Carathéodory bounds for integer cones

Carathéodory bounds for integer cones

Let b ∈ Zd be an integer conic combination of a finite set of integer vectors X ⊂ Zd . In this note we provide upper bounds on the size of a smallest subset X̃ ⊆ X such that b is an integer conic combination of elements of X̃ . We apply our bounds to general integer programming and to the cutting stock problem and provide an NP certificate for the latter, whose existence has not been known so far.

Tight Bounds for Approximate Carathéodory and Beyond

We give a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope’s vertices. Our result provides a constructive proof for the Approximate Carathéodory Problem [2], which states that any point inside a polytope contained in the lp ball of radius D can be approximated to within ǫ in lp norm by a convex combi...

Polyhedra with the Integer Carathéodory Property

Article history: Received 27 April 2010 Available online 6 May 2011

Diophantine Approximations and Integer Points of Cones

The purpose of this note is to present a relation between directed best approximations of a rational vector and the elements of the minimal Hilbert basis of certain rational pointed cones. Furthermore, we show that for a special class of these cones the integer Carathéodory property holds true.

Lower bounds for integer programming problems