Related papers: Largest bipartite sub-matchings of a random ordere…
In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges. The goal is to maintain a maximum…
We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and $O(log^2 n)$ depth. Previously, only an exponential upper bound was known on the size of…
We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…
A matching of a graph is a subset of edges no two of which share a common vertex, and a maximum matching is a matching of maximum cardinality. In a $b$-matching every vertex $v$ has an associated bound $b_v$, and a maximum $b$-matching is a…
A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an…
Sampling without replacement is a natural online rounding strategy for converting fractional bipartite matching into an integral one. In Online Bipartite Matching, we can use the Balance algorithm to fractionally match each online vertex,…
For two families $\mathcal{A}, \mathcal{B} \subseteq \mathcal{P}([k])$, we write $\mathcal{A}\vdash\mathcal{B}$ if $A\not\supseteq B$ for each two sets $A \in \mathcal{A}$ and $B \in \mathcal{B}$. $\mathcal{A}$ and $\mathcal{B}$ are called…
Given a sequence $A$ of $n$ numbers and an integer (target) parameter $1\leq i\leq n$, the (exact) selection problem asks to find the $i$-th smallest element in $A$. An element is said to be $(i,j)$-mediocre if it is neither among the top…
In modern data center networks, thousands of hosts contend for shared link capacity; the scale of these systems makes centralized scheduling impractical. This article models such scheduling as a bipartite matching problem under…
Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but…
We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements,…
Three well-studied types of subgraph-restricted matchings are induced matchings, uniquely restricted matchings, and acyclic matchings. While it is hard to determine the maximum size of a matching of each of these types, whether some given…
The draw of some knockout tournaments requires finding a perfect matching in a balanced bipartite graph. The problem becomes challenging with draw constraints: the two draw procedures used in sports are known to be non-uniformly distributed…
A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this…
The maximum matching width is a width-parameter that is defined on a branch-decomposition over the vertex set of a graph. The size of a maximum matching in the bipartite graph is used as a cut-function. In this paper, we characterize 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 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…
In Bipartite Correlation Clustering (BCC) we are given a complete bipartite graph $G$ with `+' and `-' edges, and we seek a vertex clustering that maximizes the number of agreements: the number of all `+' edges within clusters plus all `-'…
We study the "set parameterized matching" problem, a generalization of the classical parameterized matching problem introduced by Baker. In set parameterized matching, both the pattern and text are sequences where each position contains a…
The acyclic matching number of a graph $G$ is the largest size of an acyclic matching in $G$, that is, a matching $M$ in $G$ such that the subgraph of $G$ induced by the vertices incident to an edge in $M$ is a forest. We show that the…