Bin-packing with fragile objects and frequency allocation in cellular networks Nikhil Bansal Æ Zhen Liu Æ Arvind Sankar Published online: 28 November 2007 Ó Springer Science+Business Media, LLC 2007 Abstract We consider two related optimization prob- lems: bin-packing with fragile objects and frequency allocation in cellular networks. The former is a general- ization 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. Keywords Wireless communication  Cellular network  Frequency allocation  Bin-packing  Approximation 1 Introduction Consider a base station in a cellular network that has several users communicating with it on various frequency channels. In Code Division Multiple Access (CDMA), a commonly used technology in wireless systems to assign users to frequency channels, a wide frequency channel with capacity much larger than an individual user’s information rate is allocated and multiple users share a single channel. To utilize the bandwidth efficiently, it is necessary to assign as many users as possible to a single channel. However, the trade off in assigning many users to a par- ticular channel is that there is loss of quality (given by signal to noise ratio (SNR), usually denoted by b), due to interference among users sharing the channel. The goal of the frequency allocation scheme is to minimize the number of channels used while guaranteeing that each user achieves a minimum SNR of b. We call this the frequency allocation problem (FAP). Formally, consider n users communicating with a cen- tral base station. If user i transmits with power p i , then the signal received by the base station is s i ¼ p i g i , where g i is the channel gain for user i. Let U be the set of users transmitting to the base station on the same frequency The preliminary version of the paper appeared in Second IFIP International Conference on Theoretical Computer Science 2002, 38–46. N. Bansal (&) IBM T. J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA e-mail: nikhil@us.ibm.com Z. Liu IBM T. J. Watson Research Center, PO Box 704, Yorktown Heights, NY 10598, USA e-mail: zhenl@us.ibm.com A. Sankar Goldman, Sachs & Co, 85 Broad St, 4th Floor, New York, NY 10004, USA e-mail: arvind.sankar@gs.com 123 Wireless Netw (2009) 15:821–830 DOI 10.1007/s11276-007-0081-2