Results 1  10
of
21
PseudoRandom Graphs
 IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15
"... ..."
(Show Context)
Concentration
, 1998
"... Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This ‘method of bounded differences’ has over the last dozen or so years had a great impac ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This ‘method of bounded differences’ has over the last dozen or so years had a great impact in probabilistic methods in discrete mathematics and in the mathematics of operational research and theoretical computer science. Recently Talagrand introduced an exciting new method for bounding probabilities of large deviations, which often proves superior to the bounded differences approach. In this paper we
Exact Expectations and Distributions or the Random Assignment Problem
, 1999
"... A generalization of the random assignment problem asks the expected cost of the minimumcost matching of cardinality k in a complete bipartire graph Kmr*, with independent random edge weights. With weights drawn from the exponential(l) distribution, the answer has been conjectured 1 to be ]1,5_ ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
A generalization of the random assignment problem asks the expected cost of the minimumcost matching of cardinality k in a complete bipartire graph Kmr*, with independent random edge weights. With weights drawn from the exponential(l) distribution, the answer has been conjectured 1 to be ]1,5_>0, i+5< k (i)('5) ' Here, we prove the conjecture for k < 4, k = rn = 5, and k = rn = n = 6, using a structured, automated proof technique that results in proofs with relatively few cases. The method yields not only the minimum assignment cost's expectation but the Laplace transform of its distribution as well. From the Laplace transform we compute the variance in these cases, and conjecture that with k = rn = n  e<>, the variance is 2/n+ O (log n/n 2 ). We also include some asymptotic properties of the expectation and variance when k is fixed.
On the value of a random minimum weight Steiner tree
 Combinatorica
, 2004
"... Consider a complete graph on n vertices with edge weights chosen randomly and independently from, for example, an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Consider a complete graph on n vertices with edge weights chosen randomly and independently from, for example, an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1 + o(1))(k 1)(log n log k)=n when k = o(n) and n !1.
Deterministic random walks
 In Proceedings of the Workshop on Analytic Algorithmics and Combinatorics
, 2006
"... structure of the vacant set induced by a ..."
(Show Context)
On the Random 2Stage Minimum Spanning Tree
, 2004
"... It is known [6] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to (3) = . Here we consider the following stochastic twostage version of this op ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
It is known [6] that if the edge costs of the complete graph K n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to (3) = . Here we consider the following stochastic twostage version of this optimization problem. There are two sets of edge costs c M : E ! R and c T : E ! R , called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs c M (e) and c T (e) are independent random variables, uniformly distributed in [0; 1]. The Monday costs are revealed rst. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c M (e), or to wait until its Tuesday price c T (e) appears. The set of edges XM bought on Monday is then completed by the set of edges X T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost (3)=2 + o(1). We show that in the case of twostage optimization, the expected value of the optimal cost exceeds (3)=2 by an absolute constant > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than and completes them on Tuesday in an optimal way, and show that the optimal choice for is = 1=n with the expected cost (3) 1=2 + o(1). The threshold heuristic is shown to be suboptimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning outarborescence rooted at a xed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 1=e + o(1).
Exact formulas and limits for a class of random optimization problems
 LINKÖPING STUDIES IN MATHEMATICS
, 2005
"... ..."
Evaluation of Janson’s constant for the variance in the random minimum spanning tree problem. Linköping Studies in Mathematics 7
 School of Computer Science McGill University Montreal, Quebec Canada
, 2005
"... ..."
On the Rank of Random
 Matrices”, Random Structures and Algorithms 16(2
, 2000
"... structure induced by a random walk on a ..."