  In a heterogeneous network (HetNet) with a large number of low power base
stations (BSs), proper user-BS association and power control is crucial to
achieving desirable system performance. In this paper, we systematically study
the joint BS association and power allocation problem for a downlink cellular
network under the max-min fairness criterion. First, we show that this problem
is NP-hard. Second, we show that the upper bound of the optimal value can be
easily computed, and propose a two-stage algorithm to find a high-quality
suboptimal solution. Simulation results show that the proposed algorithm is
near-optimal in the high-SNR regime. Third, we show that the problem under some
additional mild assumptions can be solved to global optima in polynomial time
by a semi-distributed algorithm. This result is based on a transformation of
the original problem to an assignment problem with gains $\log(g_{ij})$, where
$\{g_{ij}\}$ are the channel gains.
