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Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces. (arXiv:1708.02222v2 [cs.DS] UPDATED)

We derandomize the famous Isolation Lemma by Mulmuley, Vazirani, and Vazirani for polytopes given by totally unimodular constraints. That is, we construct a weight assignment such that one vertex in such a polytope is isolated, i.e., there is a unique minimum weight vertex. Our weights are quasi-polynomially bounded and can be constructed in quasi-polynomial time. In fact, our isolation technique works even under the weaker assumption that every face of the polytope lies in an affine space defined by a totally unimodular matrix. This generalizes the recent derandomization results for bipartite perfect matching and matroid intersection. We prove our result by associating a lattice to each face of the polytope and showing that if there is a totally unimodular kernel matrix for this lattice, then the number of near-shortest vectors in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of near-shortest circ查看全文

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