Kőnig's theorem through min-cut/max-flow duality

Revision en1, by adamant, 2022-08-06 21:57:14

Hi everyone!

There is a somewhat well-known result about bipartite graphs which is formulated as follows:

In any bipartite graph, the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover.

Now, the result is well-known and knowing it you can easily find the size of a minimum vertex cover in a given bipartite graph. But what about actually recovering such cover? Well... There is an algorithm to it, but it doesn't seem to be very motivated, hence I constantly forget its details. However, there is a simple way to "recover" it in your memory using the duality between maximum flow and minimum cut.

It is most likely well known to those familiar with the topic, but still might be of interest to someone.

Recall that given a flow network $$$G=(V, E)$$$, finding minimum cut between $$$s$$$ and $$$t$$$ is done as follows:

  • Find a maximum flow $$$f$$$ from $$$s$$$ to $$$t$$$ in the network;
  • Find $$$X$$$, the set of all vertices reachable from $$$s$$$ in the residual network;
  • Minimum cut is formed by $$$S=X$$$ and $$$T=V \setminus X$$$, that is the minimum cut is formed by the edges going from $$$S$$$ to $$$T$$$.

Now, recall that bipartite matching between sets of vertices $$$A$$$ and $$$B$$$ may be found as maximum flow in the following network:

  • There is an edge of capacity $$$1$$$ going from $$$s$$$ to every vertex from $$$A$$$;
  • There is an edge of capacity $$$1$$$ going from every vertex from $$$B$$$ to $$$t$$$;
  • There is an edge of capacity $$$+\infty$$$ going between some vertices of $$$A$$$ and $$$B$$$, as defined by the bipartite graph.

Now... Is there anything special about minimum cut in such network?


Minimum cut $$$(S, T)$$$ in a flow network induced by a bipartite graph

Generally, some vertices from $$$A$$$ and some vertices from $$$B$$$ will be with the source in the min

Tags tutorial, konigs theorem, bipartite matching

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en15 English adamant 2022-08-07 14:51:32 57 Tiny change: 'first one.\n\nIt is like' -> 'first one. It is like'
en14 English adamant 2022-08-07 03:35:16 4
en13 English adamant 2022-08-07 02:03:52 383 Tiny change: 'nstruct a flow network, ' -> 'nstruct a network, '
en12 English adamant 2022-08-07 02:00:57 9
en11 English adamant 2022-08-07 02:00:31 2167 + Hall lemma
en10 English adamant 2022-08-06 23:35:52 10 Tiny change: '$ or $B_S$;\n- Minim' -> '$ or $B_S$ (or both);\n- Minim'
en9 English adamant 2022-08-06 23:07:05 16 Tiny change: ' no edges between $A_S$ and $B_T$ (ot' -> ' no edges from $A_S$ to $B_T$ (ot'
en8 English adamant 2022-08-06 23:05:45 19 Tiny change: 'and $B_T$ or $A_T$ and $B_S$ (otherwi' -> 'and $B_T$ (otherwi'
en7 English adamant 2022-08-06 22:35:31 36 Tiny change: 'ver**.\n\nNow, the result is well-known and knowing it ' -> 'ver**.\n\nKnowing it '
en6 English adamant 2022-08-06 22:32:30 16
en5 English adamant 2022-08-06 22:31:16 10
en4 English adamant 2022-08-06 22:29:00 216
en3 English adamant 2022-08-06 22:24:33 8
en2 English adamant 2022-08-06 22:16:36 828 Tiny change: 'g" width="600px">\n<b' -> 'g" width="700px">\n<b' (published)
en1 English adamant 2022-08-06 21:57:14 1972 Initial revision (saved to drafts)