Forwarding and optical indices of 4-regular circulant networks

An all-to-all routing in a graph $G$ is a set of oriented paths of $G$, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing $R$ is the number of times it is used (in either direction) by paths of $R$, and the maximum load of an edge is denoted by $\pi(G,R)$. The edge-forwarding index $\pi(G)$ is the minimum of $\pi(G,R)$ over all possible all-to-all routings $R$, and the arc-forwarding index $\overrightarrow{\pi}(G)$ is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by $w(G,R)$ the minimum number of colours required to colour the paths of $R$ such that any two paths having an edge in common receive distinct colours. The optical index $w(G)$ is defined to be the minimum of $w(G,R)$ over all possible $R$, and the directed optical index $\overrightarrow{w}(G)$ is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for $4$-regular circulant graphs with connection set $\{\pm 1,\pm s\}$, $1<s<n/2$. We give approximation algorithms with performance ratio a small constant for the corresponding forwarding index and routing and wavelength assignment problems for some families of $4$-regular circulant graphs.Comment: 19 pages, no figure in Journal of Discrete Algorithms 201

Approximable 1-Turn Routing Problems in All-Optical Mesh Networks

In all-optical networks, several communications can be transmitted through the same fiber link provided that they use different wavelengths. The MINIMUM ALL-OPTICAL ROUTING problem (given a list of pairs of nodes standing for as many point to point communication requests, assign to each request a route along with a wavelength so as to minimize the overall number of assigned wavelengths) has been paid a lot of attention and is known to be N P–hard. Rings, trees and meshes have thus been investigated as specific networks, but leading to just as many N P–hard problems. This paper investigates 1-turn routings in meshes (paths are allowed one turn only). We first show the MINIMUM LOAD 1-TURN ROUTING problem to be N P–hard but 2-APX (more generally, the MINIMUM LOAD k-CHOICES ROUTING problem is N P–hard but k-APX), then that the MINIMUM 1-TURN PATHS COLOURING problem is 4-APX (more generally, any d-segmentable routing of load L in a hypermesh of dimension d can be coloured with 2d(L−1)+1 colours at most). >From there, we prove the MINIMUM ALL-OPTICAL 1-TURN ROUTING problem to be APX

Optical All-to-All Communication for Some Product Graphs (Extended Abstract)

The problem of all-to-all communication in a network consists of designing directed paths between any ordered pair of vertices in a symmetric directed graph and assigning them minimum number of colours such that every two dipaths sharing an edge have distinct colour. We prove several exact results on the number of colours for some Cartesian product graphs, including 2-dimensional (toroidal) square meshes of odd side, which completes previous results for even sided square meshes