Search results
Results from the WOW.Com Content Network
In the above figure, only part (b) shows a perfect matching. A perfect matching is also a minimum-size edge cover. Thus, the size of a maximum matching is no larger than the size of a minimum edge cover: () . A graph can only contain a perfect matching when the graph has an even number of vertices. A near-perfect matching is one in which ...
A perfect matching is also a minimum-size edge cover. If there is a perfect matching, then both the matching number and the edge cover number equal | V | / 2. A perfect matching can only occur when the graph has an even number of vertices. A near-perfect matching is one in which exactly one vertex is
As shown by Mulmuley, Vazirani and Vazirani, [8] the problem of minimum weight perfect matching is converted to finding minors in the adjacency matrix of a graph. Using the isolation lemma, a minimum weight perfect matching in a graph can be found with probability at least 1 ⁄ 2.
The algorithm addresses the problem that T is not a tour by identifying all the odd degree vertices in T; since the sum of degrees in any graph is even (by the Handshaking lemma), there is an even number of such vertices. The algorithm finds a minimum-weight perfect matching M among the odd-degree ones.
Given a bipartite graph G = (A ∪ B, E), the goal is to find the maximum cardinality matching in G that has minimum cost. Let w: E → R be a weight function on the edges of E. The minimum weight bipartite matching problem or assignment problem is to find a perfect matching M ⊆ E whose total weight is minimized. The idea is to reduce this ...
Creating a matching Using a shortcut heuristic on the graph created by the matching above. The algorithm of Christofides and Serdyukov follows a similar outline but combines the minimum spanning tree with a solution of another problem, minimum-weight perfect matching. This gives a TSP tour which is at most 1.5 times the optimal.
Kőnig had announced in 1914 and published in 1916 the results that every regular bipartite graph has a perfect matching, [11] and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it can be partitioned) equals its maximum degree [12] – the latter statement is known as Kőnig's ...
The following figure shows examples of minimum edge coverings (again, the set C is marked with red). Note that the figure on the right is not only an edge cover but also a matching. In particular, it is a perfect matching: a matching M in which every vertex is incident with exactly one edge in M. A perfect matching (if it exists) is always a ...