Topology Design with Minimal Cost Subject to Network Reliability Constraint

This paper addresses an NP-hard problem, referred to as Network Topology Design with minimum Cost subject to a Reliability constraint (NTD-CR), to design a minimal-cost communication network topology that satisfies a pre-defined reliability constraint. The paper describes a dynamic programming (DP) scheme to solve the NTD-CR problem, and proposes a DP approach, called Dynamic Programming Algorithm to solve NTD-CR (DPCR-ST), to generate the topology using a selected sequence of spanning trees of the network, STX_min. The paper shows that our DPCR-ST approach always provides a feasible solution, and produces an optimal topology given an optimal order of spanning trees. The paper proves that the problem of optimally ordering the spanning trees is NP-complete, and proposes three greedy heuristics to generate and order only k spanning trees of the network. Each heuristic allows the DPCR-ST approach to generate STX_min using only k spanning trees, which improves the time complexity while producing a near optimal topology. Simulations based on fully connected networks that contain up to 2.3×10⁹ spanning trees show the merits of using the ordering methods and the effectiveness of our algorithm vis-à-vis to four existing state-of-the-art techniques. Our DPCR-ST approach is able to generate 81.5% optimal results, while using only 0.77% of the spanning trees contained in networks. Further, for a typical 2 × 100 grid network that contains up to 1.899¹⁰² spanning trees, DPCR-ST approach requires only k=1214 spanning trees to generate a topology with a reliability no larger than 5.05% off from optimal.