On ordered languages and the optimization of linear functions by greedy algorithms
Ulrich Faigle · Journal of the ACM · 1985
The optimization problem for linear functions on finite languages is studied, and an (almost) complete characterization of those functions for which a primal and a dual greedy algorithm work well with respect to a canonically associated linear programming problem is given. The discussion in this paper is within the framework of ordered languages, and the characterization uses the notion of rank feasibility of a weighting with respect to an ordered language. This yields a common generalization of a sufficient condition, obtained recently by Korte and Lovász for greedoids, and the greedy algorithm for ordered sets in Faigel's paper [6]. Ordered greedoids are considered the appropriate generalization of greedoids, and the connection is established between ordered languages, polygreedoids, and Coxeteroids. Answering a question of Björner, the author shows in particular that a polygreedoid is a Coxeteroid if and only if it is derived from an integral polymatroid.