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 =

A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and \Delta the maximum degree.We show that every graph has a domatic partition with (1-o(1))(ffi + 1) / ln n dominatingsets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1)) ln n approximation algorithm for domatic number, since the domaticnumber is always at most ffi + 1. We also show this to be essentially best possible. Namely,extending the approximation hardness of set cover by combining multi-prover protocols with zero-knowledge techniques, we show that for every ffl> 0, a (1- ffl) ln n-approximation impliesthat N P ` DT IM E(nO(log log n)). This makes domatic number the first natural maximiza-tion problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better.We also show that every graph has a domatic partition with (1-o(1))(ffi + 1) / ln \Delta dominating sets, where the " o(1) " term goes to zero as \Delta increases. This can be turned intoan efficient algorithm that produces a domatic partition of \Omega ( ffi / ln \Delta) sets.

Abstract. Job-shop scheduling is a classical NP-hard problem. Shmoys, Stein, and Wein presented the first polynomial-time approximation algorithm for this problem that has a good (polylogarithmic) approximation guarantee. We improve the approximation guarantee of their work and present further improvements for some important NP-hard special cases of this problem (e.g., in the preemptive case where machines can suspend work on operations and later resume). We also present NC algorithms with improved approximation guarantees for some NP-hard special cases.

. The Lov'asz Local Lemma (LLL) is a powerful tool that is increasingly playing a valuable role in computer science. The original lemma was nonconstructive; a breakthrough of Beck and its generalizations (due to Alon and Molloy & Reed) have led to constructive versions. However, these methods do not capture some classes of applications of the LLL. We make progress on this, by providing algorithmic approaches to two families of applications of the LLL. The first provides constructive versions of certain applications of an extension of the LLL (modeling, e.g., hypergraph-partitioning and low-congestion routing problems); the second provides new algorithmic results on constructing disjoint paths in graphs. Our results can also be seen as constructive upper bounds on the integrality gap of certain packing problems. One common theme of our work is a "gradual rounding" approach.

The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. We consider three classes of NP-hard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are sparse (e.g., VLSI routing using short paths, problems on hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan, to constructivize our packing and covering results. 1

We provide a method to produce an efficient algorithm to find an object whose existence is guaranteed by the Lov'asz Local Lemma. We feel that this method will apply to the vast majority of applications of the Local Lemma, unless the application has one of four problematic traits. However, proving that the method applies to a particular application may require proving two (possibly difficult) concentration-like properties.

The following file distribution problem is considered: Given a network of processors represented by an undirected graph G = (V; E), and a file size k, an arbitrary file w of k bits is to be distributed among all nodes of G. To this end, each node is assigned a memory device such that, by accessing the memory of its own and of its adjacent nodes, the node can reconstruct the contents of w. The objective is to minimize the total size of memory in the network. This paper presents a file distribution scheme which realizes this objective for k AE log \Delta G , where \Delta G stands for the maximum degree in G: For this range of k, the total memory size required by the suggested scheme approaches an integer programming lower bound on that size. The scheme is also constructive in the sense that, given G and k, the memory size at each node in G, as well as the mapping of any file w into the node memory devices, can be computed in time complexity which is polynomial in k and jV j. Furthermore...

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A central issue in the design of modern communication networks is that of providing pe:formance guarantees. This issue is particularly important if the networks support real-time traffic such as voice and video. The most critical pe:formance parameter to bound is the delay experienced by a packet as it travels fi'om its source to its destination.