| Research Abstract |
The aim of this research is to consider transformations between discrete problems and continuous/nonlinear worlds, and to develop efficient algorithms for discrete problems using the good properties of continuous models. By considering good transformations to continuous models, combinatorial explosion encountered in the original discrete settings might be sometimes overcome or at least weakened in some cases.
In this research, we mainly focus on the interior-point method for linear programming and its application to special or generalized cases such as network programming and integer programming. Concerning linear programming, in mid 1980's, variants of the old interior-point method was considered as a possible alternative, especially for large-scale problems, to the state-of-the-art simplex method. Through this research project, we have clarified the time complexity of the multiplicative penalty function method for linear programming which was proposed by Iri and Imai. Furthermore, applying the interior-point method to network flow problems has been proposed, and its use in designing parallel algorithms has been demonstrated.
Also, by considering the geometric properties of linear programming in detail, randomized algorithms for linear programming and its related problems have been investigated. Especially, the rounding stage arising in the application of the interior-point method for integer programming has been studied from deterministic and probabilistic viewpoints. Parallelization of several rounding procedures has been also investigated.
An efficient greedy-type algorithm has been developed for the one-dimensional matching problems for point sets, partially using the technique of computational geometry.
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