Tarjan's big achievement together with Goldberg was one algorithm for efficient computation for min cost flow problem by push-relabel method, which is a network flow problem. I am not able to read from your question the exact kind of problem you are dealing with. However, finding min-cost flows might be a subtask to your problems solution. As for the overall complexity of the algorithm you described it seems to be still polynomial, but practically even algorithms with quadratic complexity can get rather slow on little bit bigger problem instances. Thus your algorithm of complexity at least N^5 will be practically not so much useful for instances of interesting size. Maybe you could consider some heuristic procedure from which you could derive quality bounds proving it to be an approximation algorithm to your original problem