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Vertex Covers and MatchingsPowerPoint Presentation

Vertex Covers and Matchings

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## PowerPoint Slideshow about ' Vertex Covers and Matchings' - allen-perez

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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.

- 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.

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.

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.

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|.

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.

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

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] = .

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.

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