Affiliation:
1. EPFL
2. Eötvös Loránd University (ELTE)
3. MIT
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 three 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 three permutations.
This connection yields an alternative method to establish an
O
(log
n
) bound on the additive integrality gap of the 3-partition. Conversely, making use of a recent example of three permutations, for which a discrepancy of Ω(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.
Funder
Deutsche Forschungsgemeinschaft
Alexander von Humboldt-Stiftung
Office of Naval Research
Magyar Tudományos Akadémia
Division of Computing and Communication Foundations
Swiss National Science Foundation
Országos Tudományos Kutatási Alapprogramok
Feodor Lynen program
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
13 articles.
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