Max-Coloring Paths: Tight Bounds and Extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V,E) with vertex weights w such that ∑k i=1 maxv∈Ci w(vi) is minimized, where C1, . . . , Ck are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring skinny trees, a broad class of trees that includes paths and spiders. For these graphs, we show that max-coloring can be solved in time O(|V | + time for sorting the vertex weights). When vertex weights are real numbers, we show a matching lower bound of Ω(|V | log |V |) in the algebraic computation tree model.