| Research Abstract |
We examined the practicality and applicability of the algorithm for the minimum norm point problem on a polytope to the submodular function minimization problem, which is a fundamental problem in combinatorial optimization problems with submodular structures. The computational experiments showed a good performance and practicality of the algorithm. However, we still have certain technical computational problem concerning the algorithm, which is left for future research. Furthermore, we proposed a concept of combinatorial hull, which is a generalization of the concept of convex hull. This gives a basis for developing a new method for solving the submodular function minimization problem in a purely combinatorial manner. We are making a progress in this direction.
Also, we developed a scaling technique for finding a maximum mean cut in a network, which is an important component in algorithms for the minimum cost flow problem. This approach can be extended to the submodular flow problem, a generalization of the minimum cost flow problem, and furnishes an efficient algorithm that is new and runs in polynomial time.
Moreover, through the two-year survey research, we examined, from the point of view of submodular and supermodular systems, the combinatorial optimization problems with submodular structures related to : base polyhedra, greedy algorithm, crossing families, generalized polymatroids, neoflows, submodular flows, independent flows, polymatroid flows, submodular analysis, submodular programs, lexicographically optimal bases, the resource allocation problem with submodular constraints etc. We gave a unifying approach to these problems and the results will be published as a monograph. Also, in the course of this survey research we found an error in the bi-truncation algorithm recently proposed by Frank and Tardos, related to the submodular function minimization problem, and showed a correct version of it.
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