  Joint allocation of spectrum and user association is considered for a large
cellular network. The objective is to optimize a network utility function such
as average delay given traffic statistics collected over a slow timescale. A
key challenge is scalability: given $n$ Access Points (APs), there are $O(2^n)$
ways in which the APs can share the spectrum. The number of variables is
reduced from $O(2^n)$ to $O(nk)$, where $k$ is the number of users, by
optimizing over local overlapping neighborhoods, defined by interference
conditions, and by exploiting the existence of sparse solutions in which the
spectrum is divided into $k+1$ segments. We reformulate the problem by
optimizing the assignment of subsets of active APs to those segments. An
$\ell_0$ constraint enforces a one-to-one mapping of subsets to spectrum, and
an iterative (reweighted $\ell_1$) algorithm is used to find an approximate
solution. Numerical results for a network with 100 APs serving several hundred
users show the proposed method achieves a substantial increase in total
throughput relative to benchmark schemes.
