k-Colour Partitions of Acyclic Tournaments

Let G1 be the acyclic tournament with the topological sort 0 < 1 < 2 < · · · < n < n + 1 defined on node set N ∪ {0, n + 1}, where N = {1, 2, . . . , n}. For integer k ≥ 2, let Gk be the graph obtained by taking k copies of every arc in G1 and colouring every copy with one of k different colours. A k-colour partition of N is a set of k paths from 0 to n + 1 such that all arcs of each path have the same colour, different paths have different colours, and every node of N is included in exactly one path. If there are costs associated with the arcs of Gk, the cost of a k-colour partition is the sum of the costs of its arcs. For determining minimum cost k-colour partitions we describe an O(k2n2k) algorithm, and show this is an NP-hard problem. We also study the convex hull of the incidence vectors of k-colour partitions. We derive the dimension, and establish a minimal equality set. For k > 2 we identify a class of facet inducing inequalities. For k = 2 we show that these inequalities turn out to be equations, and that no other facet defining inequalities exists besides the trivial nonnegativity constraints.