Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

Rohit Gurjar, Thomas Thierauf, Nisheeth K. Vishnoi

We deterministically construct quasi-polynomial weights in quasi-polynomial time, such that in a given polytope with totally unimodular constraints, one vertex is isolated, i.e., there is a unique minimum weight vertex. More precisely, the property that we need is that every face of the polytope lies in an affine space defined by a totally unimodular matrix. This derandomizes the famous Isolation Lemma by Mulmuley, Vazirani, and Vazirani for this setting, generalizes the recent derandomization results for bipartite perfect matching and matroid intersection and resolves the problem of derandomizing the isolation lemma for polytopes with totally unimodular constraints.

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 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.