Abstract
In the Bin Packing problem one is given n items with weights w1, …, wn and m bins with capacities c1, …, cm. The goal is to find a partition of the items into sets S1, …, Sm such that w(Sj) ≤ cj for every bin j, where w(X) denotes Σi∊xwi.
Björklund, Husfeldt and Koivisto (SICOMP 2009) presented an time algorithm for Bin Packing. In this paper, we show that for every m ∊ ℕ there exists a constant σm > 0 such that an instance of Bin Packing with m bins can be solved in 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 ∊ > 0 such that if |{X ⊆ {1, …, n} : w(X) = v}| ≥ 2(1–∊)n for some v then |{w(X) : X ⊆ {1, …, n}}| ≤ 2δn.
Björklund, Husfeldt and Koivisto (SICOMP 2009) presented an time algorithm for Bin Packing. In this paper, we show that for every m ∊ ℕ there exists a constant σm > 0 such that an instance of Bin Packing with m bins can be solved in 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 ∊ > 0 such that if |{X ⊆ {1, …, n} : w(X) = v}| ≥ 2(1–∊)n for some v then |{w(X) : X ⊆ {1, …, n}}| ≤ 2δn.
| Original language | English |
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| Title of host publication | Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021 |
| Editors | Dániel Marx |
| Publisher | SIAM |
| Pages | 1682-1701 |
| Number of pages | 20 |
| DOIs | |
| Publication status | Published - 2021 |