  This work considers the problem of user association to small-cell base
stations (SBSs) in a heterogeneous and small-cell network (HetSNet). Two
optimization problems are investigated, which are maximizing the set of
associated users to the SBSs (the unweighted problem) and maximizing the set of
weighted associated users to the SBSs (the weighted problem), under
signal-to-interference-plus-noise ratio (SINR) constraints. Both problems are
formulated as linear integer programs. The weighted problem is known to be
NP-hard and, in this paper, the unweighted problem is proved to be NP-hard as
well. Therefore, this paper develops two heuristic polynomial-time algorithms
to solve both problems. The computational complexity of the proposed algorithms
is evaluated and is shown to be far more efficient than the complexity of the
optimal brute-force (BF) algorithm. Moreover, the paper benchmarks the
performance of the proposed algorithms against the BF algorithm, the
branch-and-bound (B\&B) algorithm and standard algorithms, through numerical
simulations. The results demonstrate the close-to-optimal performance of the
proposed algorithms. They also show that the weighted problem can be solved to
provide solutions that are fair between users or to balance the load among
SBSs.
