# Max matching algorithm lemma approximation algorithm

We have this algorithm which is supposed to find max matchings.


𝛭 = ∅;          //the matching is initialized to zero

for (𝑒 = {𝑥, 𝑦} ∈ 𝐸) {   //for each edge in the graph

if ( x and y vertices are free)  //check if M U e is a matching

𝛭 = 𝑀 ∪ 𝑒;
}


1. Give an example where this algorithm does not calculate a max matching
2. Prove that the algorithm is 2-approximation(The algorithm always calculates a matching M which has size |M| >= |M*|/2, where M* a max matching of the initial graph)
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– D.W.
May 17 at 4:07

For the lower bound take a path of three edges $$P_3$$. If the algorithm considers the middle edge first, it gets a matching of size one, meanwhile the matching consisting of the first and last edges has size two.
To prove this, we use notion of vertex cover. A vertex cover is a set of vertices that is incident to all edges in the graph. Note that if a graph $$G$$ has a matching of size $$k$$, then the smallest vertex cover of the graph must have size at lest $$k$$, since these edges do not share endpoints and we need to choose one endpoint of each such edge in the matching. So the vertex cover number of a graph $$\nu(G)$$ (the smallest size of a vertex cover) is at least the matching number $$\mu(G)$$ of this graph (the size of a maximum matching).
On the other hand, note that for a maximal matching $$M$$, the set $$S$$ containing both endpoints of each edge in $$M$$ is a vertex cover, since otherwise, there must exist an edge with both endpoints not in $$S$$, but such an edge can be added to $$M$$ to get a larger matching which contradicts the assumption that $$M$$ is maximal. Hence $$2|M| \geq |S| \geq |\nu(G)|\geq |\mu(G)|$$.