Ordered optimal solutions and parametric minimum cut problems

In this paper, we present an algebraic sufficient condition for the existence of a selection of optimal solutions in a parametric optimization problem that are totally ordered, but not necessarily monotone. Based on this result, we present necessary and sufficient conditions that ensure the existence of totally ordered selections of minimum cuts for some classes of parametric maximum flow problems. These classes subsume the class studied by Arai et al. [Discrete Appl. Math. 41 (1993) 69-74] as a special case.