Theory of linear and integer programming
Theory of linear and integer programming
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We study the vertices and facets of the polytopes of partitions of numbers. The partition polytope Pn is the convex hull of the set of incidence vectors of all partitions n = x1 + 2x2+...+nxn. We show that the sequence P1, P2, ..., Pn, ...can be treated as an embedded chain. The dynamics of behavior of the vertices of Pn, as n increases, is established. Some sufficient and some necessary conditions for a point of Pn to be its vertex are proved. Representation of the partition polytope as a polytope on a partial algebra--which is a generalization of the group polyhedron in the group theoretic approach to the integer linear programming--allows us to prove subadditive characterization of the nontrivial facets of Pn. These facets Σi=1n pixi ≥ p0 correspond to extreme rays of the cone of subadditive functions p : {1, 2, ..., n} → R with additional requirements p0 = pn and pi + pn-i = pn, 1 ≤ i n. The trivial facets are explicitly indicated. We also show how all vertices and facets of the polytopes of constrained partitions--in which some numbers are forbidden to participate--can be obtained from those of the polytope Pn. All vertices and facets of Pn for n ≤ 8 and n ≤ 6, respectively, are presented.