从加权图中选择边，这样每个顶点都是一个边的端点，并且边权重的总和被最小化

Question

For simplication, we can assume that the graph G=(V,E) has 2N vertexes and the answer has N edges.

I have learned that if the graph is bipartite, Hungarian algorithm works well. However, I wonder if there is any nontrivial solution(ie a polynomial one) for a general graph.

Any polynomial solutions, as well as a proof of NP Complexity, are welcome.

Answer 1

If you want every vertex be incident to exactly one edge, then you need to find perfect matching. But perfect matching not always exists even for a graph with even number of vertices.

You can see example in this answer .