Most of the past research on the problem of deadlock avoidance for complex resource allocation systems (RAS) has acknowledged the fact that the computation of the maximally permissive deadlock avoidance policy (DAP) possesses super-polynomial complexity for most RAS classes, and therefore, it has resorted to solutions that trade off maximal permissiveness for computational tractability. In this paper, we distinguish between the off-line and the on-line computation that is required for the effective implementation of the maximally permissive DAP, and we seek to develop representations of this policy that will require minimal on-line computation. The particular representation that we adopt is that of a compact classifier that will effect the underlying dichotomy of the reachable state space into safe and unsafe subspaces. Furthermore, in this first study of the aforementioned problem, we restrict our attention to a particular RAS class that is motivated by an ongoing project of ours called Gadara, and accepts separation of the safe and unsafe subspaces of its instantiations through a set of linear inequalities. Through a series of reductions of the derived classification problem, we are also able to attain extensive reductions in the computational complexity of the off-line task of the construction of the sought classifier. We formally establish completeness and optimality properties for the proposed design procedures. We also offer heuristics that, if necessary, can alleviate the computational effort that is necessary for the construction of the sought classifier. Finally, we demonstrate the efficacy of the developed approaches through a series of computational experiments. To the best of our knowledge, these experiments also establish the ability of the proposed methodology to effectively compute tractable implementations of the maximally permissive DAP for problem instances significantly beyond the capacity of any other approach currently available in the literature.