Approximation Algorithms for the Graph Orientation Minimizing the Maximum Weighted Outdegree

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}, where k 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 a factor 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 factor 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.

This work is partially supported by Grant-in-Aid for Scientific Research on Priority Areas No. 16092223, and by Grant-in-Aid for Young Scientists (B) No. 17700022, No. 18700014 and No. 18700015.