1. EMIS 8374 Max-Flow Applications: Matchings, Covers, and Independent Sets Updated 19 April 2008

2. Vertex Covers, Matchings, and Independent Sets
Let G=(V, E) be an undirected graph.
A vertex cover of G is a subset of the vertices, C, such that for every edge (i, j) in E either node i or node j, or both, is in C.
-A matching in G is a subset, M, of the edges such that no two edges in M share the same vertex.
An independent set is a subset of vertices, I, such that if i and j are elements of I, then there is no edge (i, j) in E.

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

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

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

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

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

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

9. s-t Cuts in G' N1 N2
i  N2  S
N1  S 1 i s t 1
N2  T j
j  N1  T

10. s-t Cuts in G' Which arcs will be in an s-t cut in G'?
Arcs (i, j) where i is in S and j is in T.
Let (i, j) be an arc in [S, T].
There are 3 possible cases:
If i = s, j  N1, then uij = 1.
If i  N1, j  N2, then uij = .
If i  N2, j = t, then uij = 1.

11. s-t Cuts in G': S = {s} N1 N2 1 2 4 6 8 1 1 3 s t 5 7 u[S, T] = 4
capacity = 

12. s-t Cuts in G': S = {s} S = {s}, T = {1, 2, 3, 4, 5, 6, 7, 8, t}
Arcs in [S, T]:
(s, 1), (s, 3), (s, 5), (s, 7)
u[S, T] = = 4
|{j T  N1}| = 4
|{(i, j): i S  N1, j T  N2}| = 0
|{i S  N2}| = 0

13. s-t Cuts in G': T = {t} N1 N2 1 2 4 6 8 1 1 3 s t 5 7 u[S, T] = 4
capacity = 

14. s-t Cuts in G': T = {t} S = {s, 1, 2, 3, 4, 5, 6, 7, 8}, T = { }
Arcs in [S, T]:
(2, t), (4, t), (6, t), (8, t)
u[S, T] = = 0
|{j T  N1}| = 0
|{(i, j): i S  N1, j T  N2}| = 0
|{i S  N2}| = 4

15. s-t Cuts in G': S = {s, 1, 3, 2, 4} N1 N2 1 2 4 6 8 1 1 3 s t 5 7
u[S, T] = 
capacity = 

16. s-t Cuts in G': S = {s, 1, 3, 2, 4}
S = {s, 1, 2, 3, 4}, T={ 5, 6, 7, 8, t}
Arcs in [S, T]:
(s, 5), (s, 7), (1, 6), (3, 6), (2, t), (4, t),
u[S, T] = 2 + (2) + 2 = 
|{j T  N1}| = |{5,7}| = 2
|{(i, j): i S  N1, j T  N2}| = 2
|{i S  N2}| = |{2,4}|=2

17. Observations S = {s} is always a finite cut in G' with u[S, T] = |N1|.
|{(i, j): i S  N1, j T  N2}| = 0 in any minimum s-t cut in G'.
{S  N1}  {T  N2} is an independent set.

18. Implication
Let [S, T] be a minimum s-t cut in G' and let (a, b) in an edge in G.
W.L.O.G assume a N1 and b  N2.
|{(i, j): i S  N1, j T  N2}| = 0.
If a  S, then b also  S. So, b  S  N2.
If b T, then a also  T. So, a  T  N1.
Every edge (a, b) in G touches a node in {S  N2} and/or a node in {T  N1} .
{S  N2} {T  N1} is a vertex cover of G.

19. 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].
Then C = {i  N1  T}  {i  N2  S} is a minimum cardinality vertex cover.

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

21. 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

22. Vertex Cover & Independent Set in G1 2
{N2  S} 3 4
{N1  S} 5 6
{N2  T} 7 8
{N1  T}
T ={t, 4, 7, 8}
S ={s, 1, 2, 3, 5, 6}

23. Vertex Covers and Matchings in Bipartite Graphs
Let [S, T] be a finite capacity cut in G.
C = {i  N1  T}  {i  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.
M* = {(1,2), (3,6), (7,8)} in our example
C* = {2, 6, 7} in our example

24. Independent Sets and Vertex Covers in Bipartite Graphs
Let [S, T] be a finite capacity cut in G.
C = {i  N1  T}  {i  N1  S} is a vertex cover such that |C| = u[S, T].
I = {i  N1  S}  {i  N1  T} is an independent set such that |I|=|V| - |C|.

25. Independent Sets and Vertex Covers in Bipartite Graphs
Theorem for Bipartite Graphs: The cardinality of a maximum-size independent set is equal to the total number of vertices minus cardinality of a minimum-size vertex cover.
C* = {2, 6, 7} in our example = {2, 6, 7} in our example
I* = {1,3,4,5,8} in our example
|I*| = |V| - |C*| = 8 – 3 = 5