Approximation algorithms for the graph orientation minimizing the maximum Weighted outdegree
Yuichi Asahiro, Jesper Jansson, Eiji Miyano, Hirotaka Ono, Kouhei Zenmyo, Ono, Hirotaka, Kouhei Zenmyo · 2007
Abstract. Given an undirected graph G =(V,E) and a weight function w: E → Z +, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. In this paper (1) we prove that the problem is strongly NP-hard if all edge weights belong to the set {1,k}, wherek is any integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1 + 1/k) unless P=NP; (2) we present a polynomial time algorithm that approximates the general version of the problem within afactorof(2 − 1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a factor of 3/2 fork = 2 (note that this matches the inapproximability bound above), and (2 − 2/(k +1)) for anyk ≥ 3, respectively, in polynomial time.