Base polytopes of series-parallel posets: Linear description and optimization

We define the base polytope B(P, g) of a partially ordered set P and a supermodular function g on the ideals of P as the convex hull of the incidence vectors of all linear extensions of P. This new class of polytopes contains, among others, the base polytopes of supermodular systems and permutahedra as special cases. After introducing the notion of compatibility for g, we give a complete linear description of B(P, g) for series-parallel posets and compatible functions g. In addition, we describe a greedy-type procedure which exhibits Sidney's job sequencing algorithm to minimize the total weighted completion time as a natural extension of the matroidal greedy algorithm from sets to posets.