Better Bin Packing Approximations via Discrepancy Theory.

For bin packing, the input consists of n items with sizes s(1), ..., s(n) is an element of [0, 1] which have to be assigned to a minimum number of bins of size 1. The seminal Karmarkar-Karp algorithm from 1982 produces a solution with at most OPT vertical bar O(log(2) OPT) bins. We provide the first improvement in over three decades and show that one can find a solution of cost OPT + O(logOPT center dot log logOPT) in polynomial time. This is achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation using the partial coloring method from discrepancy theory. The result is constructive via the algorithms of Bansal and Lovett-Meka.