SIAM Journal on Computing, Volume 52, Issue 6, Page 1369-1412, December 2023.
Abstract. In the Bin Packing problem one is given [math] items with weights [math] and [math] bins with capacities [math]. The goal is to partition the items into sets [math] such that [math] for every bin [math], where [math] denotes [math]. Björklund, Husfeldt, and Koivisto [SIAM J. Comput., 39 (2009), pp. 546–563] presented an [math] time algorithm for Bin Packing (the [math] notation omits factors polynomial in the input size). In this paper, we show that for every [math] there exists a constant [math] such that an instance of Bin Packing with [math] bins can be solved in [math] randomized time. Before our work, such improved algorithms were not known even for [math]. A key step in our approach is the following new result in Littlewood–Offord theory on the additive combinatorics of subset sums: For every [math] there exists an [math] such that if [math] for some [math], then [math].

中文翻译：

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通过加性组合学实现恒定箱数装箱的更快指数时间算法

SIAM 计算杂志，第 52 卷，第 6 期，第 1369-1412 页，2023 年 12 月。
摘要。在装箱问题中，给出了具有重量 [math] 的 [math] 物品和具有容量 [math] 的 [math] 箱。目标是将项目划分为集合 [math]，以便每个 bin [math] 都有 [math]，其中 [math] 表示 [math]。Björklund、Husfeldt 和 Koivisto [SIAM J. Comput., 39 (2009), pp. 546–563] 提出了一种用于 Bin Packing 的[数学]时间算法（[数学]符号省略了输入大小中的多项式因子）。在本文中，我们表明，对于每个 [math] 都存在一个常数 [math]，这样可以在 [math] 随机时间内解决带有 [math] bin 的 Bin Packing 实例。在我们的工作之前，甚至对于[数学]来说，这种改进的算法也不为人所知。我们方法的一个关键步骤是 Littlewood–Offford 理论关于子集和的加性组合的以下新结果：对于每个 [math] 都存在一个 [math]，这样如果 [math] 对于某些 [math]，则 [math] ]。