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.