Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

TL;DR

This paper introduces a geometric method to derandomize the Isolation Lemma for certain polytopes, using lattice theory and properties of totally unimodular matrices, extending previous results to broader classes of polytopes.

Contribution

It provides a new geometric approach to derandomize the Isolation Lemma for 0/1 polytopes with totally unimodular faces, generalizing prior work on bipartite perfect matching and matroid intersection.

Findings

Produces a quasi-polynomial family of integer weights for vertex isolation.

Shows the number of vectors near the shortest vector in associated lattices is polynomially bounded.

Extends derandomization techniques to a broader class of polytopes with totally unimodular faces.

Abstract

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.