Related papers: Critical Thresholds for Maximum Cardinality Matchi…

R\"odl, Ruci\'nski, and Szemer\'edi determined the minimum $(k-1)$-degree threshold for the existence of fractional perfect matchings in $k$-uniform hypergrahs, and K\"uhn, Osthus, and Townsend extended this result by asymptotically…

We consider the MINGREEDY strategy for Maximum Cardinality Matching. MINGREEDY repeatedly selects an edge incident with a node of minimum degree. For graphs of degree at most $\Delta$ we show that MINGREEDY achieves approximation ratio at…

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with…

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…

We analyse the scaling limit of the sizes of the largest components of the Random Intersection Graph $G(n,m,p)$ close to the critical point $p=\frac{1}{\sqrt{nm}}$, when the numbers $n$ of individuals and $m$ of communities have different…

In 1977, Erd\H{o}s asked the following question: for any integers $t,n \in \mathbb{N}$, if $G_1 , \dots , G_n$ are complete graphs such that each $G_i$ has at most $n$ vertices and every pair of them shares at most $t$ vertices, what is the…

This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the…

A central theme in extremal combinatorics is the study of the maximum number of edges in an $r$-uniform hypergraph ($r$-graph) with matching number at most $s$ (the Erd\H{o}s Matching Conjecture) or with pairwise intersection at least $t$…

We consider the maximum weight $b$-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from $[1,W]$, we present a $2 - \frac{1}{2W} + \varepsilon$ approximation algorithm, using a memory…

Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In…

We study fully dynamic algorithms for maximum matching. This is a well-studied problem, known to admit several update-time/approximation trade-offs. For instance, it is known how to maintain a 1/2-approximate matching in $\log^{O(1)} n$…

This paper studies the maximum cardinality matching problem in stochastically evolving graphs. We formally define the arrival-departure model with stochastic departures. There, a graph is sampled from a specific probability distribution and…

We present an algorithm that finds a maximum cardinality $f$-matching of a simple graph in time $O(n^{2/3} m)$. Here $f:V\to \mathbb{N}$ is a given function, and an $f$-matching is a subgraph wherein each vertex $v\in V$ has degree $\le…

Define a \textit{$t$-matching} of size $m$ in a $k$-uniform family as a collection $\{A_1, A_2, \ldots, A_m\} \subseteq \binom{[n]}{k}$ such that $|A_i \cap A_j| < t$ for all $1 \leq i < j \leq m$. Let $\mathcal{F}\subseteq \binom{[n]}{k}$.…

The MaxClique problem, finding the largest complete subgraph in an Erd{\"o}s-R{\'e}nyi $G(N,p)$ random graph in the large $N$ limit, is a well-known example of a simple problem for which finding any approximate solution within a factor of…

Let $k$ and $n$ be two integers, with $k\geq 3$, $n\equiv 0\pmod k$, and $n$ sufficiently large. We determine the $(k-1)$-degree threshold for the existence of a rainbow perfect matchings in $n$-vertex $k$-uniform hypergraph. This implies…

Random graph matching refers to recovering the underlying vertex correspondence between two random graphs with correlated edges; a prominent example is when the two random graphs are given by Erd\H{o}s-R\'{e}nyi graphs $G(n,\frac{d}{n})$.…

We present a $(1- \varepsilon)$-approximation algorithms for maximum cardinality matchings in disk intersection graphs -- all with near linear running time. We also present estimation algorithm that returns $(1\pm…

We establish thresholds for the feasibility of random multi-graph alignment in two models. In the Gaussian model, we demonstrate an "all-or-nothing" phenomenon: above a critical threshold, exact alignment is achievable with high…

A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…