Vertex Covers and Matchings

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

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

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

10. Residual Network S ={S, 1, 2, 3, 5, 6} 1 2 3 4 s t 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 N1 N2 j in N2  S N1  S 1 j s t 1 N2  T i can’t be in min cut i in N1  T

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.