  In an interference limited network, joint power and admission control (JPAC)
aims at supporting a maximum number of links at their specified signal to
interference plus noise ratio (SINR) targets while using a minimum total
transmission power. Various convex approximation deflation approaches have been
developed for the JPAC problem. In this paper, we propose an effective
polynomial time non-convex approximation deflation approach for solving the
problem. The approach is based on the non-convex $\ell_q$-minimization
approximation of an equivalent sparse $\ell_0$-minimization reformulation of
the JPAC problem where $q\in(0,1).$ We show that, for any instance of the JPAC
problem, there exists a $\bar q\in(0,1)$ such that it can be exactly solved by
solving its $\ell_q$-minimization approximation problem with any $q\in(0, \bar
q]$. We also show that finding the global solution of the $\ell_q$
approximation problem is NP-hard. Then, we propose a potential reduction
interior-point algorithm, which can return an $\epsilon$-KKT solution of the
NP-hard $\ell_q$-minimization approximation problem in polynomial time. The
returned solution can be used to check the simultaneous supportability of all
links in the network and to guide an iterative link removal procedure,
resulting in the polynomial time non-convex approximation deflation approach
for the JPAC problem. Numerical simulations show that the proposed approach
outperforms the existing convex approximation approaches in terms of the number
of supported links and the total transmission power, particularly exhibiting a
quite good performance in selecting which subset of links to support.
