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.
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