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 \(w_1,\ldots,w_n\) 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 \(|\{ w(X): X \subseteq \{1,\ldots,n\} \}|\leq 2^{\delta n}\).