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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. A special case of it is the assignment problem , in which the input is restricted to be a bipartite graph , and the matching constrained to be have cardinality that of the ...
The algorithm has several stages. First, find a solution using greedy algorithm. In each iteration of the greedy algorithm the tentative solution is added the set which contains the maximum residual weight of elements divided by the residual cost of these elements along with the residual cost of the set.
The matching problem can be generalized by assigning weights to edges in G and asking for a set M that produces a matching of maximum (minimum) total weight: this is the maximum weight matching problem. This problem can be solved by a combinatorial algorithm that uses the unweighted Edmonds's algorithm as a subroutine. [6]
In an undirected graph G(V, E) and a function w : E → R, let S be the set of all spanning trees T i. Let B(T i) be the maximum weight edge for any spanning tree T i. We define subset of minimum bottleneck spanning trees S′ such that for every T j ∈ S′ and T k ∈ S we have B(T j) ≤ B(T k) for all i and k. [2]
In this graph, the widest path from Maldon to Feering has bandwidth 29, and passes through Clacton, Tiptree, Harwich, and Blaxhall. In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path.
Given any positive integer k≥3, the k-set packing problem is a variant of set packing in which each set contains at most k elements. When k =1, the problem is trivial. When k =2, the problem is equivalent to finding a maximum cardinality matching , which can be solved in polynomial time.
If there are multiple maximum cliques, one of them may be chosen arbitrarily. [14] In the weighted maximum clique problem, the input is an undirected graph with weights on its vertices (or, less frequently, edges) and the output is a clique with maximum total weight. The maximum clique problem is the special case in which all weights are equal ...
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.