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

翻译：在装箱问题中，给定 $n$ 个物品（权重分别为 $w_1,\ldots,w_n$）和 $m$ 个箱子（容量分别为 $c_1,\ldots,c_m$）。目标是找到将物品划分为集合 $S_1,\ldots,S_m$ 的方案，使得对每个箱子 $j$ 满足 $w(S_j) \leq c_j$，其中 $w(X)$ 表示 $\sum_{i \in X} w_i$。Björklund、Husfeldt 和 Koivisto（SICOMP 2009）提出了一个时间复杂度为 $\mathcal{O}^\star(2^n)$ 的装箱问题算法。本文证明：对任意 $m \in \mathbf{N}$，存在常数 $\sigma_m > 0$，使得包含 $m$ 个箱子的装箱问题实例可在 $\mathcal{O}(2^{(1-\sigma_m)n})$ 随机化时间内求解。在本工作之前，即便对 $m=4$ 的情况也未有此类改进算法。我们方法的关键步骤是 Littlewood-Offord 理论中关于子集和加性组合学的以下新结果：对任意 $\delta > 0$，存在 $\varepsilon > 0$，使得若对某个 $v$ 有 $|\{ X\subseteq \{1,\ldots,n \} : w(X)=v \}| \geq 2^{(1-\varepsilon)n}$，则 $|\{ w(X): X \subseteq \{1,\ldots,n\} \}|\leq 2^{\delta n}$。