Approximating Bin Packing within O(log OPT * log log OPT) bins
Abstract
For bin packing, the input consists of n items with sizes s_1,...,s_n in [0,1] which have to be assigned to a minimum number of bins of size 1. The seminal KarmarkarKarp algorithm from '82 produces a solution with at most OPT + O(log^2 OPT) bins. We provide the first improvement in now 3 decades and show that one can find a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is achieved by rounding a fractional solution to the GilmoreGomory LP relaxation using the Entropy Method from discrepancy theory. The result is constructive via algorithms of Bansal and LovettMeka.
 Publication:

arXiv eprints
 Pub Date:
 January 2013
 arXiv:
 arXiv:1301.4010
 Bibcode:
 2013arXiv1301.4010R
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics