CS6234: Lecture 1
 Matching in Graph
 Matching in Bipartite Graph [PS82]-Ch10
 Matching in General Graphs [PS82]-Ch10
 Weighted Matching in Bipartite Graph
[PS82]-Ch 11.1—11.2
 Additional topics:

Reading/Presentation by students
Lecture notes modified from those by Lap Chi LAU, CUHK.
(CS6234, Spring 2009) Page 1
Hon Wai Leong, NUS
Copyright © 2009 by Leong Hon Wai

2

Bipartite Matching
A graph is bipartite if its vertex set can be partitioned into two subsets A and B so that each edge has one endpoint in A and the other endpoint in B.
A B
A matching M is a subset of edges so that every vertex has degree at most one in M.
3

Maximum Matching
The bipartite matching problem:
Find a matching with the maximum number of edges.
A perfect matching is a matching in which every vertex is matched.
The perfect matching problem: Is there a perfect matching?
4

First Try
• Greedy method?
(add an edge with both endpoints unmatched )
5

Key Questions
• How to tell if a graph does not have a (perfect) matching?
• How to determine the size of a maximum matching?
• How to find a maximum matching efficiently?
6

Existence of Perfect Matching
Hall’s Theorem [1935]:
A bipartite graph G=(A,B;E) has a matching that “saturates” A if and only if |N(S)| >= |S| for every subset S of A.
N(S)
S
7

Bound for Maximum Matching
What is a good upper bound on the size of a maximum matching?
Min-max theorem
Implies Hall’s theorem.
8

Algorithmic Idea?
Any idea to find a larger matching?
9

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.
10

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
1.
Consider
2.
Every vertex in has degree at most 2
3.
A component in is an even cycle or a path
4.
Since , an M-augmenting path!
11

Key: M is maximum  no M-augmenting path
12

M


Orient the edges (edges in M go up, others go down)
An M -augmenting path  a directed path between two unmatched vertices
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 At most n iterations
 An augmenting path in time by a DFS or a BFS
 Total running time
14

Minimum Vertex Cover
Hall’s Theorem [1935]:
A bipartite graph G=(A,B;E) has a matching that “saturates” A if and only if |N(S)| >= |S| for every subset S of A.
König [1931]:
In a bipartite graph, the size of a maximum matching is equal to the size of a minimum vertex cover.
Idea: consider why the algorithm got stuck…
15

Faster Algorithms
Observation: Many short and disjoint augmenting paths.
Idea: Find augmenting paths simultaneously in one search.
16

Application of Bipartite Matching
Isaac Jerry Darek Tom
Marking Tutorials Solutions Newsgroup
Job Assignment Problem:
Each person is willing to do a subset of jobs.
Can you find an assignment so that all jobs are taken care of?
17

Application of Bipartite Matching
With Hall’s theorem, now you can determine exactly when a partial chessboard can be filled with dominos.
18

Application of Bipartite Matching
Latin Square: a nxn square, the goal is to fill the square with numbers from 1 to n so that:
• Each row contains every number from 1 to n.
• Each column contains every number from 1 to n.
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Application of Bipartite Matching
Now suppose you are given a partial Latin Square.
Can you always extend it to a Latin Square?
With Hall’s theorem, you can prove that the answer is yes.
20

CS6234:
 Matching in Graph
 Matching in Bipartite Graph
 Matching in General Graphs
 Weighted Matching in Bipartite Graph
 Additional topics:

Reading/Presentation by students
Lecture notes modified from those by Lap Chi LAU, CUHK.
(CS6234, Spring 2009) Page 21
Hon Wai Leong, NUS
Copyright © 2009 by Leong Hon Wai

22

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
23

Today’s Plan
• Min-max theorems [no proofs]
• Polynomial time algorithm
• Chinese postman [skip]
Follow the same structure for bipartite matching.
24

Characterization of Perfect Matching
Hall’s Theorem [1935]:
A bipartite graph G=(A,B;E) has a matching that “saturates” A if and only if |N(S)| >= |S| for every subset S of A.
Tutte’s Theorem [1947]:
A graph has a perfect matching if and only if o(G-S) <= |S| for every subset S of V.
25

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 =
26

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.
27

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
1.
Consider
2.
Every vertex in has degree at most 2
3.
A component in is an even cycle or a path
4.
Since , an M-augmenting path!
28

Key: M is maximum  no M-augmenting path
How to find efficiently?
29

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.
30

M
• Don’t know how to orient the edges so that: an M -augmenting path  a directed path between two free vertices
…alternating path segment (before & after augmentation):
…  v1  v2  v3  v4  v5  v6  v7  …
Repeated
Vertex !!!
31

• Just find an alternating path without repeated vertices?!
Either we may exclude some possibilities,
Or we need to trace the path…
32

Note: When performing search for M-augmenting path, we may encounter M-blossom!
33

(Edmonds 1965) General matching algorithm
Shrink the blossoms!
34

• (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
35

 an M/C -augmenting path 
 an M -augmenting path
Note: One of the two paths around the blossom will be the correct one.
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 an M -augmenting path 
 an M/C -augmenting path
Note: Many cases to be considered (see [PS82]-Ch10.)
Case 0: M-augmenting path p does not “intersect” M-blossom C
Case 1: p enters/leaves M-blossom C with matched edge
37

 an M -augmenting path 
 an M/C -augmenting path
Case 2: p enters M-blossom C with free edge
2(a): p leaves via matched edge (1 st example [in black])
2(b): p leaves C with free edge (2 nd example [in red])
38

Key: M is maximum  no M-augmenting path
How to find efficiently?
39

M
• 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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• At most n augmentations.
• Each augmentation does at most n contractions.
• An alternating walk can be found in O ( m ) time.
• Total running time is O( mn 2 ) .
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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.
An O(n3) algorithm by Pape and Conradt, 1980
Ref: [SDK83] Syslo, Deo, Kowalik, Prentice-Hall, 1983
Discrete Optimization Algorithms, Ch 3.7
42

Proving Min-Max Theorem
Tutte-Berge formula [1958]:
The size of a maximum matching =
Comments by LeongHW:
Skip next few slides on mainly combinatorial results…
(Anyone wants to do this for reading assignment?)
43

Larger forest
M-alternating forest
M-augmenting path a blossom
We make progress in any of the above cases.
44

M-alternating forest
Otherwise we find the set in Tutte-Berge formula.
45

Edmonds-Gallai Decomposition
Can be obtained from M-alternating forest.
Contains a lot of information about matching.
46

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.
47

Tutte Matrix
• Skew symmetric matrix, a(u,v)=x(e) and a(v,u)=-x(e)
• Full rank  perfect matching
• Randomized algorithm
48

CS6234:
 Matching in Graph
 Matching in Bipartite Graph
 Matching in General Graphs
 Weighted Matching in Bipartite Graph
 Additional topics:

Reading/Presentation by students
Lecture notes modified from those by Lap Chi LAU, CUHK.
(CS6234, Spring 2009) Page 49
Hon Wai Leong, NUS
Copyright © 2009 by Leong Hon Wai

50

Weighted Bipartite Matching
Given a weighted bipartite graph, find a matching with maximum total weight.
A B
Not necessarily a maximum size matching.
Wlog, assume perfect matching on complete bipartite graph;
(add fictitious edges with weights 0)
51

Today’s Plan
Three algorithms
• negative cycle algorithm
• augmenting path algorithm
• primal dual algorithm
Applications
52

First Algorithm
Find maximum weight perfect matching
How to know if a given matching M is optimal?
Idea: Imagine there is a larger matching M* and consider the union of M and M*
53

First Algorithm

Orient the edges (edges in M go up, others go down)
 edges in M having positive weights, otherwise negative weights
Then M is maximum if and only if there is no negative cycles
54

Key: M is maximum  no negative cycle
How to find efficiently?
55

• At most nW iterations
• A negative cycle in O( n 3 ) time by Floyd Warshall
• Total running time O( n 4 W )
• Can choose minimum mean cycle to avoid W
56

Comments by LeongHW:
Skip next few slides on Augmenting Path Algorithm
(Anyone wants to do this for reading assignment?)
57

Augmenting Path Algorithm

Orient the edges (edges in M go up, others go down)
 edges in M having positive weights, otherwise negative weights
Find a shortest path M-augmenting path at each step
58

Augmenting Path Algorithm
Theorem: Each matching is an optimal k-matching.
Let the current matching be M
Let the shortest M-augmenting path be P
Let N be a matching of size |M|+1
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• At most n iterations
• A shortest path in O ( nm ) time by Bellman Ford
• Total running time O ( n 2 m )
• Can be speeded up by using Dijkstra: O ( m + n log n )
• Total running time O ( nm + n 2 log n )
• Kuhn (based on proof by Egervary) — Hungarian method
60

Primal Dual Algorithm
What is an upper bound of maximum weighted matching?
What is a generalization of minimum vertex cover?
weighted vertex cover:
Minimum weighted vertex cover  Maximum weighted matching
61

Primal Dual Algorithm
Minimum weighted vertex cover  Maximum weighted matching
Primal Dual (Overview)
• Maintain a weighted matching M
• Maintain a weighted vertex cover y
• Either increase M or decrease y until they are equal.
62

Primal Dual Algorithm
Minimum weighted vertex cover  Maximum weighted matching
1.
Consider the subgraph formed by tight edges (the equality subgraph).
2.
If there is a tight perfect matching, done.
3.
Otherwise, there is a vertex cover.
4.
Decrease weighted cover to create more tight edges, and repeat.
Detailed algorithm will be covered later
(after LP Primal-Dual algorithms)
63

• Augment at most n times, each O ( m )
• Decrease weighted cover at most m times, each O ( m )
• Total running time: O ( m 2 ) = O ( n 4 )
• Egervary, Hungarian method
History: The algorithm was developed by Kuhn who based it on work of Egervary [1931], (“latent work of D. Konig and J.
Egervary”) and was given the name “Hungarian method”.
See [Schr02]-Ch17 for more details.
64

Quick Summary
1.
First algorithm, negative cycle, useful idea to consider symmetric difference
2.
Augmenting path algorithm, useful algorithmic technique
3.
Primal dual algorithm, a very general framework
1.
Why primal dual?
2.
How to come up with weighted vertex cover?
3.
Reduction from weighted case to unweighted case
65

Faster Algorithms
66

Application of Bipartite Matching
Isaac Jerry Darek Tom
Marking Tutorials Solutions Newsgroup
Job Assignment Problem:
Each person is willing to do a subset of jobs.
Can you find an assignment so that all jobs are taken care of?
• Ad-auction, Google, online matching
• Fingerprint matching
67

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