Milena Mihail and Umesh Vazirani
I thank Nati Linial, Dan Spielman and Karim Adiprasito for telling me about the news.
The Mihail-Vazirani conjecture for matroids and the Feder-Mihail’s theorem
Consider a collection of vectors. A basis is a subset of of linearly independent vectors that span the subspace spanned by . We can define a graph whose vertices are all bases and two bases are adjacent if their symmetric difference has two elements. Mihail and Vazirani conjectured that for every set of vertices in this graph the number of edges between to its complement is at least .
If consists of the elements of the standard basis in and their negatives then the graph we obtain is the graph of the discrete dimensional discrete cube and the assertion of the Mihail-Vazirani conjecture is well known.
The Mihail-Vazirani conjecture was formulated (and have now been proved) for arbitrary matroids. Think about a matroid as a collection of subsets (the independent sets of the matroid) of some ground set that closed under taking subsets (hence a simplicial complex) with the following property: For every , the induced simplicial complex on denoted by is pure. In other words, for every , all maximal independent subsets of have the same cardinality.
In a pioneering 1992 paper Tomás Feder and Milena Mihail proved the conjecture for balanced matroids.
Mihail and Vazirani conjecture for 0-1 polytopes.
An even more general conjecture by Mihail and Vazirani is still open. It asserts that for every 0-1 polytope, for every set of vertices, the number of edges between to its complement is at least .
The new breakthrough
Here is the link to the paper.
Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid by Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant
Let me mention that there was simultaneously another paper by a subset of the authors on related questions, another paper by Brändén and Huh who independently proved the strong log-concavity of several of the polynomials that appear in the ALOV paper, and there will be several forthcoming papers by these two groups.
Here is the abstract: We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0<q<1. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if X is a pure simplicial complex with positive weights on its maximal faces, we can associate with X a multiaffine homogeneous polynomial such that the eigenvalues of the localized random walks on X correspond to the eigenvalues of the Hessian of derivatives of .
Spectral negative dependence and Hodge-Riemann
A key paragraph regarding the method is the following:
“Our approach has a close connection to the original plan of Feder and Mihail (1992) who used the negative correlation property of balanced matroids to show that the bases exchange walk mixes rapidly. Unfortunately, most interesting matroids do not satisfy negative correlation. But it was observed [AKH18; HW16; AOV18] that all matroids satisfy a spectral negative dependence property.” AKH18 (a typo, it should be AHK18) is the solution for the Rota-Heron conjecture by Adiprasito, Huh and Katz that relies on the Hodge-Riemann property for matroids that we mentioned in this post. (I still feel in debt for a more detailed blog post about Adiprasito, Huh and Katz’ breakthrough!).”
I think that high dimensional expanders and random walks on them also plays a role in the new development.