Bin Packing via Discrepancy of Permutations

TL;DR

This paper reveals a connection between bin packing and permutation discrepancy, establishing bounds on the integrality gap and demonstrating limitations of rounding-based algorithms like Karmarkar-Karp.

Contribution

It links bin packing approximation bounds to permutation discrepancy, providing new lower bounds on algorithm performance limits.

Findings

The integrality gap in bin packing relates to permutation discrepancy.

A lower bound of Omega(log n) discrepancy implies no improvement over O(log^2 n) bounds for rounding algorithms.

The results apply to a broad class of algorithms including Karmarkar-Karp.

Abstract

A well studied special case of bin packing is the 3-partition problem, where n items of size > 1/4 have to be packed in a minimum number of bins of capacity one. The famous Karmarkar-Karp algorithm transforms a fractional solution of a suitable LP relaxation for this problem into an integral solution that requires at most O(log n) additional bins. The three-permutations-problem of Beck is the following. Given any 3 permutations on n symbols, color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols is roughly the same. The necessary difference is called the discrepancy. We establish a surprising connection between bin packing and Beck's problem: The additive integrality gap of the 3-partition linear programming relaxation can be bounded by the discrepancy of 3 permutations. Reversely, making use of a recent example of 3 permutations, for which a discrepancy of Omega(log n) is necessary, we prove the following: The O(log^2 n) upper bound on the additive gap for bin packing with arbitrary item sizes cannot be improved by any technique that is based on rounding up items. This lower bound holds for a large class of algorithms including the Karmarkar-Karp procedure.