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The Lovász Local Lemma (LLL), introduced by Erd˝ os and Lovász in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n " bad " events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since(More)
Given a weighted bipartite graph, the maximum weight matching (MWM) problem is to find a set of vertex-disjoint edges with maximum weight. We present a new scaling algorithm that runs in O(m √ n log N) time, when the weights are integers within the range of [0, N ]. The result improves the previous bounds of O(N m √ n) by Gabow and O(m √ n log (nN)) by(More)
The maximum cardinality and maximum weight matching problems can be solved in time˜O(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this " m √ n barrier " is extremely fragile, in the following sense. For any > 0, we give an algorithm that(More)
a r t i c l e i n f o a b s t r a c t Given a tree T with weight and length on each edge, as well as a lower bound L and an upper bound U , the so-called length-constrained maximum-density subtree problem is to find a maximum-density subtree in T such that the length of this subtree is between L and U. In this study, we present an algorithm that runs in O(More)
Graph coloring is a central problem in distributed computing. Both vertex-and edge-coloring problems have been extensively studied in this context. In this paper we show that a (2∆ − 1)-edge-coloring can be computed in time smaller than log n for any > 0, specifically, in e O(√ log log n) rounds. This establishes a separation between the (2∆ −(More)
In this paper, we study the problems of enumerating cuts of a graph by non-decreasing weights. There are four problems, depending on whether the graph is directed or undirected, and on whether we consider all cuts of the graph or only s-t cuts for a given pair of vertices s, t. Efficient algorithms for these problems with˜O(n 2 m) delay between two(More)
The (∆ + 1)-coloring problem is a fundamental symmetry breaking problem in distributed computing. We give a new randomized coloring algorithm for (∆ + 1)-coloring running in O(√ log ∆) + 2 O(√ log log n) rounds with probability 1 − 1/n Ω(1) in a graph with n nodes and maximum degree ∆. This implies that the (∆ + 1)-coloring problem is easier than the(More)