Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Friedland's Lower Matching Conjecture asserts that if $G$ is a $d$--regular bipartite graph on $v(G)=2n$ vertices, and $m_k(G)$ denotes the number of matchings of size $k$, then $$m_k(G)\geq {n \choose k}^2\left(\frac{d-p}{d}\right)^{n(d-p)}(dp)^{np},$$ where $p=\frac{k}{n}$. When $p=1$, this conjecture reduces to a theorem of Schrijver which says that a $d$--regular bipartite graph on $v(G)=2n$ vertices has at least $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n$$ perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that $$\frac{\ln m_k(G)}{v(G)}\geq \frac{1}{2}\left(p\ln \left(\frac{d}{p}\right)+(d-p)\ln \left(1-\frac{p}{d}\right)-2(1-p)\ln (1-p)\right)+o_{v(G)}(1).$$ In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra $c_p\sqrt{n}$ factor compared to the conjecture if $p$ is separated away from $0$ and $1$, and is tight up to a constant factor if $p$ is separated away from $1$. We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to $(a,b)$--biregular bipartite graphs

FPRAS for computing a lower bound for weighted matching polynomial of graphs

We give a fully polynomial randomized approximation scheme to compute a lower bound for the matching polynomial of any weighted graph at a positive argument. For the matching polynomial of complete bipartite graphs with bounded weights these lower bounds are asymptotically optimal.Comment: 16 page

Towards Resistance Sparsifiers

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+\epsilon)$-resistance sparsifier of size $\tilde O(n/\epsilon)$, and conjecture this bound holds for all graphs on $n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $\Omega(n/\epsilon^2)$ edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers