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.

Example

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

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.

A Bipartite Graph

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N1 N2

Max Flow Network G’

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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|.

Max Flow in Network G’

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Matching in G

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Residual Network

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s t

S ={S, 1, 2, 3, 5, 6}

T ={T, 4, 7, 8}

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.

Vertex Cover in G

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S ={S, 1, 2, 3, 5, 6}

N1 N2

T ={T, 4, 7, 8}

Correctness of Vertex Cover Result

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i in N1 T

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can’t be in min cut

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.