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 @math be a random graph on @math vertices with a planted clique of size @math . We show that no algorithm that makes at most @math adaptive queries to the adjacency matrix of @math is likely to find the planted clique. On the other hand, when @math there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making @math adaptive queries. For detection, the additive @math term is not necessary: the number of queries needed to detect the presence of a planted clique is @math (up to logarithmic factors).
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 @cite_5 and finding long paths @cite_10 , while Conlon, Fox, Grinshpun, and He @cite_12 studied finding a copy of a fixed target graph (such as a constant size clique). All of these works focus on sparse random graphs.
@cite_5 @cite_10 @cite_12
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We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of G(n,p) in order to typically find a subgraph possessing a given target property. We show that if p≥lnn+lnlnn+ω(1)n, then one can find a Hamilton cycle with high probability after exposing (1+o(1))n edges. Our result is tight in both p and the number of exposed edges. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016 We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition probabilities (STP), for the average number of discovered edges as a function of time. We show that for STP walks site and edge exploration obey the same scaling nλ as a function of time n. Therefore, edge exploration on graphs with many loops is always lagging compared to site exploration, the revealed graph being sparse until almost all nodes have been discovered. We then introduce the edge explorer model (EEM), which presents a novel class of adaptive walks, that perform faithful network discovery even on dense networks. We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1 2), and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k>cn0.5, for any fixed c>0, thus improving the trivial case k>cn0.5(log n)0.5. The algorithm is based on the spectral properties of the graph. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 457–466, 1998