Maximum Bipartite Matching

1
Maximum Bipartite Matching

• In a graph G, if no M-augmenting path exists,
  then M is a maximum matching in G.
• In a bipartite graph, we iteratively seek
  augmenting paths to enlarge the current matching.
  If we dont find an augmenting path, we can find
  a vertex cover with the same size as the current
  matching, then the current matching has maximum
  size.

2
Algorithm 3.2.1

• Input An X,Y-bigraph G, a matching M in G, and
  the set U of M-unsaturated vertices in X.
• Output Output an M-augmenting path or a vertex
  cover of size M.

3
Algorithm 3.2.1

• Exploring M-alternating paths from U, letting S?X
  and T?Y be the sets of vertices reached. Marking
  vertices of S that have been explored for path
  extensions.

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Algorithm 3.2.1 (2/2)

• 1. SU and T?.
• 2. If S has no unmarked vertex, stop and report
  T?(X-S) as a minimum cover and M as a maximum
  matching.
• 3. Explore alternating path from an unmark x?S.
• 3.1. If there exists unsaturated y?N(x) such
  that xy?M, then stop and report an M-augmenting
  path from U to y.
• 3.2. For each xy?E(G), include y in T and
  include w in S, where yw?E(G). Mark x and Goto
  Step 2.

5
Theorem 3.2.2

• Repeatedly applying Algorithm 3.2.1 to a
  bipartite graph produces a matching and a vertex
  cover of equal size
• 1. Let QT?(X-S) and M be the matching when
  algorithm 3.2.1 stops at step 2.
• 2. No edge joins S to Y-T. ? Q is a vertex cover.
• 3. Each y?T is M-saturated.
• 4. Each vertex of X-S is M-saturated.
• 5. No edge in M joins T to X-S.
• 3.4. 5. ? M has at least TX-S edges. ? Q is
  a minimum vertex cover and M is a maximum
  matching.

6
Time Complexity of Algorithm 3.2.1

• Let G be an X,Y-bigraph with n vertices and m
  edges.
• 1. Algorithm 3.2.1 executes at most n/2 times
  since ?(G)ltn/2.
• 2. Each execution of Algorithm 3.2.1 considers
  each edge at most once.
• ? The execution time of Algorithm 3.2.1 is O(mn).

7
Maximum Weighted Bipartite Matching and Minimum
Weighted Cover (Dual Problems)

• A farming company owns n farms and n processing
  plants. The profit that results from sending the
  output of farm i to plant j is wi,j. The company
  wants to select edges forming a matching to
  maximize total profit.
• The government will pay ui if the company agrees
  not to use farm i and vj if it agrees not to use
  plant j. The government must offer amounts such
  that uivjgtwi,j for all i, j. Otherwise, the
  company makes more by using edge xiyj than taking
  the government payments for those vertices. The
  government wants to minimize ?ui?vj.

8
Lemma 3.2.7

• For a perfect matching M and weighted cover (u,v)
  in a weighted bipartite graph G, (1)
  c(u,v)gtw(M), and (2) c(u,v) w(M) iff M consists
  of edges xiyj such that uivjwi,j. In (2), M and
  (u,v) are optimal.
• (1) c(u,v)gtw(M) since uivjgtwi,j for all i, j.
• (2) Trivial.

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Equality Subgraph

• Equality Subgraph The equality subgraph Gu,v for
  a weighted cover (u,v) is the spanning subgraph
  of Kn,n whose edges are the pairs xiyj such that
  uivjwi,j. In the cover, the excess for i,j is
  uivj-wi,j.

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Algorithm 3.2.9

• Input A matrix of weights on the edges of Kn,n
  with bipartition X,Y.
• Output A maximum weighted matching.
• Idea Iteratively adjusting the cover (u,v) until
  the equality subgraph Gu,v has a perfect
  matching.
• Initialization Let (u,v) be a cover, such that
  uimaxj wi,j and vj0.

11
Algorithm 3.2.9
12
Algorithm 3.2.9

• 1. Find a maximum matching M in Gu,v. If M is
  perfect matching, stop and report M as a maximum
  weighted matching.
• 2. Let Q be a vertex cover of size M in Gu,v.
  Let RX?Q and TY?Q. Leteminuivj- wi,j
  xi?X-R, yj?Y-T.
• 3. Decreases ui byefor xi ?X-R, and increases vj
  byefor yj ?Y.
• 4. Form the new equality graph and Goto Step 1.

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Theorem 3.2.11

• Algorithm 3.2.9 finds a maximum weight matching
  and a minimum cost cover
• 1. Let (u,v) be the current cover and suppose
  that the equality subgraph has no perfect
  matching. Let (u,v) be the new lists of numbers
  assigned to the vertices.
• (1) uivj uivj for edges xiyj from X-R to
  T or from R to Y-T.
• (2) uivj uivje for edges xiyj from R to
  T.
• (3) uivj uivj-e for edges xiyj from X-R
  to Y-T.
• ? (u,v) is a cover since eminuivj- wi,j
  xi?X-R, yj?Y-T.

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Theorem 3.2.11

• 2. It terminates only when the equality subgraph
  has a perfect matching. ? equal value for the
  current matching and cover.
• 3. M is not a perfect matching. ? QMltX. ?
  RTltX-RR. ? TltX-R. ? at each
  iteration the cost of the cover is reduced at
  least e. ? algorithm 3.2.9 terminates since the
  cost starts at some value and is bounded below by
  the weight of a perfect matching.