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A parallel algorithmic version of the local lemma
, 1991
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
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Cited by 60 (10 self)
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The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.
An Extension of the Lovász Local Lemma, and its Applications to Integer Programming
 In Proceedings of the 7th Annual ACMSIAM Symposium on Discrete Algorithms
, 1996
"... 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 mea ..."
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Cited by 31 (6 self)
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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 NPhard 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 nonzero 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
Algorithmic aspects of acyclic edge colorings
 Algorithmica
"... A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a ′ (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a ′ (G) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in ..."
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Cited by 10 (0 self)
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A proper coloring of the edges of a graph G is called acyclic if there is no 2colored cycle in G. The acyclic edge chromatic number of G, denoted by a ′ (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a ′ (G) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in G. It is known that a ′ (G) ≤ ∆ + 2 for almost all ∆regular graphs, including all ∆regular graphs whose girth is at least c ∆ log ∆. We prove that determining the acyclic edge chromatic number of an arbitrary graph is an NPcomplete problem. For graphs G with sufficiently large girth in terms of ∆(G), we present deterministic polynomial time algorithms that color the edges of G acyclically using at most ∆(G) + 2 colors. 1
Nonconstructive proofs in combinatorics
 Proc. of the International Congress of Mathematicians, Kyoto
, 1990
"... One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the m ..."
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Cited by 6 (3 self)
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One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the modern combinatorial techniques, including Topological methods, Algebraic methods, and Probabilistic methods, merely supply existence proofs and do not yield efficient (deterministic or randomized) algorithms for the corresponding problems. We describe some representing nonconstructive proofs of this type, demonstrating the applications of Topological, Algebraic and Probabilistic methods in Combinatorics, and discuss the related algorithmic problems. 1 Topological methods The application of topological methods in the study of combinatorial objects like partially ordered sets, graphs, hypergraphs and their coloring have become in the last ten years part of the mathematical machinery commonly used in combinatorics. Many interesting examples appear in [12]. Some of the more recent results of this type deal with problems that are closely related to certain algorithmic problems. While the topological tools provide a powerful technique for proving the
Probabilistic methods in coloring and decomposition problems
 Discrete Math
, 1994
"... Numerous problems in Graph Theory and Combinatorics can be formulated in terms of the existence of certain colorings of graphs or hypergraphs. Many of these problems can be solved or partially solved by applying probabilistic arguments. In this paper we discuss several examples that illustrate the m ..."
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Cited by 2 (0 self)
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Numerous problems in Graph Theory and Combinatorics can be formulated in terms of the existence of certain colorings of graphs or hypergraphs. Many of these problems can be solved or partially solved by applying probabilistic arguments. In this paper we discuss several examples that illustrate the methods used. This is mainly a survey paper, but it contains some new results as well.
NearOptimal List Colourings
 Random Structures and Algorithms
, 2000
"... We show that a simple variant of a naive colouring procedure can be used to list colour the edges of a kuniform linear hypergraph of maximum degree \Delta provided every vertex has a list of at least \Delta +c(log \Delta) 4 \Delta 1\Gamma 1 k available colours (where c is a constant which depe ..."
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Cited by 1 (0 self)
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We show that a simple variant of a naive colouring procedure can be used to list colour the edges of a kuniform linear hypergraph of maximum degree \Delta provided every vertex has a list of at least \Delta +c(log \Delta) 4 \Delta 1\Gamma 1 k available colours (where c is a constant which depends on k). We can extend this to colour hypergraphs of maximum codegree o(\Delta) with \Delta + o(\Delta) colours. This improves earlier results of Kahn and our techniques are quite similar. We also develop efficient algorithms to obtain such colourings when \Delta is constant. 1 A Colouring Problem and a Colouring Procedure A hypergraph H consists of a set V (H) (or simply V ) of vertices and a set E(H) (or simply E) of edges, each of which is a subset of V (H). The chromatic index of a hypergraph is the minimum number of colours needed to colour its edges so that no two edges which intersect receive the same colour. Given a list of colours for each edge of H, we say that an edge colouring...
On Colorings of graph fractional powers
, 812
"... For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k, which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n. In this regard, we investigate chromatic nu ..."
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For any k ∈ N, the k−subdivision of graph G is a simple graph G 1 k, which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n−subdivision of G, as a fractional power of G, denoted by G m n. In this regard, we investigate chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(G m n) = ω(G m n) provided that G is a connected graph with < 1. It is also shown that this conjecture is true in some special cases. ∆(G) ≥ 3 and m