Topological additive numbering of directed acyclic graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f a labeling of its vertices with positive integers; denote by S(v) the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering if S(u) < S(v) for each arc (u, v) of the digraph. The problem asks to find the minimum number k for which D has a topological additive numbering with labels belonging to {1, . . . , k}, denoted by ηt(D). We characterize when a digraph has topological additive numberings, give a lower bound for ηt(D), and provide an integer programming formulation for our problem, characterizing when its coefficient matrix is totally unimodular. We also present some families for which ηt(D) can be computed in polynomial time. Finally, we prove that this problem is NP-Hard even when its input is restricted to planar bipartite digraphs.