36,795 research outputs found
The -matching problem on bipartite graphs
The -matching problem on bipartite graphs is studied with a local
algorithm. A -matching () on a bipartite graph is a set of matched
edges, in which each vertex of one type is adjacent to at most matched edge
and each vertex of the other type is adjacent to at most matched edges. The
-matching problem on a given bipartite graph concerns finding -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 -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 -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.Comment: 15 pages, 3 figure
Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure
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