  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.
