We present an efficient algorithm to compute all the extreme elements of a max-plus or tropical polyhedron. This algorithm relies on a combinatorial characterization of these extreme elements, when the polyhedron is defined by inequalities. We show that checking the extremality of an element of such a polyhedron reduces to computing the least model of a compact Horn formula, the latter being a ...

We present an algorithm for determining the shortest path between any two points along the surface of a polyhedron which need not be convex. This algorithm also computes for any source point on the surface of a polyhedron the inward layout and the subdivision of the polyhedron which can be used for processing queries of shortest paths between the source point and any destination point. Our algo...

We present a simple and practical data structure for storing a (not necessarily convex) polyhedron P which can, given a query surface S cutting the polyhedron, determine the volume and the area of the portion of the polyhedron above S. The queries are answered in a time linear in the size (complexity) of S. The space and preprocessing time for this data structure are linear in the size of P . W...

A Klein polyhedron is defined as the convex hull of nonzero lattice points inside an orthant of R n. It generalizes the concept of continued fraction. In this paper facets and edge stars of vertices of a Klein polyhedron are considered as multidimensional analogs of partial quotients and quantitative characteristics of these " partial quotients " , so called determinants, are defined. It is pro...

Linear state constraints define a polyhedron in the state space of a dynamical system. If this polyhedron can be made positively invariant by state feedback, the constraints are satisfied all along the trajectory for any initial point in this polyhedron. In this paper, some algebraic invariance relations are established for continuous-time linear systems. The existence of a solution to the corr...

Collision detection among complex polyhedron models is an open question because of its algorithmic complexity. To satisfy the real-time collision detection requirement of polyhedrons, we presented a parallel algorithm to perform collision detection for polyhedrons models. The algorithm decomposes nonconvex polyhedron into tetrahedrons, then detect thetetrahedrons in parallel, and boxtree of the...

As a step towards characterizing the graphs of orthogonally convex polyhedra, we show that for any simple orthogonally convex polyhedron there is an orthoball that is equivalent in the sense that it has the same graph and the same face normals. An orthoball is a simple orthogonally convex polyhedron with a point inside that sees the whole interior (informally, it is “round”). The consequence fo...

Murota (1995) introduced an M-convex function as a quantitative generalization of the set of integral vectors in an integral base polyhedron as well as an extension of valuated matroid over base polyhedron. Just as a base polyhedron can be transformed through a network, an M-convex function can be induced through a network. This paper gives a constructive proof for the induction of an M-convex ...

The cutting-plane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several decades until relatively recently, when a paper of Andersen, Louveaux, Weismantel and Wolsey gene...