted Clique

   Bresler, Guy; Jiang, Tianze (July 2023, Proceedings of Thirty Sixth Conference on Learning Theory)

   Planted Dense Subgraph (PDS) problem is a prototypical problem with a computational-statistical gap. It also exhibits an intriguing additional phenomenon: different tasks, such as detection or re- covery, appear to have different computational limits. A detection-recovery gap for PDS was sub- stantiated in the form of a precise conjecture given by Chen and Xu (2014) (based on the parameter values for which a convexified MLE succeeds), and then shown to hold for low-degree polynomial algorithms by Schramm and Wein (2022) and for MCMC algorithms for Ben Arous et al. (2020). In this paper we demonstrate that a slight variation of the Planted Clique Hypothesis with secret leakage (introduced in Brennan and Bresler (2020)), implies a detection-recovery gap for PDS. In the same vein, we also obtain a sharp lower bound for refutation, yielding a detection-refutation gap. Our methods build on the framework of Brennan and Bresler (2020) to construct average-case reductions mapping secret leakage Planted Clique to appropriate target problems. 

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4. Algorithms Approaching the Threshold for Semi-random Planted Clique

   https://doi.org/10.1145/3564246.3585184  

   Buhai, Rares-Darius; Kothari, Pravesh K.; Steurer, David (January 2023, ACM Symposium on Theory of Computing, STOC)

   We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian 2001. The previous best algorithms for this model succeed if the planted clique has size at least n2/3 in a graph with n vertices (Mehta, Mckenzie, Trevisan 2019 and Charikar, Steinhardt, Valiant 2017). Our algorithms work for planted-clique sizes approaching n1/2 -- the information-theoretic threshold in the semi-random model (Steinhardt 2017) and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige 2019 and Steinhardt 2017. Our algorithms are based on higher constant degree sum-of-squares relaxation and rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite Erdős--Rényi random graphs into algorithms for semi-random planted clique. The use of a higher-constant degree sum-of-squares is essential in our setting: we prove a lower bound on the basic SDP for certifying bicliques that shows that the basic SDP cannot succeed for planted cliques of size k=o(n2/3). We also provide some evidence that the information-computation trade-off of our current algorithms may be inherent by proving an average-case lower bound for unbalanced bicliques in the low-degree-polynomials model. 

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5. Algorithmic Decorrelation and Planted Clique in Dependent Random Graphs: The Case of Extra Triangles

   Bresler, Guy; Guo, Chenghao; Polyanskiy, Yury (November 2023, Foundations of Computer Science)

   We aim to understand the extent to which the noise distribution in a planted signal-plus-noise problem impacts its computational complexity. To that end, we consider the planted clique and planted dense subgraph problems, but in a different ambient graph. Instead of Erd\H{o}s-R\'enyi , which has independent edges, we take the ambient graph to be the \emph{random graph with triangles} (RGT) obtained by adding triangles to . We show that the RGT can be efficiently mapped to the corresponding , and moreover, that the planted clique (or dense subgraph) is approximately preserved under this mapping. This constitutes the first average-case reduction transforming dependent noise to independent noise. Together with the easier direction of mapping the ambient graph from Erd\H{o}s-R\'enyi to RGT, our results yield a strong equivalence between models. In order to prove our results, we develop a new general framework for reasoning about the validity of average-case reductions based on \emph{low sensitivity to perturbations}. 

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