Carathéodory Bounds for Integer Cones.pdf

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Carathéodory Bounds for Integer Cones

´ Caratheodory Bounds for Integer Cones Friedrich Eisenbrand, Gennady Shmonin Max-Planck-Institut fur¨ Informatik Stuhlsatzenhausweg 85 66123 Saarbruck¨ en Germany [eisen,shmonin]@mpi-inf.mpg.de 22nd September 2005 Abstract Let b d be an integer conic combination of a finite set of integer vectors X d . In this note we provide upper bounds on the size of a smallest subset X X such that b is an integer conic combination of elements of X . 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. Keywords: Caratheodory’´ s Theorem; Integer Cone; Integer Programming; Cutting Stock; Bin Pack- ing 1 Introduction The conic hull of a finite set X d is the set cone X λ x λ x t 0; x x X ; λ λ 0 1 1 t t 1 t 1 t Caratheodory’´ s theorem (see, e.g. [7]) states that if b coneX , then b cone X , where X X is a subset of X whose cardinality is bounded by the maximum number of linearly independent points of X and thus is bounded by d . The integer conic hull of a finite set X d is the set int coneX λ x λ x t 0; x x X ; λ λ

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