Theoretical Analysis of Sequential Importance Sampling Algorithms for a Class of Perfect Matching Problems

Perfect Matchings in Inhomogeneous Random Bipartite Graphs in Random Environment

In this note we study inhomogeneous random bipartite graphs in random environment. These graphs can be thought of as an extension of the classical Erdös-Rényi random graphs in a random environment. We show that the expected number of perfect matchings obeys a precise quenched asymptotic and that it can be approximated using an iterative process that converges exponentially fast.

Limit shape of perfect matchings on rail-yard graphs

We obtain limit shape of perfect matchings on a large class of rail-yard graphs with right boundary condition given by the empty partition, and left boundary condition given by either by a staircase partition with constant density or a piecewise partition with densities either 1 or 0. We prove the parametric equations for the frozen boundary, and find conditions under which the frozen boundary is a cloud curve, or a union of disjoint cloud curves.

The Minimum Perfect Matching in Pseudo-dimension $0<q<1$

It is known that for $K_{n,n}$ equipped with i.i.d. $exp(1)$ edge costs, the minimum total cost of a perfect matching converges to $π^2/6$ in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension $q \geq 1$, such as Wei(1,q) costs. In this paper we extend those results all $q>0$, confirming the Mézard-Parisi conjecture in the last remaining applicable case.

超图中的完美匹配与Feige不等式

Perfect matchings in hypergraphs and Feige's inequality

本文探讨超图中完美匹配的存在条件，揭示分数完美匹配与Feige概率不等式间的联系，并给出该猜想的简洁自包含证明。

Perfect matchings in hyperfinite graphings

We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite bipartite graphing admits a measurable perfect matching if it is one-ended or the degree is odd. We give several applications of this result, answering various open questions in the field. For instance, we extend the Lyons--Nazarov theorem by characterizing bipartite Cayley graphs which admit a factor of iid perfect matching, answering the bipartite case of a well-known question of Lyons and Nazarov, popularized by Kechris and Marks. Moreover, we show …

Perfect Matchings as IID Factors on Non-Amenable Groups

We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs.