  We consider a combinatorial generalization of the classical multi-armed
bandit problem that is defined as follows. There is a given bipartite graph of
$M$ users and $N \geq M$ resources. For each user-resource pair $(i,j)$, there
is an associated state that evolves as an aperiodic irreducible finite-state
Markov chain with unknown parameters, with transitions occurring each time the
particular user $i$ is allocated resource $j$. The user $i$ receives a reward
that depends on the corresponding state each time it is allocated the resource
$j$. The system objective is to learn the best matching of users to resources
so that the long-term sum of the rewards received by all users is maximized.
This corresponds to minimizing regret, defined here as the gap between the
expected total reward that can be obtained by the best-possible static matching
and the expected total reward that can be achieved by a given algorithm. We
present a polynomial-storage and polynomial-complexity-per-step
matching-learning algorithm for this problem. We show that this algorithm can
achieve a regret that is uniformly arbitrarily close to logarithmic in time and
polynomial in the number of users and resources. This formulation is broadly
applicable to scheduling and switching problems in networks and significantly
extends prior results in the area.
