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首先要声明一点：如果你关心的是perfect matching的话，那么最大权和最小权是同一个问题，因为你总是可以把权负过来最大就变成了最小；如果你不喜欢负权还可以给所有边加一个足够大的数把它们变正，并不影响算法的正确性。. 同样道理，你这里最大化最小权 ...
2015年12月7日 · A bipartite perfect matching (especially in the context of Hall's theorem) is a matching in a bipartite graph which involves completely one of the bipartitions. If the bipartite graph is balanced – both bipartitions have the same number of vertices – then the concepts coincide.
2015年2月18日 · So, there is a polynomial-time algorithm to find the perfect matching whose total weight is maximized. Given this, we can solve your problem (find the perfect matching whose total weight is minimized) by simply negating all of the edge weights.
However, the classic Blossom Algorithm can find a perfect matching in polynomial time--determining if a graph has a perfect matching is in $\mathsf{P}$. This seems like a conflict to me: we can reduce the perfect matching problem to Linear Programming, but its polytope must be exponentially large.
2018年2月19日 · It is easy to show that there is a perfect matching for the graph, by using flow and showing a flow of size $|V|/2$. Using flow networks, we can also use Dinic's algorithm to find the max-flow which is equal to the perfect matching (thus, deriving a perfect matching
2021年11月28日 · If we remove all vertices in this manner, we have found a perfect matching. Notice that, while the removal may disconnect the graph, it will remain acyclic and this is what matters. Disconnecting the graph essentially splits the tree into multiple trees, so the iteration continues to make sense even in this case.
2016年8月12日 · To the best of my knowledge, finding a perfect matching in an undirected graph is NP-hard. But is this also the case for directed and possibly cyclic graphs? I guess there are two possibilities to
$\begingroup$ @Guess601 If the problem (Maximum matching in hypergraphs) is NP-hard for some "restricted set" of all possible inputs (3-partite hypergraphs, as shown by the wikipedia), then it is easy to observe (consider the definition of NP-hard) that it is also NP-hard in general (on hypergraphs that may or may not be tripartite).
2019年2月8日 · A minimum-weight perfect 2-matching of a graph G is a subgraph M of minimal total edge weight, such that each vertex in G is incident by exactly 2 edges from M. matching. Share. Cite. Improve this question. asked Feb 8, 2019 at 23:02.
2. I'm trying to find pairs in a complete, weighted graph, similar to the one below (weights not shown). For each possible pair there is a weight and I would like to find pairs for including all vertices, maximizing the weight of those pairs. Many of the algorithms for finding maximum matchings are only concerned with finding them in bipartite ...
