Decreasing minimization on M-convex sets: background and structures

The present work is the first member of a pair papers concerning decreasingly-minimal (dec-min) elements set integral vectors, where vector dec-min if its largest component as small possible, within this, next and so on. This discrete notion, along with fractional counterpart, showed up earlier in literature under various names. domain we consider an M-convex set, that is, base-polyhedron. A fundamental difference between case base-polyhedron has always unique element, while admits rich structure, described here help ‘canonical chain’. As consequence, prove this arises from matroid by translating characteristic vectors bases vector. By relying on these characterizations, element only square-sum components minimum, property resulting new type min-max theorems. characterizations also give rise, shown companion paper, to strongly polynomial algorithm, several applications areas resource allocation, network flow, matroid, graph orientation problems, which actually provided major motivation investigations. In particular, conjecture orientation.