cs6234 l1 matching

8/9/2019 CS6234 L1 Matching

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Hon Wai Leong, NUS (CS6234, Spring 2009) Page 1Copyright © 2009 y Leong Hon Wai

CS6234: Lecture 1

Matching in GraphMatching in Bipartite Graph [PS82]-Ch10

Matching in General Graph [PS82]-Ch10

!eighte" Matching in Bipartite Graph

[PS82]-Ch 11#1$11#2

%""iti&nal t&pic: Reading/Presentation by students

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


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Bipartite Matching


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Bipartite Matching

A graph is bipartite if its vertex set can be partitionedinto to 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:!ind a matching ith the maximum number of edges.

Maximum Matching

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

The perfect matching problem: "s there a perfect matching#


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Key Questions

$ Ho to te(( if a graph does not have a &perfect' matching#

$ Ho to determine the si)e of a maximum matching#

$ Ho to find a maximum matching efficient(y#


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Ha((*s +heorem -/012

A bipartite graph %3&A,B45' has a matching that 6saturates7 A

if and on(y if 89&:'8 ;3 8:8 for every subset : of A.

:9&:'

Existence of Perfect Matching


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K


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%iven a matching M, an M#alternating path is a path that a(ternates

beteen edges in M and edges not in M. An M>a(ternating path

hose endpoints are unmatched by M is an M#augmenting path.

ugmenting Path


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=hat if there is no more M>augmenting path#

?rove the contrapositive2

A bigger matching an M >augmenting path

-. Consider@. 5very vertex in has degree at most @

/. A component in is an even cyc(e or a path

. :ince , an M>augmenting path

"f there is no M>augmenting path, then M is maximum

$ptimality %ondition


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AlgorithmKey: M is maximum no M -augmenting path

Ho to find efficient(y#Ho to find efficient(y#


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Finding -augmenting paths Orient the edges (edges in M go up, others go down)

An M -augmenting path

a direted path !etween two unmatched "erties


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Complexity At most n iterations

An augmenting path in time !# a $%& or a '%&

ota running time


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Ha((*s +heorem -/012

A bipartite graph %3&A,B45' has a matching that 6saturates7 A

if and on(y if 89&:'8 ;3 8:8 for every subset : of A.

K


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Ebservation2 Many short and disFoint augmenting paths.

"dea2 !ind augmenting paths simu(taneous(y in one search.

Faster lgorithms


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pplication of Bipartite Matching

=ith Ha((*s theorem, no you can determine exact(y

hen a partia( chessboard can be fi((ed ith dominos.


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

CS6234:

Matching in GraphMatching in Bipartite Graph

Matching in General Graph





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%enera( Matching


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+eneral Matching

%iven a graph &not necessari(y bipartite',

find a matching ith maximum tota( eight.

un,eighted -cardinality. 'ersion2

a matching ith maximum number of edges


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Today/s Plan

$ Min>max theorems no proofs1

$ ?o(ynomia( time a(gorithm

$ Chinese postman sip1

!o((o the same structure for bipartite matching.


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Min#Max Theorem

+utte>Berge formu(a -012

+he si)e of a maximum matching 3

K


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%iven a matching M, an M#alternating path is a path that a(ternates

beteen edges in M and edges not in M. An M>a(ternating pathhose endpoints are unmatched by M is an M#augmenting path.

ugmenting Path"

=ors for genera( graphs.


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=hat if there is no more M>augmenting path#

?rove the contrapositive2

A bigger matching an M >augmenting path

-. Consider

@. 5very vertex in has degree at most @

/. A component in is an even cyc(e or a path

. :ince , an M>augmenting path

"f there is no M>augmenting path, then M is maximum#

$ptimality %ondition"


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Agorithm*

Key: M is maximum no M -augmenting path

Ho to find efficient(y#


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Finding ugmenting Path in Bipartite +raphs

Erient the edges &edges in M go up, others go don'

edges in M having positive eights, otherise negative eights

"n a bipartite graph, an M>augmenting path corresponds to a directed path.


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o repeated "erties. &o*

+ /ust ind an aternating path without repeated "erties*

ither we ma# eude some possi!iities,

r we need to trae the path


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e# emma

+ (dmonds) M is a maimum mathing in G

M/C is a maimum mathing in G/C


+ ∃ 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

9ote2 $ne of the to paths around the b(ossomi(( be the correct one.


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Agorithm




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%inding an M -augmenting path

+ %ind an aternating wa !etween two ree "erties.

+ his an !e done in inear time !# a $%& or a '%&.

+ ither an M -augmenting path or a !ossom an !e ound.

+ a !ossom is ound, shrin it, and (reursi"e#) ind an M/C -augmenting path P in G/C , and then

epand P to an M -augmenting path in G.


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:ompeit#

+ At most n augmentations.

+ ah augmentation does at most n ontrations.

+ An aternating wa an !e ound in O(m) time.

+ ota running time is (mn2) .


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5istorical 6emar7s

"n his famous paper 6paths, trees, and f(oers7, Gac 5dmonds2

$ "ntroduced the notion of a 6good7 a(gorithm.

$ 5dmonds so(ved eighted genera( matching, prima( dua(.

$ Linear programming description, a breathrough in po(yhedra( combinatorics.

n $-n8. algorithm by Pape and %onradt9 4;3

6ef: *yslo9 =eo9 Ko,ali79 Prentice#5all9 4;8

=iscrete $ptimi0ation lgorithms9 %h 8?2


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+utte>Berge formu(a -012

+he si)e of a maximum matching 3

Pro'ing Min#Max Theorem

Comments by LeongH=2

:ip next fe s(ides on main(ycombinatoria( resu(tsD

&Anyone ants to do this forreading assignment#'


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M#alternating forest

Larger forest

M>augmenting path a b(ossom

=e mae progress in any of the above cases.


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M#alternating forest

Etherise e find the set in +utte>Berge formu(a.


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Edmonds#+allai =ecomposition

Can be obtained from M>a(ternating forest.Contains a (ot of information about matching.


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%hinese Postman

%iven an undirected graph, a vertex v, a (ength (&e' on each edge e,

find a shortest tour to visit every edge once and come bac to v.

isit every edge exact(y once if 5u(erian.

Etherise, find a minimum eighted perfect matching beteen odd vertices.


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CS6234


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=eighted Bipartite Matching


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Today/s Plan

+hree a(gorithms

$ negative cyc(e a(gorithm

$ augmenting path a(gorithm

$ prima( dua( a(gorithm

App(ications


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First lgorithm

Ho to no if a given matching M is optima(#

"dea2 "magine there is a (arger matching MS

and consider the union of M and MS

!ind maximum eight perfect matching


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%irst Agorithm

Key: M is maximum no negative cycle



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:ompeit#

+ At most nW iterations

+ A negati"e #e in (n3) time !# %o#d ;arsha

+ ota running time (n4W )

+ :an hoose minimum mean #e to a"oid W


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Comments by LeongH=2

:ip next fe s(ides on Augmenting ?ath A(gorithm

&Anyone ants to do this for reading assignment#'


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ugmenting Path lgorithm

Theorem: 5ach matching is an optima( >matching.

Let the current matching be M

Let the shortest M>augmenting path be ?

Let 9 be a matching of si)e 8M8T-


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Primal =ual lgorithm

=hat is an upper bound of maximum eighted matching#

=hat is a genera(i)ation of minimum vertex cover#

eighted vertex cover2

Minimum eighted vertex cover ≥ Maximum eighted matching


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Primal =ual lgorithm

Minimum eighted vertex cover≥

Maximum eighted matching

-. Consider the subgraph formed by tight edges &the eIua(ity subgraph'.

@. "f there is a tight perfect matching, done.

/. Etherise, there is a vertex cover.

. ecrease eighted cover to create more tight edges, and repeat.

etai(ed a(gorithm i(( be covered (ater&after L? ?rima(>ua( a(gorithms'


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:ompeit#

+ Augment at most n times, eah O(m)

+ $erease weighted o"er at most m times,eah O(m)

+ ota running time O(m2) @ O(n4)

+ ger"ar#, ?ungarian method

5istory: +he a(gorithm as deve(oped by Kuhn ho based iton or of 5gervary -/-1, &6(atent or of . Konig and G.5gervary7' and as given the name 6Hungarian method7.:ee :chrQ@1>Ch-J for more detai(s.


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Quic7 *ummary

-. !irst a(gorithm, negative cyc(e, usefu( idea to consider symmetric difference

@. Augmenting path a(gorithm, usefu( a(gorithmic techniIue

/. ?rima( dua( a(gorithm, a very genera( frameor

-. =hy prima( dua(#

@. Ho to come up ith eighted vertex cover#

/. Peduction from eighted case to uneighted case


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Faster lgorithms


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pplication of Bipartite Matching

Gerry

Maring

are +om"saac

+utoria(s :o(utions 9esgroup

(ob ssignment Problem:

5ach person is i((ing to do a subset of Fobs.

Can you find an assignment so that a(( Fobs are taen care of#

$ Ad>auction, %oog(e, on(ine matching

$ !ingerprint matching


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eadingBCresentation opis

cs6234 l1 matching

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