Theory of linear and integer programming
Theory of linear and integer programming
An algorithmic characterization of antimatroids
Discrete Applied Mathematics
Submodular linear programs on forests
Mathematical Programming: Series A and B
Structural aspects of ordered polymatroids
Proceedings of the 5th Twente workshop on on Graphs and combinatorial optimization
K-submodular functions and convexity of their Lovász extension
Discrete Applied Mathematics
The affine representation theorem for abstract convex geometries
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
The affine representation theorem for abstract convex geometries
Computational Geometry: Theory and Applications - Special issue on the 19th European workshop on computational geometry - EuroCG 03
Dual greedy polyhedra, choice functions, and abstract convex geometries
Discrete Optimization
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Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as dual of antimatroids. We consider functions defined on the sets of the extreme points of a convex geometry. Faigle-Kern (Math. Programming 72 (1996) 195-206) presented a greedy algorithm to linear programming problems for shellings of posets, and Krüger (Discrete Appl. Math. 99 (2002) 125-148) introduced b-submodular functions and proved that Faigle-Kern's algorithm works for shellings of posets if and only if the given set function is b-submodular. We extend their results to all classes of convex geometries, that is, we prove that the same algorithm works for all convex geometries if and only if the given set function on the extreme sets is submodular in our sense.