Minimum polyhedron with $n$ vertices

Estimating the number of vertices of a polyhedron

Given a polyhedron P by a list of inequalities we develop unbiased estimates of the number of vertices and bases of P . The estimates are based on applying tree estimation methods to the reverse search technique. The time to generate an unbiased estimate is essentially bounded by the time taken to solve a linear program on P with the simplex method. Computational experience is reported. The met...

Generating all vertices of a polyhedron is hard 1 , 2

We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of (directed) negative cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generate...

Maximizing the Minimum Angle with the Insertion of Steiner Vertices

We consider the problem of inserting a vertex inside a star-shaped input polygon at the location that maximizes the minimum angle in the resulting triangulation. An existing polynomial-time algorithm solves for the intersection of three polynomial surfaces (a prior paper indicates that these are eighth-degree polynomials) and computes the maxima of the curve of intersection of two such surfaces...

Nonseparating vertices in tournaments with large minimum degree

A minimum broadcast graph on 26 vertices

Broadcasting is the process of information dissemination in a communication network in which a message, originated by one member, is transmitted to all members of the network. A broadcast graph is a graph which permits broadcasting from any originator in minimum time. The broadcast function B(n) is the minimum number of edges in any broadcast graph on n vertices. In this paper, we construct a b...