Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces
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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 circuits in a regular matroid. This is the technical core of the paper and relies on a variant of Seymour's decomposition theorem for regular matroids. It generalizes an influential result by Karger on the number of minimum cuts in a graph to regular matroids. Both of our results, on lattices and matroids, should be of independent interest.
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