One of the major problems faced in operating large networks is the enormous amount of processing and communications overhead required for setting up and tearing down the large number of connections maintained by the network. ATM and MPLS aim at solving these problems via the Virtual Path (VP) mechanism which is used to group together the connections. When a need for setting up a connection rises, the request and its resource allocation are processed by the VP agent and not by the network, thus reducing the processing cost significantly. An important question in the design of these networks is the amount of network resources to be dynamically allocated to and held by the VP agents; too high allocation will result with bandwidth resource waste, while too low allocation will result with heavy connection set-up and tear-down processing load. In this paper we deal with this problem, and at deriving simple-operational rules to determine the amount of bandwidth resources to be held by the various VP agents, while balancing between bandwidth waste and connection processing overhead. We formulate the resource allocation problem by accounting both for bandwidth utilization and for connection processing constraints. Recognizing the complexity of the problem, we use a decomposition approach in which we first analyze the single link problem and then propose to use this solution as a building block in constructing algorithms for the whole network. For the single link problem we realize that the pure problem is too complex and thus formulate an approximate model and derive the optimal allocation for it. The optimal rule is expressed as a closed-form square-root allocation. Extensive numerical examination shows that the rule proposed yields very efficient allocations. For the full network problem, we propose to capitalize on the closed form structure of the single link problem solution and use it in devising algorithms for the whole network.