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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 perfect matching polytope of a graph G, ... 206 By solving algorithmic problems on convex sets, one can find a minimum-weight perfect matching. [4]: ...
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 ...
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.
A matching M is called perfect if every vertex v in V is contained in exactly one hyperedge of M. This is the natural extension of the notion of perfect matching in a graph. A fractional matching M is called perfect if for every vertex v in V, the sum of fractions of hyperedges in M containing v is exactly 1.
The minimum degree of a graph, often denoted by deg(G) or δ(v), is the minimum of deg(v) over all vertices v in V. A matching in a graph is a set of edges such that each vertex is adjacent to at most one edge; a perfect matching is a matching in which each vertex is adjacent to exactly one edge. A perfect matching does not always exist, and ...