  We present a geometric approach towards derandomizing the Isolation Lemma by
Mulmuley, Vazirani, and Vazirani. In particular, our approach produces a
quasi-polynomial family of weights, where each weight is an integer and
quasi-polynomially bounded, that can isolate a vertex in any 0/1 polytope for
which each face lies in an affine space defined by a totally unimodular matrix.
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.
