Decomposing tournaments into paths

We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kühn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament (Formula presented.). There is a natural lower bound for this number in terms of the degree sequence of (Formula presented.) and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.