K-submodular functions and convexity of their Lovász extension

AbstractWe consider a class of lattice polyhedra introduced by Hoffman and Schwartz. The polyhedra are defined in terms of a kind of submodular function defined on the set of antichains of a poset. Recently, Krüger (Discrete Appl. Math. 99 (2000) 125–148) showed the validity of a greedy algorithm for this class of lattice polyhedra, which had been proved by Faigle and Kern to be valid for a less general class of polyhedra. In this paper, we investigate submodular functions in Krüger's sense and associated polyhedra. We show that the Lovász extension of a submodular function in Krüger's sense is convex, and vice versa. Furthermore, we show a polynomial-time algorithm to test whether or not a vector is an extreme point of the associated polyhedron