Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

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\u27s decomposition theorem for regular matroids. It generalizes an influential result by Karger on the number of minimum cuts in a graph to regular matroids

An Efficient Algorithm for Minimum-Weight Bibranching

Given a directed graph D = (V; A) and a set S ` V , a bibranching is a set of arcs B ` A that contains a v--(V n S) path for every v 2 S and an S--v path for every v 2 V n S. In this paper, we describe a primal-dual algorithm that determines a minimum weight bibranching in a weighted digraph. It has running time O(n 0 (m + n log n)), where m = jAj, n = jV j and n 0 = minfjSj; jV n Sjg. Thus, our algorithm obtains the best known bounds for two important special cases of the problem: bipartite edge cover and r-branching

An efficient algorithm for minimum-weight bibranching

AbstractGiven a directed graphD=(V,A) and a setS⊆V, a bibranching is a set of arcsB⊆Athat contains av−(V\S) path for everyv∈Sand anS−vpath for everyv∈V\S. In this paper, we describe a primal–dual algorithm that determines a minimum weight bibranching in a weighted digraph. It has running timeO(n′(m+nlogn)), wherem=|A|,n=|V| andn′=min{|S|,|V\S|}. Thus, our algorithm obtains the best known bounds for two important special cases of the problem: bipartite edge cover andr-branching