Computing the minimum cost pipe network interconnecting one sink and many sources

In this paper, we study the problem of computing the minimum cost pipe network interconnecting a given set of wells and a treatment site, where each well has a given capacity and the treatment site has a capacity that is no less than the sum of all the capacities of the wells. This is a generalized Steiner minimum tree problem which has applications in communication networks and in groundwater treatment. We prove that there exists a minimum cost pipe network that is the minimum cost network under a full Steiner topology. For each given full Steiner topology, we can compute all the edge weights in linear time. A powerful interior-point algorithm is then used to find the minimum cost network under this given topology. We also prove a lower bound theorem which enables pruning in a backtrack method that partially enumerates the full Steiner topologies in search for a minimum cost pipe network. A heuristic ordering algorithm is proposed to enhance the performance of the backtrack algorithm. We then define the notion of k-optimality and present an efficient (polynomial time) algorithm for checking 5-optimality. We present a 5-optimal heuristic algorithm for computing good solutions when the problem size is too large for the exact algorithm. Computational results are presented.