Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $z$-matching problem on bipartite graphs   The $z$-matching problem on bipartite graphs is studied with a local
algorithm. A $z$-matching ($z \ge 1$) on a bipartite graph is a set of matched
edges, in which each vertex of one type is adjacent to at most $1$ matched edge
and each vertex of the other type is adjacent to at most $z$ matched edges. The
$z$-matching problem on a given bipartite graph concerns finding $z$-matchings
with the maximum size. Our approach to this combinatorial optimization are of
two folds. From an algorithmic perspective, we adopt a local algorithm as a
linear approximate solver to find $z$-matchings on general bipartite graphs,
whose basic component is a generalized version of the greedy leaf removal
procedure in graph theory. From an analytical perspective, in the case of
random bipartite graphs with the same size of two types of vertices, we develop
a mean-field theory for the percolation phenomenon underlying the local
algorithm, leading to a theoretical estimation of $z$-matching sizes on
coreless graphs. We hope that our results can shed light on further study on
algorithms and computational complexity of the optimization problem.

14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn