ABSTRACT: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph Gn,1�2 Ž., 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 ˇ, for

Alon, Krivelevich and Sudakov (Random Structures and Algorithms, 1998) designed an algorithm based on spectral techniques that almost surely finds a clique of size \Omega\Gamma p n) hidden in an otherwise random graph. We show that a different algorithm, based on the Lov'asz theta function, almost surely both finds the hidden clique and certifies its optimality. Our algorithm has an additional advantage of being more robust: it also works in a semi-random hidden clique model, in which an adversary can remove edges from the random portion of the graph. 1 Introduction A clique in a graph G is a subset of the vertices every two of which are connected by an edge. The maximum clique problem, that is, finding a clique of maximum size in a graph, is fundamental in the area of combinatorial optimization, and is closely related to the independent set problem (clique on the edge complement graph G), the vertex cover problem (the vertex complement of the independent set) and chromatic...

independent set

For many random Constraint Satisfaction Problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomialtime algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 − ɛ)χ colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of {1,..., k} n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ≥ 2χ, but like an error-correcting code for k ≤ (2 − ɛ)χ. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step Replica-Symmetry-Breaking hypothesis of statistical physics for random CSPs.

In this work, we consider the problems of testing whether a distribution over {0, 1} n is k-wise (resp. (ɛ, k)-wise) independent using samples drawn from that distribution. For the problem of distinguishing k-wise independent distributions from those that are δ-far from k-wise independence in statistical distance, we upper bound the number of required samples by Õ(nk /δ 2) and lower bound it by Ω(n k−1 2 /δ) (these bounds hold for constant k, and essentially the same bounds hold for general k). To achieve these bounds, we use Fourier analysis to relate a distribution’s distance from k-wise independence to its biases, a measure of the parity imbalance it induces on a set of variables. The relationships we derive are tighter than previously known, and may be of independent interest. To distinguish (ɛ, k)-wise independent distributions from those that are δ-far from (ɛ, k)-wise independence in statistical distance, we upper bound the number of required samples by O ` k log n δ2ɛ2 ´ and lower bound it by

We present a comprehensive review of past research into adiabatic quantum computation and then propose a scalable architecture for an adiabatic quantum computer that can treat NP-hard problems without requiring local coherent operations. Instead, computation can be performed entirely by adiabatically varying a magnetic field applied to all the qubits simultaneously. Local (incoherent) operations are needed only for: (1) switching on or off certain pairwise, nearest-neighbor inductive couplings in order to set the problem to be solved and (2) measuring some subset of the qubits in order to obtain the answer to the problem.

Algorithmic Graph Coloring and Random Graphs have long become one of the most prominent branches of Combinatorics and Combinatorial Optimization. It is thus very natural to expect that their mixture will produce quite many very attractive, diverse and challenging problems. And indeed, the last thirty or so years witnessed rapid growth of the field of Algorithmic Random Graph Coloring, with many researchers working in this area and bringing there their experience from different directions of Combinatorics, Probability and Computer Science. One of the most distinctive features of this field is indeed the diversity of tools and approaches used to tackle its central problems. This survey is not intended to be a very detailed, monograph-like coverage of Algorithmic Random Graph Coloring. Instead, our aim is to acquaint the reader with several of the main problems in the field and to show several of the approaches that proved most fruitful in attacking those problems. We do not and we simply cannot...

We construct a new public key encryption based on two assumptions: 1. One can obtain a pseudorandom generator with small locality by connecting the outputs to the inputs using any sufficiently good unbalanced expander. 2. It is hard to distinguish between a random graph that is such an expander and a random graph where a (planted) random logarithmic-sized subset S of the outputs is connected to fewer than |S | inputs. The validity and strength of the assumptions raise interesting new algorithmic and pseudorandomness questions, and we explore their relation to the current state-of-art. 1

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Abstract. There are a number of scenarios where users wishing to communicate, share a weak secret. Often, they are also part of a common social network. Connections (edges) from the social network are represented as shared link keys between participants (vertices). We propose mechanisms that utilise the graph topology of such a network, to increase the entropy of weak pre-shared secrets. Our proposal is based on using random walks to identify a chain of common acquaintances between Alice and Bob, each of which contribute entropy to the final key. Our mechanisms exploit one-wayness and convergence properties of Markovian random walks to, firstly, maximize the set of potential entropy contributors, and second, to resist any contribution from dubious sources such as Sybill sub-networks. 1