For a polyhedron @math P let @math B(P) denote the polytopal complex that is formed by all bounded faces of @math P. If @math P is the intersection of @math n halfspaces in @math RD, but the maximum dimension @math d of any face in @math B(P) is much smaller, we show that the combinatorial complexity of @math P cannot be too high; in particular, that it is independent of @math D. We show that the number of vertices of @math P is @math O(nd) and the total number of bounded faces of the polyhedron is @math O(nd2). For inputs in general position the number of bounded faces is @math O(nd). We show that for certain specific values of @math d and @math D, our bounds are tight. For any fixed @math d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a number of linear programs that is polynomial in @math n.
We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational results. In a one-machine nonpreemptive scheduling problem, the feasible schedules may be defined by the vector of the corresponding job completion times. For given positive processing times, the associated simple scheduling polyhedronP is the convex hull of these feasible completion time vectors. The main result of this paper is a complete description of the minimal linear system definingP. We also give a complete, combinatorial description of the face lattice ofP, and a simple, O(n logn) separation algorithm. This algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems.  A characterization is given to the distance between subtrees of a tree defined as the shortest path length between subtrees. This is a generalization of the four-point condition for tree metrics. For this, we use the theory of the tight span and obtain an extension of the famous result by Dress that a metric is a tree metric if and only if its tight span is a tree. Given a monomial k[x1,. . . ,xn]-module M in the Laurent polynomial ring k[x1±1, . . . , xn±1], the hull complex is defined to be the set of bounded faces of the convex hull of the points ta| xa ∈ M for sufficiently large t. Bayer and Sturmfels conjectured that the faces of this polyhedron are of bounded complexity in the sense that every such face is affinely isomorphic to a subpolytope of the (n – 1)-dimensional permutohedron, which in particular would imply that these faces have at most n! vertices. In this paper we prove that the latter statement is true, and give a counterexample to the stronger conjecture.