Routing permutations and involutions on optical ring networks: complexity results and solution to an open problem

Given a network G and a demand D of communication requests on G  , a routing for (G,D)(G,D) is a set of directed paths of G, each from the source to the destination of one request of D. The Routing and Wavelength Assignment Problem seeks a routing RR for (G,D)(G,D) and an assignment of wavelengths to the directed paths in RR such that the number of wavelengths used is minimized, subject to that any two directed paths with at least one common arc receive distinct wavelengths. In the case where G is a ring, this problem is known as the Ring Routing and Wavelength Assignment Problem (RRWA). If in addition D   is symmetric (that is, (s,t)∈D(s,t)∈D implies (t,s)∈D(t,s)∈D) and the directed paths for requests (s,t)(s,t) and (t,s)(t,s) are required to be reverse of each other, then the problem is called the Symmetric Ring Routing and Wavelength Assignment Problem (SRRWA). A demand is called a permutation demand if, for each vertex v of G, the number of requests with source v and the number of requests with destination v   are the same and are equal to 0 or 1. A symmetric permutation demand is called an involution demand. In this paper we prove that both RRWA and SRRWA are NPNP-complete even when restricted to involution demands. As a consequence RRWA is NPNP-complete when restricted to permutation demands. For general demands we prove that RRWA and SRRWA can be solved in polynomial time when the number of wavelengths is fixed. Finally, we answer in the negative an open problem posed by Gargano and Vaccaro and construct infinitely many counterexamples using involution demands.