Bin-packing with fragile objects and frequency allocation in cellular networks

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 also give improved approximation algorithms for frequency allocation problem (which is a special case of bin-packing with fragile objects). Finally, we consider a probabilistic setting and show that our algorithm for frequency allocation approaches optimality as the number of users increases.