Ori Gurel-Gurevich

Finding Hidden Cliques in Linear Time with High Probability

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability 1/2. This random graph model is denoted G(n, 1/2, k). The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when k = c√n for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [14]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [14] gives success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2), and has a failure probability that is less than polynomially small.

Recurrence of random walk traces

We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings.

Optimal discovery strategies in white space networks

The whitespace-discovery problem describes two parties, Alice and Bob, trying to discovery one another and establish communication over one of a given large segment of communication channels. Subsets of the channels are occupied in each of the local environments surrounding Alice and Bob, as well as in the global environment (Eve). In the absence of a common clock for the two parties, the goal is to devise time-invariant (stationary) strategies minimizing the discovery time. We model the problem as follows. There are N channels, each of which is open (unoccupied) with probability p1, p2, q independently for Alice, Bob and Eve respectively. Further assume that N ≫ 1/(p1p2q) to allow for sufficiently many open channels. Both Alice and Bob can detect which channels are locally open and every time-slot each of them chooses one such channel for an attempted discovery. One aims for strategies that, with high probability over the environments, guarantee a shortest possible expected discovery time depending only on the pis and q. Here we provide a stationary strategy for Alice and Bob with a guaranteed expected discovery time of O(1/(p1p2q2)) given that each party also has knowledge of p1, p2, q. When the parties are oblivious of these probabilities, analogous strategies incur a cost of a poly-log factor, i.e. O(1/(p1p2q2)). Furthermore, this performance guarantee is essentially optimal as we show that any stationary strategies of Alice and Bob have an expected discovery time of at least Ω(1/(p1p2q2)).

Nonconcentration of return times

We show that the distribution of the first return time τ to the origin, v, of a simple random walk on an infinite recurrent graph is heavy tailed and nonconcentrated. More precisely, if dv is the degree of v, then for any t≥1 we have Pv(τ≥t)≥cdvt√ and Pv(τ=t∣τ≥t)≤Clog(dvt)t for some universal constants c>0 and C<∞. The first bound is attained for all t when the underlying graph is Z, and as for the second bound, we construct an example of a recurrent graph G for which it is attained for infinitely many t’s. Furthermore, we show that in the comb product of that graph G with Z, two independent random walks collide infinitely many times almost surely. This answers negatively a question of Krishnapur and Peres [Electron. Commun. Probab. 9 (2004) 72–81] who asked whether every comb product of two infinite recurrent graphs has the finite collision property.

The diameter of a random Cayley graph of ℤ q

Abstract Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order . This answers a question of Benjamini. The same also holds when the generating set is taken to be a symmetric set of size 2k.

Pursuit–Evasion Games with incomplete information in discrete time

Pursuit–Evasion Games (in discrete time) are stochastic games with nonnegative daily payoffs, with the final payoff being the cumulative sum of payoffs during the game. We show that such games admit a value even in the presence of incomplete information and that this value is uniform, i.e. there are

Cutpoints and resistance of random walk paths.

{\epsilon}

Stationary map coloring

-optimal strategies for both players that are