Greedy Fans: A geometric approach to dual greedy algorithms

The purpose of this paper is to understand greedily solvable linear programs in a geometric way. Such linear programs have recently been considered by Queyranne, Spieksma and Tardella, Faigle and Kern, and Krüger for antichains of posets, and by Frank for a class of lattice polyhedra, and by Kashiwabara and Okamoto for extreme points of abstract convex geometries. Our guiding principle is that solving linear programs is equivalent to finding a normal cone of a polyhedron which contains a given cost vector. Motivated by this observation, we introduce and investigate a class of simplicial subdivisions, called greedy fans, whose membership problem can be greedily solvable. Our approach sheds a new perspective on greediness and submodularity in terms of theory of regular triangulations. Furthermore we introduce a well-behaved special class of greedy fans, named acyclic greedy fans, which can be obtained by some poset. In particular, its close relationship to reverse lexicographic triangulations is revealed. We show that the set of acyclic greedy fans on fixed vertices can be naturally regarded as a certain kind of a polyhedral subdivision like secondary fans. We establish the relationship between our approach and Frank’s and Kashiwabara and Okamoto’s models.