Graph Orientations Optimizing the Number of Light or Heavy Vertices

Abstract

This paper introduces four graph orientation problems named MaximizeW-Light, MinimizeW-Light, MaximizeW-Heavy, and MinimizeW-Heavy, where W can be any fixed non-negative integer. In each of these problems, the input is an undirected graph G and the objective is to assign a direction to each edge in G so that the number of vertices with outdegree at most W or at least W in the resulting directed graph is maximized or minimized. We derive a number of results on the computational complexity and polynomial-time approximability of these problems for different values of W and various special classes of graphs. In particular, we show that Maximize 0-Light and Minimize 1-Heavy are equivalent to Maximum Independent Set and Minimum Vertex Cover, respectively, so by allowing the value of W to vary, we obtain a new, natural generalization of the two latter problems.