A multilevel approach to minimal cost network flows

Abstract

This thesis presents an exploration of the application of multigrid/multilevel techniques to a non-geometric long transportation problem. An introduction to multigrid is given, and specifics of how it is applied to this minimum cost network flow problem are explored. This research shows that multilevel techniques can be applied to network optimization problems. Further, since a previous restriction is removed by transferring the problem from a physical space to a cost space, the techniques can be applied to a broader range of problems. Both a multilevel V-cycle and a Full Multigrid (FMG) algorithm are implemented. Various strategies for restriction and local relaxation are discussed, and comparisons between the methods are made. Experimental results are given. Directions for future work include investigation of graph theoretic aspects of the problem, implementation of a regular grid overlay of the domain, exploration of a fast adaptive composite (FAC) grid algorithm, and development of a full approximation scheme (FAS) algorithm.