Constrained Matroidal Bottleneck Problems

Algorithms for the constrained bottleneck matroid base problem and the constrained bottleneck independent set problem are presented. The algorithms have improved complexities upon the best known general algorithm. Sufficient conditions are obtained when the proposed algorithms have linear complexity and are therefore the best possible ones. Several specific classes of matroidal problems are considered, including problems on partition, graphic and uniform matroids, special variants of the assignment and scheduling problems. The proposed approach is based on shrinking iteratively the dimensions of auxiliary problems. It is demonstrated that the approach cannot be extended beyond the class of matroidal problems, and the assumption of matroidal structure is not only sufficient but also necessary for the applicability of the approach.