Approximating Bin Packing within O(log OPT * Log Log OPT) Bins

For bin packing, the input consists of n items with sizes between 0 and 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(log² 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.