Let A be an arbitrary matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we show that the smoothed precision necessary to solve Ax = b, for any b, using Gaussian elimination without pivoting is logarithmic. Moreover, when A is an all-zero square matrix, our results significantly improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997).

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We consider two related optimization problems: bin-packing with fragile objects and frequency allocation in cellular networks. The former is a generalization of the classical bin-packing problem and is motivated by the latter. The problem is as follows: each object has two attributes, weight and fragility. The goal is to pack objects into bins such that, for every bin, the sum of weights of objects in that bin is no more than the fragility of any object in that bin. We consider approximation algorithms for this problem. We provide a 2-approximation to the problem of minimizing the number of bins. We also show a lower bound of 3/2 on the approximation ratio. Unlike for the classical bin-packing problem, this lower bound holds in the asymptotic case. We then consider the approximation with respect to fragility and provide a 2-approximation algorithm (i.e., our algorithm uses the same number of bins as the optimum, but the weight of objects in a bin can exceed the fragility by a factor of 2).We then consider the frequency allocation problem (which is a special case of bin-packing with fragile objects) and give improved approximation algorithms for it. Finally, we consider a probabilistic setting and show that our algorithm for frequency allocation approaches optimality as the number of users increases.

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We consider an extension of the classical bin packing problem, motivated by a frequency allocation problem arising in cellular networks. The problem is as follows: Each object has two attributes, weight and fragility. The goal is to pack objects into bins such that, for every bin, the sum of weights of objects in that bin is no more than the fragility of the most fragile object in that bin. We look for approximation algorithms for this problem. We provide a 2-approximation to the problem of minimizing the number of bins. We also show a lower bound of 3/2. Unlike in traditional bin packing, this bound holds in the asymptotic case. We then consider the approximation with respect to fragility and provide a 2-approximation algorithm. Our algorithm uses the same number of bins as the optimum but the weight of objects in a bin can exceed the fragility by a factor of 2. R. Baeza-Yates et al. (eds.