Let P be a property of graphs. An-test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than n 2 edges… (More)
We continue the study of combinatorial property testing, initiated by Goldreich, Goldwasser and Ron in 7]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language L 2 f0; 1g and an integer n there exists a randomized algorithm which always accepts a word w of length n if w 2 L, and rejects it with high… (More)
We consider the following probabilistic model of a graph on n labeled vertices. Ž. First choose a random graph G n, 1r2 , 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… (More)
Let P be an optimization problem, and let A be an approximation algorithm for P. The domination ratio domr(A, n) is the maximum real q such that the solution x(I) obtained by A for any instance I of P of size n is not worse than at least a fraction q of the feasible solutions of I. We describe a deterministic, polynomial time algorithm with domination ratio… (More)
We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be… (More)
A code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector's bits. We show that the Reed-Muller codes of constant order are locally testable. Specifically, we describe an efficient randomized algorithm to test if a given vector of length n =… (More)
In this article we study Hamilton cycles in sparse pseudo-random graphs. We prove that if the second largest absolute value of an eigenvalue of a d-regular graph G on n vertices satisfies —————————————————— ðlog log nÞ 2 1000 log n ðlog log log nÞ d and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for… (More)
It is proven that for k ≥ 4 and n > k every k-color-critical graph on n vertices has at least k−1 2 + k−3 2(k 2 −2k−1) n edges, thus improving a result of Gallai from 1963.