Matching and duality in bipartite Graphs

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Matching and duality in bipartite Graphs

• Alireza Fathi

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Introduction-Matching

• G (V,E).
• Matching ? M.
• A matching M is a subset of edges E for all
  vertices v in V.
• At most one edge of M is incident on v.
• v is matched by matching M if one edge in M is
  incident on v, otherwise, v is unmatched.
• A maximum matching is a matching with maximum
  cardinality.
• A matching M such that for every matching M ,
  M gt M
• Perfect matching is a matching that contains all
  vertexes in V

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Introduction-bipartite graph, practical
applications

• Bipartite graph is a graph which
• all vertexes can partitioned into two V (L U
  R)
• L and R are disjoint and all edges in E go
  between L and R
• every vertex in V also has one incident edge
• Practical applications
• Example
• Matching a set L of machines with a set R of
  tasks to be performed simultaneously . Presence
  of an edge (u,v) means that a particular machine
  u from L is capable to perform the task v from
  R.
• maximum matching provides tasks for as many
  machines as possible.

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Introduction-flow networks
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Corresponding flow network

• We define the corresponding graph G (V,E)
  for the bipartite graph G as follows.
• V V U S,T
• E (s,u) u in L U (u,v) u in L, v in R,
  (u,v) in E
• U (v,t) v in R
• each edge in G has unit flow capacity.
• We say that a flow f is an integer-valued if
  f(u,v) is integer for all v,u in V.

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Lemma

• Let G (V,E) be a bipartite graph with vertex
  partition V L U R, and let G (V,E) be its
  corresponding flow network.
• If M is matching in G, then there is an
  integer-valued flow f in G with value f M.
• Conversely, if f is an integer-valued flow in G,
  then there is a matching M in G with cardinality
  M f

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Cardinality of a maximum matching M in a
bipartite graph G is the value of a maximum flow
f in its corresponding flow network

• Proof we use the LEMMA .
• suppose that M is the maximum flow network in G
  but f is not the maximum flow in G.
• then there is a maximum flow network f in G
  which f gt f
• since the capacities in G are integer-valued f
  has an integer value too.
• so there is corresponding matching M for f in
  G with cardinality
• M f gt f M contradicting our
  assumption which M is the maximum matching in G.

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Halls theorem

• Definition a vertex cover in G is a set of S
  vertices such that S contains at least one
  endpoint of every edge in G. the vertices in S
  cover the Edges in G.
• Halls theorem
• If G is a bipartite graph with bipartition X, Y
  , then G has a matching of X into Y if and only
  if N(S) gt S for all S from X

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Maximum matching minimum covering

• Theorem If G is a bipartite graph , then the
  maximum size of a matching in G equals the
  minimum size of a vertex cover in G.

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Weighted bipartite matching
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Resources

• Douglas B.West, Introduction to Graph
  theory,1996,Prentice-Hall.
• CLRS,Introduction to algorithms,2002
• Papers
• Algorithms for bipartite matching, Mustafa Nabil
• Matching on bipartite graphs, Brandenburg
• How to find a maximum matching on a bipartite
  graph, unknown!