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),... ,w(n) and m bins with capacities c1 , . . . , cm. The goal is to partition the items into sets S1 , . . . , Sm such that w(S-j)<= c(j) for every bin j, where w(X) denotes & sum;(i is an element of X )w(i) Bjorklund, Husfeldt, and Koivisto [SIAM J. Comput. , 39 (2009), pp. 546--563] presented an O-star(2(n)) time algorithm for Bin Packing (the O-star(.) notation omits factors polynomial in the input size). In this paper, we show that for every m is an element of N there exists a constant sigma(m) > 0 such that an instance of Bin Packing with m bins can be solved in O(2((1-sigma m)n)) randomized time. Before our work, such improved algorithms were not known even for m = 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 subset of{1,& mldr;,n}:w(X)=v}|>= 2((1-epsilon)n) for some v then |{w(X):X subset of{1,& mldr;,n}}|<= 2(delta n)