In most combinatorial (and real world) applications the convex sets one encounters are polyhedra. Often these polyhedra have “simple” vertices and facets. It turns out that the knowledge of such additional information on the convex sets in question extends the power of the ellipsoid method considerably. In particular, optimum solutions can be calculated exactly, boundedness and full-dimensionality assumptions can be dropped, and dual solutions can be obtained. In the case of explicitly given linear programs this was the main contribution of Khachiyan to the ellipsoid method. If the linear programs are given by some oracle — which is often the case in combinatorial optimization — then these additional goals can still be achieved, albeit with more involved techniques. In particular, we have to make use of the simultaneous diophantine approximation algorithm described in Chapter 5.
KeywordsPolynomial Time Valid Inequality Dual Solution Strong Separation Ellipsoid Method
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