  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.
