Abstract
We study the problem of minimizingc · x subject toA · x =b. x Ćș 0 andx integral, for a fixed matrixA. Two cost functionsc andcĆș are considered equivalent if they give the same optimal solutions for eachb. We construct a polytopeSt(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Grobner bases associated withA. The union of the reduced Grobner bases asc varies (called the universal Grobner basis) consists precisely of the edge directions ofSt(A). We present geometric algorithms for computingSt(A), the Graver basis, and the universal Grobner basis.
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