algorithm – 二分图中的最大匹配

2023年1月10日 135次阅读

使用以下启发式算法：

M = NULL
while E != NULL do {
if ((∃u vertex) and (gr(u) == 1)) then 
    e ← the incident edge with u
  else 
    e ← an incident edge with a vertex with the most incident edges
M ← M ∪ {e}
E ← E - (all the incident edges with e)
}
return M //return the matching

其中：M,E – 边缘; gr(u) – u的等级(与u的入射边数);

我们被问到的是什么：

  a) Prove that this algorithm returns the maximum matching for a tree.
  b) Prove that if there is a perfect matching M0 then the algorithm returns it, for any bipartite graph.
  c) Prove that |M| ≥ (v(G)/2), for any bipartite graph. 
  //G is the graph, v(G) is the matching number, size of the maximum matching.

我几乎可以肯定这个算法类似于我未能找到的一些经典算法,或者解决方案可以完全基于二分图的定理和性质.

能不能给我一个起点..我错过了什么？

我认为a)很容易..我仍然试图找到正确的证据,我认为它可能完全基于树和二分图的属性.
对于b)和c)..我还没有任何想法.

最佳答案 这与贪心匹配算法非常相似.有关更多信息,请参见
the wikipedia article.

至于问题……

a) Show that the matching you get is maximal (there are no larger matchings containing it). What does this imply on a tree?
b) Show that if M0 is a valid matching that can be found in M ∪ E in a given step, that it can be found in M ∪ E in the next. By induction, the statement holds.
c) Consider a maximum matching M1. Why must at least one of the vertices adjacent to any given edge in M1 appear as an endpoint for some edge in the matching the algorithm outputs? What does this tell you about a lower bound for its size?
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