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- Shlomo Hoory, Nathan Linial, Avi Wigderson, Ran Gilad-Bachrach, Danny Harnik, Boaz Barak +15 others
- 2006

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the… (More)

We analyze the eigenvalue gap for the adjacency matrices of sparse random graphs. Let λ 1 ≥. .. ≥ λ n be the eigenvalues of an n-vertex graph, and let λ = max[λ 2 , |λ n |]. Let c be a large enough constant. For graphs of average degree d = c log n it is well known that λ 1 ≥ d, and we show that λ = O(√ d). For d = c it is no longer true that λ = O(√ d),… (More)

A simple nonconstructive argument shows that most ¿CNF formulas with Ò clauses (where is a large enough constant) are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most formulas with Ò clauses proves that they are not satisfiable. We present a polynomial time algorithm that for most ¿CNF formulas with Ò… (More)

We study the complexity of the following problem that we call Max edge t-coloring: given a multigraph G and a parameter t, color as many edges as possible using t colors, such that no two adjacent edges are colored with the same color. (Equivalently, find the largest edge induced subgraph of G that has chromatic index at most t). We show that for every… (More)

We consider random 3CNF formulas with n variables and m clauses. It is well known that when m > cn (for a sufficiently large constant c), most formulas are not satisfi-able. However, it is not known whether such formulas are likely to have polynomial size witnesses that certify that they are not satisfiable. A value of m n 3/2 was the forefront of our… (More)

Let φ be a 3CNF formula with n variables and m clauses. A simple nonconstructive argument shows that when m is sufficiently large compared to n, most 3CNF formulas are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most such formulas proves that they are not satisfiable. A possible approach to refute a… (More)

One of the fundamental problems in distributed computing is how to efficiently perform routing in a faulty network in which each link fails with some probability. This paper investigates how big the failure probability can be, before the capability to efficiently find a path in the network is lost. Our main results show tight upper and lower bounds for the… (More)

We consider the problem of finding a maximum independent set in a random graph. The random graph G is modelled as follows. Every edge is included independently with probability d n , where d is some sufficiently large constant. Thereafter, for some constant α, a subset I of αn vertices is chosen at random, and all edges within this subset are removed. In… (More)

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