General Matching

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General Matching

• Lecture 5 Jan 24

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General Matching
Given a graph (not necessarily bipartite), find a
matching with maximum total weight.
unweighted (cardinality) version a matching
with maximum number of edges
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Todays Plan

• Min-max theorems
• Polynomial time algorithm
• Chinese postman

Follow the same structure for bipartite matching.
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Characterization of Perfect Matching
Halls Theorem 1935 A bipartite graph
G(A,BE) has a matching that saturates A if
and only if N(S) gt S for every subset S of A.
Tuttes Theorem 1947 A graph has a perfect
matching if and only if o(G-S) lt S for every
subset S of V.
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Min-Max Theorem
König 1931 In a bipartite graph, the size of
a maximum matching is equal to the size of a
minimum vertex cover.
Tutte-Berge formula 1958 The size of a
maximum matching
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Augmenting Path?
Given a matching M, an M-alternating path is a
path that alternates between edges in M and edges
not in M. An M-alternating path whose endpoints
are unmatched by M is an M-augmenting path.
Works for general graphs.
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Optimality Condition?
What if there is no more M-augmenting path?
If there is no M-augmenting path, then M is
maximum?

• Prove the contrapositive
• A bigger matching ? an M-augmenting path
• Consider
• Every vertex in has degree at most 2
• A component in is an even cycle or a path
• Since , an
  M-augmenting path!

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Algorithm?

• Key M is maximum ? no M-augmenting path

How to find efficiently?
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Finding Augmenting Path in Bipartite Graphs

• Orient the edges (edges in M go up, others go
  down)
• edges in M having positive weights, otherwise
  negative weights

In a bipartite graph, an M-augmenting path
corresponds to a directed path.
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Finding M-augmenting paths

• Dont know how to orient the edges so that
• An M-augmenting path ?
• a directed path between two
  free vertices

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No repeated vertices

• Just find an alternating path without repeated
  vertices?!

Either we may exclude some possibilities, Or we
need to trace the path
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Blossom
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Key idea
(Edmonds 1965) General matching algorithm

• Shrink the blossoms!

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Key Lemma

• (Edmonds) M is a maximum matching in G ?
• M/C is a maximum
  matching in G/C
• Key M is maximum ? no M-augmenting path
• an M-augmenting path in G ?
• an M/C-augmenting path in G/C

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An M/C-augmenting path ? an
M-augmenting path
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An M-augmenting path ? an
M/C-augmenting path
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Algorithm

• Key M is maximum ? no M-augmenting path

How to find efficiently?
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Finding an M-augmenting path

• Find an alternating walk between two free
  vertices.
• This can be done in linear time by a DFS or a
  BFS.
• Either an M-augmenting path or a blossom can be
  found.
• If a blossom is found, shrink it, and
  (recursively)
• find an M/C-augmenting path P in G/C, and
  then
• expand P to an M-augmenting path in G.

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Complexity

• At most augmentations.
• Each augmentation does at most
  contractions.
• An alternating walk can be found in
  time.
• Total running time is .

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Historical Remarks

• In his famous paper paths, trees, and flowers,
  Jack Edmonds
• Introduced the notion of a good algorithm.
• Edmonds solved weighted general matching, primal
  dual.
• Linear programming description, a breakthrough
  in polyhedral combinatorics.

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Proving Min-Max Theorem
Tutte-Berge formula 1958 The size of a
maximum matching
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M-alternating forest
Larger forest
M-augmenting path
a blossom
We make progress in any of the above cases.
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M-alternating forest
Otherwise we find the set in Tutte-Berge formula.
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Edmonds-Gallai Decomposition
Can be obtained from M-alternating
forest. Contains a lot of information about
matching.
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Chinese Postman
Given an undirected graph, a vertex v, a length
l(e) on each edge e, find a shortest tour to
visit every edge once and come back to v.
Visit every edge exactly once if Eulerian.
Otherwise, find a minimum weighted perfect
matching between odd vertices.
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Tutte Matrix

• Skew symmetric matrix, a(u,v)x(e) and
  a(v,u)-x(e)
• Full rank ? perfect matching
• Randomized algorithm