Cliques In Random Graphs Research Articles

Random cliques in random graphs and sharp thresholds for F$$ F $$‐factors

AbstractWe show that for each , in a density range extending up to, and slightly beyond, the threshold for a ‐factor, the copies of in the random graph are randomly distributed, in the (one‐sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus Jeff Kahn's recent asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem implies a corresponding bound for the threshold for to contain a ‐factor. The case is more difficult, and has been settled by Annika Heckel. We also prove a corresponding result for ‐factors in random ‐uniform hypergraphs, as well as (in some cases weaker) generalizations replacing by certain other (hyper)graphs.

Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups

The clique number of a random graph in the Erdös–Rényi model [Formula: see text] yields a random variable which takes values asymptotically almost surely (as [Formula: see text]) within one of an explicit logarithmic function [Formula: see text]. We show that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint cliques with [Formula: see text] vertices. Such a result is motivated by, and applied to, the multi-tasking version of Farber’s topological model to study the motion planning problem in robotics. Indeed, we study the behavior of all the higher topological complexities of Eilenberg–MacLane spaces of type [Formula: see text], where [Formula: see text] is a random right angled Artin group.

Dense subgraphs in random graphs

For a constant γ∈[0,1] and a graph G, let ωγ(G) be the largest integer k for which there exists a k-vertex subgraph of G with at least γk2 edges. We show that if 0<p<γ<1 then ωγ(Gn,p) is concentrated on a set of two integers. More precisely, with α(γ,p)=γlogγp+(1−γ)log1−γ1−p, we show that ωγ(Gn,p) is one of the two integers closest to 2α(γ,p)(logn−loglogn+logeα(γ,p)2)+12, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to handle the more complicated ways in which these “quasi-cliques” may overlap.

Finding a large submatrix of a Gaussian random matrix

We consider the problem of finding a $k\times k$ submatrix of an $n\times n$ matrix with i.i.d. standard Gaussian entries, which has a large average entry. It was shown in [Bhamidi, Dey and Nobel (2012)] using nonconstructive methods that the largest average value of a $k\times k$ submatrix is $2(1+o(1))\sqrt{\log n/k}$, with high probability (w.h.p.), when $k=O(\log n/\log\log n)$. In the same paper, evidence was provided that a natural greedy algorithm called the Largest Average Submatrix ($\mathcal{LAS}$) for a constant $k$ should produce a matrix with average entry at most $(1+o(1))\sqrt{2\log n/k}$, namely approximately $\sqrt{2}$ smaller than the global optimum, though no formal proof of this fact was provided. In this paper, we show that the average entry of the matrix produced by the $\mathcal{LAS}$ algorithm is indeed $(1+o(1))\sqrt{2\log n/k}$ w.h.p. when $k$ is constant and $n$ grows. Then, by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a $k\times k$ matrix with asymptotically the same average value $(1+o(1))\sqrt{2\log n/k}$ w.h.p., for $k=o(\log n)$. Since the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor $\sqrt{2}$ performance gap suffered by both algorithms might be very challenging. Surprisingly, we construct a very simple algorithm which produces a $k\times k$ matrix with average value $(1+o_{k}(1)+o(1))(4/3)\sqrt{2\log n/k}$ for $k=o((\log n)^{1.5})$, that is, with the asymptotic factor $4/3$ when $k$ grows. To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically $(1+o(1))\alpha\sqrt{2\log n/k}$ for a fixed value $\alpha\in[1,\sqrt{2}]$. The overlap corresponds to the number of common rows and the number of common columns for pairs of matrices achieving this value (see the paper for details). We discover numerically an intriguing phase transition at $\alpha^{*}\triangleq5\sqrt{2}/(3\sqrt{3})\approx1.3608\ldots\in[4/3,\sqrt{2}]$: when $\alpha \alpha^{*}$, appropriately defined. We conjecture that the OGP observed for $\alpha>\alpha^{*}$ also marks the onset of the algorithmic hardness—no polynomial time algorithm exists for finding matrices with average value at least $(1+o(1))\alpha\sqrt{2\log n/k}$, when $\alpha>\alpha^{*}$ and $k$ is a mildly growing function of $n$.

On the Integrality Gap of Degree-4 Sum of Squares for Planted Clique

The problem of finding large cliques in random graphs and its “planted” variant, where one wants to recover a clique of size ω &gt; log ( n ) added to an Erdős-Rényi graph G ∼ G ( n ,1/2), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size ω = Ω (√ n ). By contrast, information theoretically, one can recover planted cliques so long as ω &gt; log ( n ). In this work, we continue the investigation of algorithms from the Sum of Squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson [2] and Deshpande and Montanari [25]. Our main result is that degree four SoS does not recover the planted clique unless ω &gt; √ n / polylog n , improving on the bound ω &gt; n 1/3 due to Reference [25]. An argument of Kelner shows that the this result cannot be proved using the same certificate as prior works. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by “correcting” the certificate of References [2, 25, 27].

On comparing algorithms for the maximum clique problem

Several algorithms for the exact solution of the maximum clique problem are available in the literature. Some have been proposed with the aim of bounding the worst case complexity of the problem, while others focus on practical performance as evaluated experimentally. These two groups of works are somewhat independent, in the sense that little experimental investigation is available in the former group, and little theoretical analysis exists for the latter. Moreover, the experimental results seem to be much better than could be expected from the theoretical results. We show that a broad class of branch and bound algorithms for the maximum clique problem display sub-exponential average running time behavior, and also show how this helps to explain the apparent discrepancy between the theoretical and experimental results. We also propose a more structured methodology for the experimental analysis of algorithms for the maximum clique problem, which takes into account the peculiarities of cliques in random graphs, bringing the theoretical and experimental approaches closer together in the search for better algorithms. As a proof of concept, we apply the proposed methodology to thirteen algorithms from the literature.

Infinite paths and cliques in random graphs

We study the thresholds for the emergence of various properties in random subgraphs of $(\mathbb N, <)$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific ass

Threshold dominating cliques in random graphs and interval routing

The existence of (shortest-path) interval routing schemes for random graphs that use at most one interval label per edge is an open problem posed in [C. Gavoille, D. Peleg, The compactness of interval routing for almost all graphs, SIAM J. Computing 31 (3) (2001) 706–721]. In this paper, we show that for any random graph G(n,p) with edge probability p>0.765, there exists an interval routing scheme that uses at most one label per edge and has an additive stretch 1. In doing so, we provide an interesting construction of such an interval routing scheme for graphs that have a 12-threshold dominating clique, and establish a general result on the existence of threshold dominating cliques in random graphs.

Hiding cliques for cryptographic security

We demonstrate how a well studied combinatorial optimizationproblem may be used as a new cryptographic primitive. The problemin question is that of finding a "large" clique in a randomgraph. While the largest clique in a random graph with nvertices and edge probability p is very likely tobe of size about \(2\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n\), it is widely conjecturedthat no polynomial-time algorithm exists which finds a cliqueof size \( \geqslant (1 + \varepsilon )\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n\) with significantprobability for any constant ∈ > 0. We presenta very simple method of exploiting this conjecture by “hiding”large cliques in random graphs. In particular, we show that ifthe conjecture is true, then when a large clique—of size,say, \((1 + 2\varepsilon )\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n{\mathbf{---}}\)is randomlyinserted (“hidden”) in a random graph, finding a clique ofsize \( \geqslant (1 + \varepsilon )\log _{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$p$}}} n\) remains hard.Our analysis also covers the case of high edge probabilitieswhich allows us to insert cliques of size up to \(n^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {4 - \varepsilon }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${4 - \varepsilon }$}}} (\varepsilon >0)\). Our result suggests several cryptographicapplications, such as a simple one-way function.

Small cliques in random graphs

AbstractFor d ≥ 1, a d‐clique in a graph G is a complete d‐vertex subgraph not contained in any larger complete subgraph of G. We investigate the limit distribution of the number of d‐cliques in the binomial random graph G(n, p), p = p(n), n→∞.

Cliques in random graphs

Let 0 &lt; p &lt; 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i &lt; j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.