maximum matching in general graphs

Maximum Matching in General Graphs

Ahmad Khayyat

Department of Electrical & Computer Engineering

[email protected]

Course ProjectCISC 879 — Algorithms and Applications

Queen’s University

November 19, 2008

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Maximum Matching in General Graphs • 1

Introduction Paths, Trees and Flowers Efficient Implementation Reachability Approach Conclusion

Outline

1 Introduction

2 Paths, Trees and Flowers

3 Efficient Implementation of Edmonds’ Algorithm

4 Reachability Problem Approach

5 Conclusion



Outline

1 IntroductionTerminologyBerge’s TheoremBipartite Matching




5 Conclusion



Terminology

Maximum Matching

G = (V,E) is a finite undirected graph: n = |V |,m = |E|.A matching M in G, (G,M), is a subset of its edges suchthat no two meet the same vertex.

M is a maximum matching if no other matching in Gcontains more edges than M .

A maximum matching is not necessarily unique.

Given (G,M), a vertex is exposed if it meets no edge in M .

M



Terminology

Augmenting Paths

An alternating path in (G,M) is a simple path whose edgesare alternately in M and not in M .

An augmenting path is an alternating path whose ends aredistinct exposed vertices.

v1 v2 v3

v4v5v6

v1 v2 v3

v4v5v6



Berge’s Theorem

Berge’s Theorem

Berge’s Theorem (1957)

A matched graph (G,M) has an augmenting path if and only if Mis not maximum.

v1 v2 v3

v4v5v6

v1 v2 v3

v4v5v6

Unique?

An Exponential Algorithm:Exhaustively search for an augmenting path starting from anexposed vertex.



Bipartite Matching

Bipartite Graphs

A bipartite graph G = (A,B,E) is a graph whose vertices canbe divided into two disjoint sets A and B such that everyedge connects a vertex in A to one in B.

Equivalently, it is a graph with no odd cycles.

1 A

C

D

E

B

4

3

2

5

1

2

3

A

B

4

5

D

C

E



Bipartite Matching

Bipartite Graph Maximum Matching

O(nm)

for all v ∈ A, v is exposed doSearch for simple alternating paths starting at vif path P ends at an exposed vertex u ∈ B thenP is an augmenting path {Update M}

end ifend forCurrent M is maximum {No more augmenting paths}

1

2

3

A

B

4

5

D

C

E

1

2

3

A

B

4

5

D

C

E



Bipartite Matching

Non-Bipartite Matching

Problem: Odd cycles . . .

1 2 3

4

5

6

1 2 3

46

5



Outline

1 Introduction

2 Paths, Trees and FlowersBlossomsThe Algorithm



5 Conclusion



Blossoms

Blossoms

Blossoms

A blossom B in (G,M) is an odd cycle with a unique exposedvertex (the base) in M ∩B.

1 2 3

4

5

6

1 2 3

46

5



Blossoms

Edmonds’ Blossoms Lemma

Blossoms Lemma

Let G′ and M ′ be obtained by contracting a blossom B in (G,M)to a single vertex.The matching M of G is maximum iff M ′ is maximum in G′.

1 2 3

4

5

6

1 2 B 6



Blossoms

Detecting Blossoms

Performing the alternating path search of the bipartitematching algorithm (starting from an exposed vertex):

ë Label vertices at even distance from the root as “outer”;ë Label vertices at odd distance from the root as “inner”.

If two outer vertices are found adjacent, we have a blossom.

1 2 3

4

5

6O I O

I

O



The Algorithm

Edmonds’ Algorithm (1965)

O(n2)O(n3)

for all v ∈ V , v is exposed doSearch for simple alternating paths starting at v

Shrink any found blossomsif path P ends at an exposed vertex thenP is an augmenting path {Update M}

else if no augmenting paths found thenIgnore v in future searches

end ifend forCurrent M is maximum {No more augmenting paths}

Complexity: O(n4)



The Algorithm

Example

|M | = 4

|M | = 5

1 2 3 4

10 9

8

5 6

7

B1B2

O I O I

O

I

O

OO

I

O

OI

B1 = 5, 6, 7B2 = B1, 8, 9 = 5, 6, 7, 8, 9



The Algorithm

Example

|M | = 4

|M | = 5

1 2 3 4

10 9

8

5 6

7

B1

B2

O I O I

O

I

O

O

O

I

O

OI

B1 = 5, 6, 7

B2 = B1, 8, 9 = 5, 6, 7, 8, 9



The Algorithm

Example

|M | = 4

|M | = 5

1 2 3 4

10

9

8

5 6

7

B1

B2

O I O I

O

I

O

OO

I

O

OI

B1 = 5, 6, 7B2 = B1, 8, 9 = 5, 6, 7, 8, 9



The Algorithm

Example

|M | = 4

|M | = 5

1 2 3 4

10 9

8

5 6

7

B1

B2

O I O I

O

I

O

OO

I

O

OI

B1 = 5, 6, 7B2 = B1, 8, 9 = 5, 6, 7, 8, 9



The Algorithm

Example

|M | = 4 |M | = 5

1 2 3 4

10 9

8

5 6

7

B1B2

O I O I

O

I

O

OO

I

O

OI

B1 = 5, 6, 7B2 = B1, 8, 9 = 5, 6, 7, 8, 9



Outline

1 Introduction


3 Efficient Implementation of Edmonds’ AlgorithmData StructuresThe AlgorithmPerformance


5 Conclusion



Data Structures

Three Arrays

u is an exposed vertex.

A vertex v is outer if there is a path P (v) = (v, v1, . . . , u),where vv1 ∈M .

1 MATE: Specifies a matching. An entry for each vertex:

ë vw ∈M ⇒ MATE (v) = w and MATE (w) = v.

2 LABEL: Provides a type and a value:

LABEL(v) ≥ 0 → v is outer

LABEL(u) → start label, P (u) = (u)

1 ≤ LABEL(v) ≤ n → vertex label

n+ 1 ≤ LABEL(v) ≤ n+ 2m → edge label

3 START (v) = the first non-outer vertex in P (v).



The Algorithm

Gabow’s Algorithm (1976)

O(n)

O(1)O(n)

O(n)O(n)

O(n)O(1)

for all u ∈ V, u is exposed dowhile ∃ an edge xy, x is outer AND

no augmenting path found doif y is exposed, y 6= u then

(y, x, . . . , u) is an augmenting pathelse if y is outer then

Assign edge labels to P (x) and P (y)else if MATE (y) is non-outer then

Assign a vertex label to MATE (y)end if

end whileend for

Complexity: O(n3)



Performance

Experimental Performance

Using an implementation in Algol W on the IBM 360/165

Worst-case graphs:

ë Efficient Implementation: run times proportional to n2.8.ë Edmond: run times proportional to n3.5.

Random graphs: times one order of magnitude faster thanworst-case graphs.

Space used is 5n+ 4m.



Outline

1 Introduction



4 Reachability Problem ApproachReachability and GraphsThe Algorithm

5 Conclusion



Reachability and Graphs

The Reachability Problem in Bipartite Graphs

Construction:

Bipartite graph + Matching → Directed graphG = (A,B,E) + M → G′ = (V ′, E′)

ë V ′ = V ∪ {s, t}ë ∀ xy ∈M,x ∈ A, y ∈ B → (x, y) ∈ E′ e ∈M ⇒ e : A→ B

ë ∀ xy /∈M,x ∈ A, y ∈ B → (y, x) ∈ E′ e /∈M ⇒ e : B → A

ë ∀ b ∈ B, b is exposed → add (s, b) to E′

ë ∀ a ∈ A, a is exposed → add (a, t) to E′

A1

A2

B1

B2

=⇒

A1

A2

B1

B2

t s

An augmenting path in G ⇔ A simple path from s to t in G′.




The Reachability Problem in General GraphsConstruction

For each v ∈ V , we introduce two nodes vA and vB

V ′ = {vA, vB|v ∈ V } ∪ {s, t} s, t /∈ V, s 6= t

e ∈M ⇒ e : A→ B, e /∈M ⇒ e : B → A

E′ = {(xA, yB), (yA, xB) | (x, y) ∈M}∪ {(xB, yA), (yB, xA) | (x, y) /∈M}∪ {(s, xB) |x is exposed} ∪ {(xA, t) |x is exposed}

1 2

4 3

=⇒1A

1B

2A

2B

3A

3B

4A

4B

s

t




The Reachability Problem in General GraphsStrongly Simple Paths

A path P in G′ is strongly simple if:

P is simple.vA ∈ P ⇒ vB /∈ P .

Theorem

There is an augmenting path in G if and only if there is a stronglysimple path from s to t in G′.

1 2

4 3

=⇒1A

1B

2A

2B

3A

3B

4A

4B

s

t




Solving the Reachability Problem

Solution: A strongly simple path from s to t in G′:

Depth-First Search (DFS) for t starting at s.

DFS finds simple paths.

We need to find strongly simple paths only.

We use a Modified Depth-First Search (MDFS) algorithm.

Example:

1 2

4 3

=⇒1A

1B

2A

2B

3A

3B

4A

4B

s

t



The Algorithm

Data Structures

Stack K

ë TOP(K): the last vertex added to the stack K.ë Vertices in K form the current path.ë In each step, the MDFS algorithm considers an edge

(TOP(K), v), v ∈ V ′.

List L(vA)ë To get a strongly simple path, vA and vB cannot be in K

simultaneously (we may ignore a vertex, temporarily).ë List L(vA) keeps track of such vertices.



The Algorithm

Hopcroft and Karp Algorithm for Bipartite Graphs (1973)

Step 1: M ← φ

Step 2: Let l(M) be the length of a shortest augmenting path of MFind a maximal set of paths {Q1, Q2, . . . , Qt} such that:

2.1 For each i, Qi is an augmenting path of M , |Qi| = l(M),Qi are vertex-disjoint.

2.2 Halt if no such paths exists.

Step 3: M ←M ⊕Q1 ⊕Q2 ⊕ · · · ⊕Qt; Go to 1.

Hopcroft and Karp Theorem

If the cardinality of a maximum matching is s, then this algorithmconstructs a maximum matching within 2b

√sc+ 2 executions of

Step 2.

Step 2 complexity: O(m)⇒ Overall complexity: O(√nm)



The Algorithm

An O(√

nm) Algorithm for General Graphs

Blum describes an O(m) implementation of Step 2 for generalgraphs, using a Modified Breadth-First Search (MBFS).

Blum’s Step 2 Algorithm:

Step 1: Using MBFS, compute G′

Step 2: Using MDFS, compute a maximal set of strongly simple pathsfrom s to t in G′.

Blum’s Theorm

A maximum matching in a general graph can be found in timeO(√nm) and space O(m+ n).



Outline

1 Introduction




5 ConclusionSummaryQuestions



Summary

Summary

1965

Edmonds

O(n4)

1976

Gabow’sIm

plementation

O(n3)

1980

Micali &

Vazirani

O(n2.5)

1989

Vazirani’sProof

1990

Blum

O(n2.5)

1991

Gabow

O(n2.5)



Summary

References

Jack EdmondsPaths, Trees, and FlowersCanadian Journal of Mathematics, 17:449–467, 1965.

http://www.cs.berkeley.edu/~christos/classics/edmonds.ps

Harold N. GabowAn Efficient Implementation of Edmonds’ Algorithm forMaximum Matching on GraphsJournal of the ACM (JACM), 23(2):221–234, 1976.

Norbert BlumA New Approach to Maximum Matching in General GraphsLecture Notes in Computer Science: Automata, Languages and Programming,

443::586–597, Springer Berlin / Heidelberg, 1990.


http://www.cs.berkeley.edu/~christos/classics/edmonds.ps


Summary

Thank You

Thank You!



Questions

Questions

???


maximum matching in general graphs

Education

ahmad khayyat maximum

introduction paths

augmenting paths

ahmad khayyatmaximum

maximum i

simple alternating paths

simple path

alternating path search