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The assignment problem consists of finding, in a weighted bipartite graph, a matching of maximum size, in which the sum of weights of the edges is minimum. If the numbers of agents and tasks are equal, then the problem is called balanced assignment, and the graph-theoretic version is called minimum-cost perfect matching.
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 3 w ( 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 3 ...
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
The first is also a perfect matching, while the second is far from it with 4 vertices unaccounted for, but has high value weights compared to the other edges in the graph. In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized.
It returns a spanning arborescence rooted at of minimum weight, where the weight of an arborescence is defined to be the sum of its edge weights, () = (). The algorithm has a recursive description. Let f ( D , r , w ) {\displaystyle f(D,r,w)} denote the function which returns a spanning arborescence rooted at r {\displaystyle r} of minimum weight.
Then ν(H r) = 1 since every two hyperedges intersect, and ν*(H r) = r – 1 + 1 / r by the fractional matching that assigns a weight of 1 / r to each hyperedge (it is a matching since each vertex is contained in r hyperedges, and its size is r – 1 + 1 / r since there are r 2 – r + 1 hyperedges).
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 ...
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 ...