The weighted link ring loading problem

In the weighted link ring loading problem, we are given an n -node undirected ring network. Each of its links is associated with a weight. Traffic demands are given for each pair of nodes in the ring. The load of a link is the sum of the flows routed through the link, and the weighted load of a link is the product of its weight and the smallest integer not less than its load. The objective of the problem is to find a routing scheme such that the maximum weighted load on the ring is minimized. In this paper we consider three variants: (i) demands may be split into two parts, and then each part is sent in a different direction; (ii) demands are allowed to be split into two parts but restricted to be integrally split; (iii) each demand must be entirely routed in either of the two directions, clockwise or counterclockwise. We first prove that the first variant is polynomially solvable. We then present a pseudo-polynomial time algorithm for the second one. Finally, for the third one, whose NP -hardness can be drawn from the result in the literature, we derive a polynomial-time approximation scheme (PTAS).