1 “graph theory” course for the master degree program “geographic information systems” yulia...

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“Graph theory”Course for the master degree program

“Geographic Information Systems”

Yulia BurkatovskayaDepartment of Computer EngineeringAssociate professor

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Bipartite Matching

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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 matching M is a subset of edges so that

every vertex has degree at most one in

M.

A B

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The bipartite matching problem:

Find a matching with the maximum number of edges.

Maximum Matching

A perfect matching is a matching in which every vertex is matched.

The perfect matching problem: Is there a perfect matching?

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Greedy method?

(add an edge with both endpoints unmatched)

First Try

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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?

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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.

SN(S)

Existence of Perfect Matching

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König [1931]:

In a bipartite graph, the size of a maximum matching

is equal to the size of a minimum vertex cover.

What is a good upper bound on the size of a maximum matching?

Min-max theorem

Implies Hall’s theorem.

Bound for Maximum Matching

König [1931]:

In a bipartite graph, the size of a maximum matching

is equal to the size of a minimum vertex cover.

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Any idea to find a larger matching?

Algorithmic Idea?

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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.

Augmenting Path

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What if there is no more M-augmenting path?

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!

If there is no M-augmenting path, then M is maximum!

Optimality Condition

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Algorithm

Key: M is maximum no M-augmenting path

How to find efficiently?How to find efficiently?

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Finding M-augmenting paths

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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Complexity

At most n iterations

An augmenting path in time by a DFS or a BFS

Total running time

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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…

Minimum Vertex Cover

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Observation: Many short and disjoint augmenting paths.

Idea: Find augmenting paths simultaneously in one search.

Faster Algorithms

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Application of Bipartite Matching

Jerry

Marking

Darek TomIsaac

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?

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Application of Bipartite Matching

With Hall’s theorem, now you can determine exactly

when a partial chessboard can be filled with dominos.

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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.

Hon Wai Leong, NUS(CS6234, Spring 2009) Page 21

Copyright © 2009 by Leong Hon Wai

Matching in Graph

Matching in Bipartite GraphMatching in General GraphsWeighted Matching in Bipartite Graph

Additional topics: Reading/Presentation by students

Lecture notes modified from those by Lap Chi LAU, CUHK.

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General Matching

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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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Today’s Plan

• Min-max theorems [no proofs]

• Polynomial time algorithm

• Chinese postman [skip]

Follow the same structure for bipartite matching.

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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.

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Min-Max Theorem

Tutte-Berge formula [1958]:

The size of a maximum matching =

König [1931]:

In a bipartite graph, the size of a maximum matching

is equal to the size of a minimum vertex cover.

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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.

Augmenting Path?

Works for general graphs.

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What if there is no more M-augmenting path?

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!

If there is no M-augmenting path, then M is maximum?

Optimality Condition?

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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 pathsin General Graphs

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 !!!

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No repeated vertices. So?

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

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Shrink the

blossoms!

Key idea(Edmonds 1965) General matching algorithm

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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

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

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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 (1st example [in black])

2(b): p leaves C with free edge (2nd example [in red])

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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 n augmentations.

Each augmentation does at most n contractions.

An alternating walk can be found in O(m) time.

Total running time is O(mn2) .

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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

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Tutte-Berge formula [1958]:

The size of a maximum matching =

Proving Min-Max Theorem

Comments by LeongHW:

Skip next few slides on mainly combinatorial results…

(Anyone wants to do this for reading assignment?)

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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

Hon Wai Leong, NUS(CS6234, Spring 2009) Page 49

Copyright © 2009 by Leong Hon Wai

CS6234:

Matching in GraphMatching in Bipartite GraphMatching in General GraphsWeighted Matching in Bipartite Graph

Additional topics: Reading/Presentation by students

Lecture notes modified from those by Lap Chi LAU, CUHK.

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Weighted Bipartite Matching

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Weighted Bipartite Matching

Given a weighted bipartite graph,

find a matching with maximum total weight.

Not necessarily a maximum size matching.

A B

Wlog, assume perfect matching on complete bipartite graph; (add fictitious edges with weights 0)

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First Algorithm

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*

Find maximum weight perfect matching

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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

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First Algorithm

Key: M is maximum no negative cycle

How to find efficiently?

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Complexity

At most nW iterations

A negative cycle in O(n3) time by Floyd Warshall

Total running time O(n4W)

Can choose minimum mean cycle to avoid W

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Comments by LeongHW:

Skip next few slides on Augmenting Path Algorithm

(Anyone wants to do this for reading assignment?)

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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

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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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Complexity At most n iterations

A shortest path in O(nm) time by Bellman Ford

Total running time O(n2m)

Can be speeded up by using Dijkstra: O(m + n log n)

Total running time O(nm + n2log n)

Kuhn (based on proof by Egervary) — Hungarian method

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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

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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.

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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)

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Complexity

Augment at most n times, each O(m)

Decrease weighted cover at most m times, each O(m)

Total running time: O(m2) = O(n4)

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.

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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

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Faster Algorithms

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Application of Bipartite Matching

Jerry

Marking

Darek TomIsaac

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