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

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.

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

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.

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

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

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

Matching in G 1 3 5 7 2 4 6 8

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

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.

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 = 

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 + 0 + 0 = 4 |{j T  N1}| = 4 |{(i, j): i S  N1, j T  N2}| = 0 |{i S  N2}| = 0

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 = 

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 + 0 + 4 = 0 |{j T  N1}| = 0 |{(i, j): i S  N1, j T  N2}| = 0 |{i S  N2}| = 4

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 = 

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

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.

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.

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.

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

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

Vertex Cover & Independent Set in G 1 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}

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

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|.

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