The Greedy Algorithm and Coxeter Matroids

• Authors: A. Vince
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2000

The notion of matroid has been generalized toCoxeter matroid by Gelfand and Serganova. To each pair (W, P)consisting of a finite irreducible Coxeter group W and parabolicsubgroup P is associated a collection of objects called Coxetermatroids. The (ordinary) matroids are a special case, the caseW = A (isomorphic to the symmetric group Sym_n+1)and P a maximal parabolic subgroup. The main result of this paperis that for Coxeter matroids, just as for ordinary matroids, thegreedy algorithm provides a solution to a naturally associatedcombinatorial optimization problem. Indeed, in many important cases,Coxeter matroids are characterized by this property. This resultgeneralizes the classical Rado-Edmonds and Gale theorems.A corollary of our theorem is that, for Coxeter matroids L, the greedy algorithm solves the L-assignment problem. Let W be a finitegroup acting as linear transformations on a Euclidean space {\bb E}, and letThe L-assignment problem is to minimize thefunction f_{\xi, \eta} on a given subset L ⊆ W.An important tool in proving the greedy result is a bijection betweenthe set W/P of left cosets and a “concrete” collection A oftuples of subsets of a certain partially ordered set. If a pair ofelements of W are related in the Bruhat order, then thecorresponding elements of A are related in the Gale (greedy)order. Indeed, in many important cases, the Bruhat order on W isisomorphic to the Gale order on A. This bijection has animportant implication for Coxeter matroids. It provides basesand independent sets for a Coxeter matroid, these notions notbeing inherent in the definition.