  In this paper, we consider several finite-horizon Bayesian multi-armed bandit
problems with side constraints which are computationally intractable (NP-Hard)
and for which no optimal (or near optimal) algorithms are known to exist with
sub-exponential running time. All of these problems violate the standard
exchange property, which assumes that the reward from the play of an arm is not
contingent upon when the arm is played. Not only are index policies suboptimal
in these contexts, there has been little analysis of such policies in these
problem settings. We show that if we consider near-optimal policies, in the
sense of approximation algorithms, then there exists (near) index policies.
Conceptually, if we can find policies that satisfy an approximate version of
the exchange property, namely, that the reward from the play of an arm depends
on when the arm is played to within a constant factor, then we have an avenue
towards solving these problems. However such an approximate version of the
idling bandit property does not hold on a per-play basis and are shown to hold
in a global sense. Clearly, such a property is not necessarily true of
arbitrary single arm policies and finding such single arm policies is
nontrivial. We show that by restricting the state spaces of arms we can find
single arm policies and that these single arm policies can be combined into
global (near) index policies where the approximate version of the exchange
property is true in expectation. The number of different bandit problems that
can be addressed by this technique already demonstrate its wide applicability.
