In this paper we investigate the problem of computing the maximum number of
entries which can be added to a partially filled latin square. The decisio
n version of this question is known to be NP-complete. We present two appro
ximation algorithms for the optimization version of this question. We first
prove that the greedy algorithm achieves a factor of 1/3. We then use insi
ghts derived from the linear relaxation of an integer program to obtain an
algorithm based on matchings that achieves a better performance guarantee o
f 1/2. These are the first known polynomial-time approximation algorithms f
or the latin square completion problem that achieve nontrivial worst-case p
erformance guarantees. Our study is motivated by applications to lightpath
assignment and switch configuration in wavelength routed multihop optical n
etworks.