DeepAI

# A Faster Exponential Time Algorithm for Bin Packing With a Constant Number of Bins via Additive Combinatorics

In the Bin Packing problem one is given n items with weights w_1,…,w_n and m bins with capacities c_1,…,c_m. The goal is to find a partition of the items into sets S_1,…,S_m such that w(S_j) ≤ c_j for every bin j, where w(X) denotes ∑_i ∈ Xw_i. Björklund, Husfeldt and Koivisto (SICOMP 2009) presented an 𝒪^⋆(2^n) 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^δ n.

12/02/2022

### Bin Packing with Partition Matroid can be Approximated within o(OPT) Bins

We consider the Bin Packing problem with a partition matroid constraint....
03/18/2022

### Tight Vector Bin Packing with Few Small Items via Fast Exact Matching in Multigraphs

We solve the Bin Packing problem in O^*(2^k) time, where k is the number...
01/19/2021

### Anticoncentration versus the number of subset sums

Let w⃗ = (w_1,…, w_n) ∈ℝ^n. We show that for any n^-2≤ϵ≤ 1, if #{ξ⃗...
11/09/2020

### An APTAS for Bin Packing with Clique-graph Conflicts

We study the following variant of the classic bin packing problem. The i...
12/04/2017

### (Gap/S)ETH Hardness of SVP

We prove the following quantitative hardness results for the Shortest ...
08/19/2019

### On bin packing with clustering and bin packing with delays

We continue the study of two recently introduced bin packing type proble...
09/15/2019

### A Primal Decomposition Algorithm for the Two-dimensional Bin Packing Problem

The Two-dimensional Bin Packing Problem calls for packing a set of recta...