  Combinatorial optimization is a fertile testing ground for statistical
physics methods developed in the context of disordered systems, allowing one to
confront theoretical mean field predictions with actual properties of finite
dimensional systems. Our focus here is on minimum matching problems, because
they are computationally tractable while both frustrated and disordered. We
first study a mean field model taking the link lengths between points to be
independent random variables. For this model we find perfect agreement with the
results of a replica calculation. Then we study the case where the points to be
matched are placed at random in a d-dimensional Euclidean space. Using the mean
field model as an approximation to the Euclidean case, we show numerically that
the mean field predictions are very accurate even at low dimension, and that
the error due to the approximation is O(1/d^2). Furthermore, it is possible to
improve upon this approximation by including the effects of Euclidean
correlations among k link lengths. Using k=3 (3-link correlations such as the
triangle inequality), the resulting errors in the energy density are already
less than 0.5% at d>=2. However, we argue that the Euclidean model's 1/d series
expansion is beyond all orders in k of the expansion in k-link correlations.
