Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

Given a simple, 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. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1, k}, where k is any fixed 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 new polynomial-time algorithm that approximates the general version of the problem within a ratio of (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 ratio of 3/2 for k = 2 (note that this matches the inapproximability bound above), and (2 -2/(k +1)) for any k ≤ 3, respectively, in polynomial time.