Efficient oblivious branching programs for threshold functions

In his survey paper on branching programs, A.A. Razborov (1991) asked the following question: Does every rectifier-switching network computing the majority of n bits have size n^1+Ω(1)? We answer this question in the negative by constructing a simple oblivious branching program of size O(n log³ n/log log n log log log n) for computing any threshold function. This improves the previously best known upper bound of O(n^3/2) due to O.B. Lupanov (1965)