Isolating A Vertex Via Lattices: Polytopes With Totally Unimodular Faces

We present a geometric approach toward derandomizing the isolation lemma of Mulmuley, Vazirani, and Vazirani. We construct a quasi-polynomial family of weights that isolate a vertex in any 0/1-polytope for which each face spans an affine space defined by a totally unimodular matrix. These polytopes are also called box-totally dual integral or principally box-integer. This includes the polytopes given by totally unimodular constraints and generalizes the recent derandomization of the isolation lemma 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 vectors of length within 3/2 of the shortest vector in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of circuits of size within 3/2 of the shortest circuit 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.