A Polynomial-Time-Delay and Polynomial-Space Algorithm for Enumeration Problems in Multi-criteria Optimization

We propose a polynomial-time-delay polynomial-space algorithm to enumerate all efficient extreme solutions of a multi-criteria minimum-cost spanning tree problem, while only the bicriteria case was studied in the literature. The algorithm is based on the reverse search framework due to Avis & Fukuda. We also show that the same technique can be applied to the multi-criteria version of the minimum-cost basis problem in a (possibly degenerated) submodular system. As an ultimate generalization, we propose an algorithm to enumerate all efficient extreme solutions of a multi-criteria linear program. When the given linear program has no degeneracy, the algorithm runs in polynomial-time delay and polynomial space. To best of our knowledge, they are the first polynomial-time delay and polynomial-space algorithms for the problems.