1. Vertex Covers and Matchings
Given an undirected graph G=(V,E),
a vertex cover of G is a subset of the vertices C such that for every edge (i,j) in E i is in C and/or j is in C.
-a matching in G is a subset of the edges M such that no two edges in M share the same vertex.

2. Example
C = {1, 2, 3, 4}
M = {(1,5), (2,3), (4,6)} 1 2 5 6 3 4

3. Maximum Cardinality Matching
Find the largest matching in a given graph G.
Tricky (put polynomial) in general, but “easy” in a bipartite graph.
A bipartite network has two sets of nodes N1 and N2 such that all edges have one endpoint in N1 and the other in N2.

4. A Bipartite Graph N1 N2 1 3 5 7 2 4 6 8

5. Max Flow Network G’
capacity =  1 2 4 6 8 1 1 3 s t 5 7

6. Mapping Flows into Matchings
Given a feasible flow in G’, let (i,j) be an edge in M if and only if xij = 1.
Observe that |M| = value of the flow.
If two edges in E, (i,j) and (k,j), share a node, then either xij = 0, or xkj = 0, or both. Otherwise the arc capacity of (j,t) will be violated.
If two edges in E, (i,j) and (i,k), share a node, then either xij = 0, or xik = 0, or both.

7. Mapping Matchings into Flows
Start with a zero flow.
If (i,j) is an edge in M, then let xsi=1, xij=1, and xjt=1.
Consider a pair of matched nodes i and j.
The flow x sends exactly one unit of flow to node i on arc (s,i) and exactly one unit of flow into the sink on arc (j,t)
Thus, a matching of size |M| gives a feasible s-t flow of value |M|.

8. Max Flow in Network G’ 1 1 2 4 6 8 1 1 3 1 s t 5 1 7

9. Matching in G 1 3 5 7 2 4 6 8

10. Residual Network 1 2 3 4 s t 5 6 7 8 S ={S, 1, 2, 3, 5, 6}
T ={T, 4, 7, 8} 7 8

11. Minimum Cardinality Vertex Covers
Find a vertex cover with a minimum number of nodes.
Hard in general, but polynomial in bipartite graphs.
Solve max flow problem as described earlier and find min cut [S,T].
C = {i in N1  T}  {i in N2  S} is a minimum cardinality vertex cover.

12. Vertex Cover in G N1 N2 1 2 3 4 5 6 7 8 T ={T, 4, 7, 8}
S ={S, 1, 2, 3, 5, 6}

13. Correctness of Vertex Cover Result
j in N2  S
N1  S 1 j s t 1
N2  T i
i in N1  T
can’t be in min cut

14. Correctness of Vertex Cover Result
Let [S,T] be a finite cut in G.
Claim: C = {i in N1  T}  {i in N2  S} is a vertex cover.
Suppose not.
This implies there is some edge (a,b) such that a is in N1 and S, and b is in N2 and T.
Since a is in N1 and b is in N2 capacity (a,b) = .
But (a,b) goes from s to t. So, u[S,T] = .

15. Correctness of Vertex Cover Result
The [S,T] be a finite cut in G.
Claim: C = {i in N1  T}  {i in N2  S} is a vertex cover such that|C|=u[S,T].
Theorem for Bipartite Graphs: The cardinality of a maximum-size matching is equal to the cardinality of a minimum-size vertex cover.