We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ɛ < 1/(k log n)). An additional advantage of our constructions is their simplicity.

We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called ffl-biased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are ffl-biased can be used to construct "almost" k-wise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k- wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using ffl-biased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...

In this paper, we prove that there exists a schedule for routing any set of packets with edge-simple paths, on any network, in O(c+d) steps, where c is the congestion of the paths in the network, and d is the length of the longest path. The result has applications to packet routing in parallel machines, network emulations, and job-shop scheduling.

The Regularity Lemma of Szemerédi is a result that asserts that every graph can be par-titioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is co-NP-complete. However, we also prove that despite this difficulty the lemma can be made constructive; we show how to obtain, for any input graph, a partition with the properties guaranteed by the lemma, efficiently. The desired partition, for an n-vertex graph, can be found in time O(M(n)), where M(n) = O(n 2.376) is the time needed to multiply two n by n matrices with 0, 1-entries over the integers. The algorithm can be parallelized and implemented in NC 1. Besides the curious phenomenon of exhibiting a natural problem in which the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if P and NP differ), our constructive version of the Regularity Lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.

Computer Science these questions take on an algorithmic tone, having proven the existence of a graph or other structure can it be constructed in polynomial time. A recent success of J. Beck allows the Lov'asz Local Lemma to be derandomized. Sometimes. We close with two forays into a land dubbed Asymptopia by David Aldous. There the asymptotic behavior of random objects are given by an infinite object, allowing powerful noncombinatorial tools to be used. 1 Chernoff, Azuma, Janson, Talagrand Let X = X 1 + . . . +Xm with the X i mutually independent and normalized so that E[X] = E[X i ] = 0. The so-called Chernoff bounds (Bernstein or antiquity might be more accurate attributions) bound the "large deviation" Pr[X ? a] ! e ] = e ] (See, e.g., the appendix of [2].) The power in the inequality is that it holds for all ? 0 and one chooses = (a) for optimal results. Suppose, for example, that jX i j 1. One can show E[e ] cosh() exp( =2), the extreme case when X i =

Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c> 0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|Gn|, with probability going to one, uniformly in p. The method from Ajtai, Komlós and Szemerédi [Combinatorica 2 (1982) 1–7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented. 1. Introduction. In

We show that for all large n, every n-uniform hypergraph with at most 0:7pn = ln n \Theta 2n edges can be 2-colored. This makes progress on a problem of Erd&quot;&quot;os (1963), improving the previous-best bound of n1=3 \Gamma o(1) \Theta 2n due to Beck (1978). We further generalize this to a

In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al [1] and Raghavan and Upfal [22] as a model for routing in all-optical networks. It is also related to questions in circuit routing. In this paper, we improve the O(ln N ) approximation result of Kleinberg and Tardos [14] for path coloring on the N \Theta N mesh. We give an O(1) approximation algorithm to the number of colors needed, and a poly(ln ln N ) approximation algorithm to the choice of paths and colors. To the best of our knowledge, these are the first sub-logarithmic bounds for any network other than trees, rings, or trees of rings. Our results are based on developing new techniques for randomized rounding. These techniques iteratively improve a fractional solution until it approaches integral...

Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map : H ! G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n \new" eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range [ 2 d 1; 2 d 1] (If true, this is tight , e.g. by the Alon-Boppana bound). Here we show that every graph of maximal degree d has a 2-lift such that all \new" eigenvalues are in the range [ c d; c d] for some constant c.

The Lovász Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its restrictions were subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06, Sri08, Mos08a]. In [Mos08b], a constructive proof was presented that works under negligible restrictions, formulated in terms of the Bounded Occurrence Satisfiability problem. In the present paper, we reformulate and improve upon these findings so as to directly apply to almost all known applications of the general Local Lemma. Key Words and Phrases. Lovász Local Lemma, constructive proof, parallelization. 1