Tight Lower Bound for the Channel Assignment Problem

We study the complexity of the CHANNEL ASSIGNMENT problem. An open problem asks whether CHANNEL ASSIGNMENT admits an O(c(n)) (times a polynomial in the bit size) time algorithm, where n is a number of the vertices, for a constant c independent of the weights on the edges. We answer this question in the negative. Indeed, we show that in the standard Word RAM model, there is no 2(o(n log n)) (times a polynomial in the bit size) time algorithm solving CHANNEL ASSIGNMENT unless the exponential time hypothesis fails. Note that the currently best known algorithm works in time O*(n!) = 2(O(n log n)), so our lower bound is tight (where the O*( ) notation suppresses polynomial factors).