Approximating Bin Packing within O(log OPT * log log OPT) bins

TL;DR

This paper improves bin packing approximation algorithms by achieving a solution within OPT + O(log OPT * log log OPT) bins, using discrepancy theory and polynomial-time algorithms, marking the first such improvement in three decades.

Contribution

It introduces a novel polynomial-time algorithm that improves the approximation bound for bin packing, reducing the additive error from O(log^2 OPT) to O(log OPT * log log OPT).

Findings

Achieves a new approximation bound for bin packing.

Uses the Entropy Method from discrepancy theory.

Provides a constructive algorithm via Bansal and Lovett-Meka methods.

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.