Finding Planted Cliques in Sublinear Time

Mardia, Jay; Asi, Hilal; Chandrasekher, Kabir Aladin

doi:10.48550/arxiv.2004.12002

Cited by 4 publications

(9 citation statements)

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“…If we were looking at a more fine-grained picture, a gap does emerge between detection and recovery. [MAC20] showed that for k = ω(n 2/3 ), planted clique detection can be solved in o(n) time. However, by results of [RS19] we know that any recovery algorithm must require Ω(n) time.…”

Section: Recoverymentioning

confidence: 99%

“…As we see in Section 1.1, a long line of work on restricted models of computation has been used to show hardness of the planted clique problem. On the other hand, this work (like [MAC20]) studies a restricted model of computation with the primary aim of making algorithmic progress and further pushing down the complexity of successful planted clique algorithms.…”

Section: Recoverymentioning

confidence: 99%

“…Our key idea, inspired by [MAC20], is to implement a better filtering step that gets more than a constant factor of improvement in each round. The filtering / thresholding of [DGGP14] does not utilise the size of the planted clique k at all, other than the fact that it is Ω( √ n).…”

Section: (Nearly) Linear Time Algorithmsmentioning

confidence: 99%

“…The filtering / thresholding of [DGGP14] does not utilise the size of the planted clique k at all, other than the fact that it is Ω( √ n). On the other hand, [MAC20] uses knowledge of k to design a single round filtering algorithm that recovers the planted clique for clique sizes ω( √ n log log n) = k = o( √ n log n) in sublinear time. By appropriately implementing this idea in our context for multiple rounds, we can utilize knowledge of the number of clique vertices in V t−1 , |V t−1 ∩ K|, to make sure that in going from V t−1 to V t the following happens.…”

Section: (Nearly) Linear Time Algorithmsmentioning

confidence: 99%

“…If we can show logarithmic space algorithms exist above the polynomial time threshold k = Ω( √ n), we will show that the statisticalcomputational gap holds even for space complexity. Also, as observed in [MAC20], the work [RS19] implies that if k = O(n 1−δ ) for some constant δ > 0, solving the planted clique problem (either detection or recovery) takes at least poly(n) time. Hence any successful planted clique algorithm requires Ω(log n) bits of space, which makes the quest for a O(log n) space algorithm also the quest for an optimal deterministic algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Is the space complexity of planted clique recovery the same as that of detection?

Mardia¹

2020

Preprint

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We study the planted clique problem in which a clique of size k is planted in an Erdős-Rényi graph G(n, 12 ), and one is interested in either detecting or recovering this planted clique. This problem is interesting because it is widely believed to show a statistical-computational gap at clique size k = Θ( √ n), and has emerged as the prototypical problem with such a gap from which average-case hardness of other statistical problems can be deduced. It also displays a tight computational connection between the detection and recovery variants, unlike other problems of a similar nature. This wide investigation into the computational complexity of the planted clique problem has, however, mostly focused on its time complexity. In this work, we ask-Do the statistical-computational phenomena that make the planted clique an interesting problem also hold when we use 'space efficiency' as our notion of computational efficiency? It is relatively easy to show that a positive answer to this question depends on the existence of a O(log n) space algorithm that can recover planted cliques of size k = Ω( √ n). Our main result comes very close to designing such an algorithm. We show that for k = Ω( √ n), the recovery problem can be solved infor any constant integer ℓ > 0, the space usage is O(log n) bits. 2. If k = Θ( √ n), the space usage is O(log * n • log n) bits. Our result suggests that there does exist an O(log n) space algorithm to recover cliques of size k = Ω( √ n), since we come very close to achieving such parameters. This provides evidence that the statistical-computational phenomena that (conjecturally) hold for planted clique time complexity also (conjecturally) hold for space complexity.

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Section: Recoverymentioning

confidence: 99%

Section: Recoverymentioning

confidence: 99%

Section: (Nearly) Linear Time Algorithmsmentioning

confidence: 99%

Section: (Nearly) Linear Time Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Is the space complexity of planted clique recovery the same as that of detection?

Mardia¹

2020

Preprint

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A Planted Clique Perspective on Hypothesis Pruning

Lin

Murali

Karaman

2022

IEEE Robot. Autom. Lett.

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Hypothesis pruning is an important prerequisite while working with outlier-contaminated data in many computer vision problems. However, the underlying random data structures are barely explored in the literature, limiting designing efficient algorithms. To this end, we provide a novel graph-theoretic perspective on hypothesis pruning exploiting invariant structures of data. We introduce the planted clique model, a central object in computational statistics, to investigate the information-theoretical and computational limits of the hypothesis pruning problem.In addition, we propose an inductive learning framework for finding hidden cliques that learns heuristics on synthetic graphs with planted cliques and generalizes to real vision problems. We present competitive experimental results with large runtime improvement on synthetic and widely used vision datasets to show its efficacy.

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Logspace Reducibility From Secret Leakage Planted Clique

Mardia

2021

Preprint

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Finding Planted Cliques in Sublinear Time

Cited by 4 publications

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Is the space complexity of planted clique recovery the same as that of detection?

Is the space complexity of planted clique recovery the same as that of detection?

A Planted Clique Perspective on Hypothesis Pruning

Logspace Reducibility From Secret Leakage Planted Clique

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