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

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].