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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 it ..."
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Cited by 38 (7 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
Acyclic edge colorings of graphs
 Journal of Graph Theory
, 2001
"... Abstract: 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 0 (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a 0 (G) D(G) ‡ 2 where D(G) is the maximum d ..."
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Cited by 33 (1 self)
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Abstract: 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 0 (G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a 0 (G) D(G) ‡ 2 where D(G) is the maximum degree in G. It is known that a 0 (G) 16 D(G) for any graph G. We prove that ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ
New Algorithmic Aspects Of The Local Lemma With Applications To Routing And Partitioning
"... . 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 metho ..."
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Cited by 32 (6 self)
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. 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., hypergraphpartitioning and lowcongestion 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.
Better approximation guarantees for jobshop scheduling
 SIAM Journal on Discrete Mathematics
, 1997
"... Abstract. Jobshop scheduling is a classical NPhard problem. Shmoys, Stein, and Wein presented the first polynomialtime approximation algorithm for this problem that has a good (polylogarithmic) approximation guarantee. We improve the approximation guarantee of their work and present further impro ..."
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Cited by 31 (2 self)
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Abstract. Jobshop scheduling is a classical NPhard problem. Shmoys, Stein, and Wein presented the first polynomialtime 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 NPhard 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 NPhard special cases.
Multicommodity flow and circuit switching
 Proceedings of the Hawaii International Conference on System Sciences
, 1998
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The Algorithmic Aspects of Uncrowded Hypergraphs
 PROC. 8TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS SODA
, 1996
"... We consider the problem of finding deterministically a large independent set of guaranteed size in a hypergraph on n vertices and with m edges. With respect to the Tur'an bound, the quality of our solutions is for hypergraphs with not too many small cycles by a logarithmic factor in the input s ..."
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Cited by 15 (14 self)
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We consider the problem of finding deterministically a large independent set of guaranteed size in a hypergraph on n vertices and with m edges. With respect to the Tur'an bound, the quality of our solutions is for hypergraphs with not too many small cycles by a logarithmic factor in the input size better. The algorithms are fast; they often have a running time of O(m) + o(n³). Indeed, the denser the hypergraphs are the more close are the running times to the linear ones. This gives for the first time for some combinatorial problems algorithmic solutions with stateoftheart quality, solutions of which so far only the existence was known. In some cases, the corresponding upper bounds match the lower bounds up to constant factors. The involved concepts are uncrowded hypergraphs.
Improved Algorithmic Versions of the Lovász Local Lemma
"... The Lovász Local Lemma is a powerful tool in combinatorics and computer science. The original version of the lemma was nonconstructive, and efficient algorithmic versions have been developed by Beck, Alon, Molloy & Reed, et al. In particular, the work of Molloy & Reed lets us automatically e ..."
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Cited by 14 (1 self)
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The Lovász Local Lemma is a powerful tool in combinatorics and computer science. The original version of the lemma was nonconstructive, and efficient algorithmic versions have been developed by Beck, Alon, Molloy & Reed, et al. In particular, the work of Molloy & Reed lets us automatically extract efficient versions of essentially any application of the symmetric version of the Local Lemma. However, with some exceptions, there is a significant gap between what one can prove using the original Lemma nonconstructively, and what is possible through these efficient versions; also, some of these algorithmic versions run in superpolynomial time. Here, we lessen this gap, and improve the running time of all these applications (which cover all applications in the Molloy & Reed framework) to polynomial. We also improve upon the parallel algorithmic version of the Local Lemma for hypergraph coloring due to Alon, by allowing noticeably more overlap among the edges.
VIRTUAL CHANNELS IN WORMHOLE ROUTERS
, 1999
"... This paper analyzes the impact of virtual channels on the performance of wormhole routing algorithms. We study wormhole routing on network in which each physical channel, i.e., communication link, can support up to B virtual channels. We show that it is possible to route any set of messages with L f ..."
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This paper analyzes the impact of virtual channels on the performance of wormhole routing algorithms. We study wormhole routing on network in which each physical channel, i.e., communication link, can support up to B virtual channels. We show that it is possible to route any set of messages with L flits each, whose paths have congestion C and dilation D in O((L+ D) C(D log D) 1 B B) flit steps, where a flit step is the time taken to transmit B flits, i.e., one flit per virtual channel, across a physical channel. We also prove a nearly matching lower bound; i.e., for any values of C, D, B, and L, where C, D B+1 and L=(1+0(1)) D, we show how to construct a network and a set of Lflit messages whose paths have congestion C and dilation D that require 0(LCD 1 B B) flit steps to route. These upper and lower bounds imply that increasing the buffering capacity and the bandwidth of each physical channel by a factor of B can speed up a wormhole routing algorithm by a superlinear factor, i.e., a factor significantly larger than B. We also present a simple randomized wormhole routing algorithm for the butterfly network. The algorithm routes any qrelation on the inputs and outputs
Edge Coloring with Delays
"... Consider the following communication problem, that leads to a new notion of edge coloring. The communication network is represented by a bipartite multigraph, where the nodes on one side are the transmitters and the nodes on the other side are the receivers. The edges correspond to messages, and eve ..."
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Consider the following communication problem, that leads to a new notion of edge coloring. The communication network is represented by a bipartite multigraph, where the nodes on one side are the transmitters and the nodes on the other side are the receivers. The edges correspond to messages, and every edge e is associated with an integer c(e), corresponding to the time it takes the message to reach its destination. A proper kedgecoloring with delays is a function f from the edges to {0, 1,..., k − 1}, such that for every two edges e1 and e2 with the same transmitter, f(e1) � = f(e2), and for every two edges e1 and e2 with the same receiver, f(e1) + c(e1) � ≡ f(e2) + c(e2) (mod k). Haxell, Wilfong and Winkler [10] conjectured that there always exists a proper edge coloring with delays using k = ∆ + 1 colors, where ∆ is the maximum degree of the graph. We prove that the conjecture asymptotically holds for simple bipartite graphs, using a probabilistic approach, and further show that it holds for some multigraphs, applying algebraic tools. The probabilistic proof provides an efficient algorithm for the corresponding algorithmic problem, whereas the algebraic method does not.