research-article

A faster exponential time algorithm for bin packing with a constant number of bins via additive combinatorics

Authors:

Jesper Nederlof,

Jakub Pawlewicz,

Céline M. F. Swennenhuis,

Karol WęgrzyckiAuthors Info & Claims

SODA '21: Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

Pages 1682 - 1701

Published: 21 March 2021 Publication History

Abstract

In the Bin Packing problem one is given n items with weights w₁, …, w_n and m bins with capacities c₁, …, c_m. The goal is to find a partition of the items into sets S₁, …, S_m such that w(S_j) ⩽ c_j for every bin j, where w(X) denotes Σ_i∈X w_i.

Björklund, Husfeldt and Koivisto (SICOMP 2009) presented an O^*(2ⁿ) time algorithm for Bin Packing. In this paper, we show that for every m ∈ N there exists a constant σ_m > 0 such that an instance of Bin Packing with m bins can be solved in O(2^(1−σ_m)n) randomized time. Before our work, such improved algorithms were not known even for m equals 4.

A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every δ > 0 there exists an ε > 0 such that if |{X ⊆ {1, …, n} : w(X) = v}| ⩾ 2^(1−ε)n for some v then |{w(X) : X ⊆ {1, …, n}}| ⩽ 2^δn.

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Cited By

Nederlof JWęgrzycki KKhuller SVassilevska Williams V(2021)Improving Schroeppel and Shamir’s algorithm for subset sum via orthogonal vectorsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451024(1670-1683)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451024

A faster exponential time algorithm for bin packing with a constant number of bins via additive combinatorics
1. Mathematics of computing
2. Theory of computation

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cover image ACM Conferences

SODA '21: Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

January 2021

3063 pages

ISBN:9781611976465

Program Chair:
Dániel Marx
CISPA Helmholtz Center for Information Security, Germany

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Society for Industrial and Applied Mathematics

United States

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Published: 21 March 2021

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SODA '21: ACM-SIAM Symposium on Discrete Algorithms

January 10 - 13, 2021

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Cited By

Nederlof JWęgrzycki KKhuller SVassilevska Williams V(2021)Improving Schroeppel and Shamir’s algorithm for subset sum via orthogonal vectorsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451024(1670-1683)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451024

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