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 $w_1,\dots ,w_n$ and $m$ bins with capacities $c_1,\dots ,c_m$. The goal is to find a partition of the items into sets $S_1,\dots ,S_m$ such that $w(S_j) \leq c_j$foreverybin$j$,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}$$thereexistsaconstant$\sigma_m >0$ 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 $\epsilon >0$ such that if $|{ X\subseteq {1,\ldots,n } : w(X)=v }| \geq 2^{(1-\varepsilon)n}$forsome$v$then$|{ w(X): X \subseteq {1,\ldots,n} }|\leq 2^{\delta n}$.