  The problem of minimizing the number of bidirectional SONET rings required to
support a given traffic demand has been studied by several researchers. Here we
study the related ``ring grooming problem'' of minimizing the number of
add/drop locations instead of the number of rings; in a number of situations
this is a better approximation to the true equipment cost. Our main result is a
new lower bound for the case of uniform traffic. This allows us to prove that a
certain simple algorithm for uniform traffic is in fact a constant-factor
approximation algorithm, and it also demonstrates that known lower bounds for
the general problem--in particular, the linear programming relaxation--are not
within a constant factor of the optimum. We also show that our results for
uniform traffic extend readily to the more practically important case of
quasi-uniform traffic. Finally, we show that if the number of nodes on the ring
is fixed, then ring grooming is solvable in polynomial time; however, whether
ring grooming is fixed-parameter tractable is still an open question.
