Decreasing Minimization on Base-Polyhedra: Relation Between Discrete and Continuous Cases

TL;DR

This paper explores the relationship between discrete and continuous decreasing minimization problems on base-polyhedra, establishing proximity results and algorithms that connect the two cases and enhance understanding of their structure.

Contribution

It provides a comprehensive analysis linking discrete and continuous dec-min problems on base-polyhedra, including proximity results and decomposition algorithms.

Findings

Proximity results show geometric closeness of solutions in discrete and continuous cases.

Relation established between principal and canonical partitions.

Decomposition algorithms for the discrete case are described.

Abstract

This paper is concerned with the relationship between the discrete and the continuous decreasing minimization problem on base-polyhedra. The continuous version (under the name of lexicographically optimal base of a polymatroid) was solved by Fujishige in 1980, with subsequent elaborations described in his book (1991). The discrete counterpart of the dec-min problem (concerning M-convex sets) was settled only recently by the present authors, with a strongly polynomial algorithm to compute not only a single decreasing minimal element but also the matroidal structure of all decreasing minimal elements and the dual object called the canonical partition. The objective of this paper is to offer a complete picture on the relationship between the continuous and discrete dec-min problems on base-polyhedra by establishing novel technical results and integrating known results. In particular, we derive proximity results, asserting the geometric closeness of the decreasingly minimal elements in the continuous and discrete cases, by revealing the relation between the principal partition and the canonical partition. We also describe decomposition-type algorithms for the discrete case following the approach of Fujishige and Groenevelt.