Approximating Bin Packing within O(log OPT * log log OPT) bins
Data Structures and Algorithms
2014-03-11 v3 Combinatorics
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 Karmarkar-Karp 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 Gilmore-Gomory LP relaxation using the Entropy Method from discrepancy theory. The result is constructive via algorithms of Bansal and Lovett-Meka.
Cite
@article{arxiv.1301.4010,
title = {Approximating Bin Packing within O(log OPT * log log OPT) bins},
author = {Thomas Rothvoss},
journal= {arXiv preprint arXiv:1301.4010},
year = {2014}
}