Search results
Results from the WOW.Com Content Network
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 .
Minimum maximal independent set a.k.a. minimum independent dominating set [4] NP-complete special cases include the minimum maximal matching problem, [3]: GT10 which is essentially equal to the edge dominating set problem (see above). Metric dimension of a graph [3]: GT61 Metric k-center; Minimum degree spanning tree; Minimum k-cut
The original application was to minimum-weight (or maximum-weight) perfect matchings in a graph. Each edge is assigned a random weight in {1, …, 2 m }, and F {\displaystyle {\mathcal {F}}} is the set of perfect matchings, so that with probability at least 1/2, there exists a unique perfect matching.
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 ...
Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: [1] In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are ...
You are free: to share – to copy, distribute and transmit the work; to remix – to adapt the work; Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made.
The minimum-weight perfect matching can have no larger weight, so w(M) ≤ w(C)/2. Adding the weights of T and M gives the weight of the Euler tour, at most 3w(C)/2. Thanks to the triangle inequality, even though the Euler tour might revisit vertices, shortcutting does not increase the weight, so the weight of the output is also at most 3w(C)/2 ...
A minimum T-join can be obtained by constructing a complete graph on the vertices of T, with edges that represent shortest paths in the given input graph, and then finding a minimum weight perfect matching in this complete graph. The edges of this matching represent paths in the original graph, whose union forms the desired T-join.