Finding Hamilton cycles in random graphs with few queries

Ferber, Asaf; Krivelevich, Michael; Sudakov, Benny; Vieira, Pedro

doi:10.1002/rsa.20679

Cited by 16 publications

(25 citation statements)

References 20 publications

(56 reference statements)

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“…The problem of finding structure in a random graph by adaptively querying the existence of edges between pairs of vertices was recently introduced by Ferber, Krivelevich, Sudakov, and Vieira [4,5].…”

Section: Related Workmentioning

confidence: 99%

“…In particular, they studied finding a Hamilton cycle [4] and finding long paths [5] in the adaptive query model. Conlon, Fox, Grinshpun, and He [3] also study this problem, which they term the subgraph query problem, focusing on finding a copy of a target graph H, and in particular studying the case when H is a small (constant size) clique.…”

Section: Related Workmentioning

confidence: 99%

“…Now probe all edges in T ′ , finding a clique of size approximately log n (w.h.p.). Together with S ′ , this gives a clique of size approximately 4 3 log n (w.h.p.). For q = Θ(n ) the same algorithm applies, with different set sizes.…”

Section: Two Roundsmentioning

confidence: 99%

“…Then S ′ has size approximately 3 log n, while T ′ has size approximately n ∕2 (w.h.p.). The largest clique in T ′ , together with S ′ , gives a clique of size approximately 4 Proof. We first present the algorithm for the case when q = Θ(n).…”

Section: Two Roundsmentioning

confidence: 99%

See 3 more Smart Citations

Finding cliques using few probes

Feige

Gamarnik

Neeman

et al. 2019

Random Struct Algorithms

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Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an n vertex graph, and need to output a clique. We show that if the input graph is drawn at random from Gn,12 (and hence is likely to have a clique of size roughly 2logn), then for every δ<2 and constant ℓ, there is an α<2 (that may depend on δ and ℓ) such that no algorithm that makes nδ probes in ℓ rounds is likely (over the choice of the random graph) to output a clique of size larger than αlogn.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Two Roundsmentioning

confidence: 99%

Section: Two Roundsmentioning

confidence: 99%

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Finding cliques using few probes

Feige

Gamarnik

Neeman

et al. 2019

Random Struct Algorithms

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show abstract

“…Several recent works consider finding structure in a random graph under such an adaptive edge query model. Ferber, Krivelevich, Sudakov, and Vieira studied finding a Hamilton cycle (Ferber et al, 2016) and finding long paths (Ferber et al, 2017), while Conlon et al (2019) studied finding a copy of a fixed target graph (such as a constant size clique). All of these works focus on sparse random graphs.…”

Section: Introductionmentioning

confidence: 99%

Finding a planted clique by adaptive probing

Rácz¹,

Schiffer²

2020

ALEA

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We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let G ∼ G(n, 1/2, k) be a random graph on n vertices with a planted clique of size k. We show that no algorithm that makes at most q = o(n 2 /k 2 + n) adaptive queries to the adjacency matrix of G is likely to find the planted clique. On the other hand, when k ≥ (2 + ε) log 2 n there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making q = O((n 2 /k 2) log 2 n + n log n) adaptive queries. For detection, the additive n term is not necessary: the number of queries needed to detect the presence of a planted clique is n 2 /k 2 (up to logarithmic factors).

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Finding paths in sparse random graphs requires many queries

Ferber

Krivelevich

Sudakov

et al. 2016

Random Struct Algorithms

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We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph G∼G(n,p) in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in G∼G(n,p) when p=1+εn for some fixed constant ε>0 . This random graph is known to have typically linearly long paths. To have ℓ edges with high probability in G∼G(n,p) one clearly needs to query at least Ω(ℓp) pairs of vertices. Can we find a path of length ℓ economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length ℓ=Ωtrue(logtrue(1εtrue)εtrue) with at least constant probability in G∼G(n,p) with p=1+εn must query at least Ωtrue(ℓpεlogtrue(1εtrue)true) pairs of vertices. This is tight up to the logtrue(1εtrue) factor. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 71–85, 2017

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Finding Hamilton cycles in random graphs with few queries

Cited by 16 publications

References 20 publications

Finding cliques using few probes

Finding cliques using few probes

Finding a planted clique by adaptive probing

Finding paths in sparse random graphs requires many queries

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