Related papers: Maximum Matchings and Popularity
It is a celebrated result in early combinatorics that, in bipartite graphs, the size of maximum matching is equal to the size of a minimum vertex cover. K\H{o}nig's proof of this fact gave an algorithm for finding a minimum vertex cover…
Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields…
In an undirected graph $G=(V,E)$, we say $(A,B)$ is a pair of perfectly matched sets if $A$ and $B$ are disjoint subsets of $V$ and every vertex in $A$ (resp. $B$) has exactly one neighbor in $B$ (resp. $A$). The size of a pair of perfectly…
The stable marriage problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. We consider a useful variation of the…
Submodular functions have many applications. Matchings have many applications. The bitext word alignment problem can be modeled as the problem of maximizing a nonnegative, monotone, submodular function constrained to matchings in a complete…
We study the classical problem of matching $n$ agents to $n$ objects, where the agents have ranked preferences over the objects. We focus on two popular desiderata from the matching literature: Pareto optimality and rank-maximality. Instead…
A matching in a bipartite graph with parts X and Y is called envy-free if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
Bipartite networks manifest as a stream of edges that represent transactions, e.g., purchases by retail customers. Many machine learning applications employ neighborhood-based measures to characterize the similarity among the nodes, such as…
The Bipartite Polarization Problem is an optimization problem where the goal is to find the highest polarized bipartition on a weighted and labelled graph that represents a debate developed through some social network, where nodes represent…
Graphs provide an efficient tool for object representation in various computer vision applications. Once graph-based representations are constructed, an important question is how to compare graphs. This problem is often formulated as a…
In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to K\"{o}nig's work in 1916 (here $m=nd$ is the…
As a common generalization of previously solved optimization problems concerning bipartite stable matchings, we describe a strongly polynomial network flow based algorithm for computing $\ell$ disjoint stable matchings with minimum total…
A maximal matching $M$ that consists of independent edges is a subgraph of a simple and undirected graph $G$ for which $G-M$ forms an independent set. A graph $G$ is called equimatchable if all maximal matchings have the same number of…
We study the the following question in Random Graphs. We are given two disjoint sets $L,R$ with $|L|=n=\alpha m$ and $|R|=m$. We construct a random graph $G$ by allowing each $x\in L$ to choose $d$ random neighbours in $R$. The question…
In a graph, a perfect matching cut is an edge cut that is a perfect matching. Perfect Matching Cut (PMC) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and…
This work will appear as a chapter in a forthcoming volume titled `Topics in Probabilistic Graph Theory'. For a given graph $G$, each partition of the vertices has a modularity score, with higher values indicating that the partition better…
Given a weighted bipartite graph $G = (L, R, E, w)$, the maximum weight matching (MWM) problem seeks to find a matching $M \subseteq E$ that maximizes the total weight $\sum_{e \in M} w(e)$. This paper presents a novel algorithm with a time…
A matching cut of a graph is a partition of its vertex set in two such that no vertex has more than one neighbor across the cut. The Matching Cut problem asks if a graph has a matching cut. This problem, and its generalization d-cut, has…
In two-sided matching markets, the agents are partitioned into two sets. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking…