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

We de®ne 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. Ó 1998 The Mathematical Programing Society, Inc. Published by Elsevier Science B.V.