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Let's consider this graph:

enter image description here

Now I take a matching M that only contains the edge 1. Clearly this matching is not maximum, because I can take the edge 3, so given that:

We can easily notice that if there exists an augmenting path p with respect to M, then M is not maximum.

I should find an augmenting path w.r.t M. Looking at the definition for a matching M:

An augmenting path is an alternating path that starts from and ends on free vertices.

and

an alternating path is a path in which the edges belong alternatively to the matching and not to the matching,

What I don't get is how to build such a path given my matching? My only possibilites are:

  • start from the top vertex of the edge 2
  • start from the right vertex of the edge 5
  • start from the bottom vertex of the edge 4

Now from there I take the edge 2 or 4 or 5, then I take the edge 1 and then I cannot go further.

So is the first quote a one-way implication (so if M is not a maximum matching then I don't always find an augmenting path)? Or did I misunderstood how to build my augmenting path?

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Looks like there are 4 maximal matchings:{3,1}, {3,4}, {3,5}, {2}. They're matchings, since they do not share any vertices, and they're maximal since by adding any additional edge (to any of those four) necessitates the existence of a shared vertex, thereby rendering them as non-matchings.

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  • $\begingroup$ I know how to find the maximum matchings in this graph. My question is, given the matching I started with, which I know that it's not a maximum matching, how can I find an augmenting path w.r.t this matching? Sorry maybe the title of the question is a bit misleading but I think it's enough clear in the body ;) $\endgroup$ – user2336315 May 22 '15 at 19:07
  • $\begingroup$ Ah, indeed! Let's see. $\endgroup$ – prime4567 May 22 '15 at 19:15
  • $\begingroup$ A path is a sequence of edges that connects the vertices, so I'm not sure 3,1,4 and 3,1,5 are paths since the vertices from 3 are not connected with the one from 1-4 or 1-5. $\endgroup$ – user2336315 May 22 '15 at 19:40
  • $\begingroup$ I think I got it. I think the trick is in the fact that you traverse 1 back and forth in one alternation. This is an undirected graph so given an edge with vertices a and b the edge is just {a,b}, no? Unlike in a directed graph where they would be distinct, i.e. (a,b) or (b,a). So the trick is that you go up the edge 1 and down again BUT that counts as a single alternation. I hope that was clear? So the augmented path starts with edge 2, then 1(going up), then 1(going back down), and then edge 4 (or 5 for another path). $\endgroup$ – prime4567 May 22 '15 at 20:07
  • $\begingroup$ But 2-1-5 and 2-1-4 are not paths. The vertices need to be distincts (except the first and the last and then you get a cycle)... If I follow the vertices from your "path", I get two times on the bottom vertice of the edge 1 and that is not my starting/end point so it's not a path. $\endgroup$ – user2336315 May 22 '15 at 20:10
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The algorithm you're looking for is called the Hungarian Maximum Matching Algorithm, and it works for any bipartite graph, for which it always gives the largest-size matching. More detailed info can be found on this Wolfram MathWorld page and this link has a neat visualisation of how the algorithm works.

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