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

TL;DR

This paper introduces a faster exponential time algorithm for the Bin Packing problem with a fixed number of bins by leveraging new results in additive combinatorics, significantly improving previous algorithms.

Contribution

It presents a novel algorithm that solves Bin Packing with m bins in faster-than-previous exponential time, using additive combinatorics techniques.

Findings

Achieves $ ilde{O}(2^{(1-\sigma_m)n})$ randomized time for fixed m bins.

Introduces a new Littlewood-Offord theory result on subset sums.

Improves upon the previous $ ilde{O}(2^n)$ algorithm for small fixed m.

Abstract

In the Bin Packing problem one is given $n$ items with weights $w_1,\ldots,w_n$ and $m$ bins with capacities $c_1,\ldots,c_m$. The goal is to find a partition of the items into sets $S_1,\ldots,S_m$ such that $w(S_j) \leq c_j$ for every bin $j$, where $w(X)$ denotes $\sum_{i \in X}w_i$. Bj\"orklund, Husfeldt and Koivisto (SICOMP 2009) presented an $\mathcal{O}^\star(2^n)$ time algorithm for Bin Packing. In this paper, we show that for every $m \in \mathbf{N}$ there exists a constant $\sigma_m >0$ such that an instance of Bin Packing with $m$ bins can be solved in $\mathcal{O}(2^{(1-\sigma_m)n})$ 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 $\delta >0$ there exists an $\varepsilon >0$ such that if $|\{ X\subseteq \{1,\ldots,n \} : w(X)=v \}| \geq 2^{(1-\varepsilon)n}$ for some $v$ then $|\{ w(X): X \subseteq \{1,\ldots,n\} \}|\leq 2^{\delta n}$.