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 $ε$-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.