A Logarithmic Additive Integrality Gap for Bin Packing

TL;DR

This paper introduces an improved approximation algorithm for bin packing with an additive gap of O(log OPT), combining discrepancy theory and a novel two-stage packing method, advancing the theoretical bounds.

Contribution

It presents a new LP-based approximation algorithm with an additive gap of O(log OPT), matching lower bounds and employing a novel two-stage packing approach.

Findings

Achieves an additive integrality gap of O(log OPT) bins.

Uses a combination of discrepancy theory and a two-stage packing method.

Provides a cleaner and more effective algorithm than previous approaches.

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. Recently, the second author gave an LP-based polynomial time algorithm that employed techniques from discrepancy theory to find a solution using at most $OPT + O(\log OPT \cdot \log \log OPT)$ bins. In this paper, we present an approximation algorithm that has an additive gap of only $O(\log OPT)$ bins, which matches certain combinatorial lower bounds. Any further improvement would have to use more algebraic structure. Our improvement is based on a combination of discrepancy theory techniques and a novel 2-stage packing: first we pack items into containers; then we pack containers into bins of size 1. Apart from being more effective, we believe our algorithm is much cleaner than the one of Rothvoss.