  It is well known that a stable matching in a many-to-one matching market with
couples need not exist. We introduce a new matching algorithm for such markets
and show that for a general class of large random markets the algorithm will
find a stable matching with high probability. In particular we allow the number
of couples to grow at a near-linear rate. Furthermore, truth-telling is an
approximated equilibrium in the game induced by the new matching algorithm. Our
results are tight: for markets in which the number of couples grows at a linear
rate, we show that with constant probability no stable matching exists.
