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
A Bipartite Graph
N2
N1
5
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 xsi1, xij1,
  and xjt1.
• 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
3
1
s
t
5
1
7
9
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
S S, 1, 2, 3, 5, 6
T T, 4, 7, 8
13
Correctness of Vertex Cover Result
N1
N2
j in N2 ? S
N1 ? S
1
j
s
t
1
N2 ? T
i
cant 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, uS,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 thatCuS,T.
• Theorem for Bipartite Graphs The
  cardinality of a maximum-size matching is equal
  to the cardinality of a minimum-size vertex cover.