Better Bin Packing Approximations via Discrepancy Theory

For bin packing, the input consists of $n$ items with sizes $s_1,\ldots,s_n \in [0,1]$ which have to be assigned to a minimum number of bins of size 1. The seminal Karmarkar--Karp algorithm from 1982 produces a solution with at most $OPT + O(\log^2 OPT)$ bins. We provide the first improvement in over three decades and show that one can find a solution of cost $OPT + O(\log OPT \cdot \log \log OPT)$ 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.