Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...

previously seemed, since they can be successfully applied to only a limited number of problems exhibiting special, amenable properties. Combinatorial optimization, neural networks, mean eld annealing. i optimization networks Key words: Summary I am greatly indebted to my supervisor, Richard Prager, for initially allowing me the freedom to explore various research areas, and subsequently providing invaluable support as my work progressed. Members of the Speech, Vision and Robotics Group at the Cambridge University Department of Engineering have provided a stimulating and friendly environment to work in: special thanks must go to Patrick Gosling and Tony Robinson for maintaining a superb computing service, and to Sree Aiyer for both setting me on the right course and for numerous helpful discussions since then. I would like to thank the Science and Engineering Research Council of Great Britain, the Cambridge University Department of Engineering and Queen

A polyhedron having vertices is called integer if all of its vertices are integer. This property is coNP-complete in general. Recognizing integral set-packing polyhedra is one of the biggest challenges of graph theory (perfectness test). Various other special cases are major problems of discrete mathematics.

Given a graph G = (V, E) and a weight function on the edges w: E ↦→ R, we consider the polyhedron P(G, w) of negative-weight flows on G, and get a complete characterization of the vertices and extreme directions of P(G, w). Based on this characterization, and using a construction developed in [11], we show that, unless P = NP, there is no output polynomial-time algorithm to generate all the vertices of a 0/1-polyhedron. This strengthens the NP-hardness result of [11] for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes [8]. As further applications, we show that it is NP-hard to check if a given integral polyhedron is 0/1, or if a given polyhedron is half-integral. Finally, we also show that it is NP-hard to approximate the maximum support of a vertex a polyhedron in R n within a factor of 12/n.