Lecture 16

Maximum Matching

Incremental Method

• Transform from a feasible solution to another feasible solution to increase (or decrease) the value of objective function.

Matching in Bipartite Graph

common.

in endpoint no have edges any twoin which edges of

subset a is matching a ),,,(graph bipartite aGiven EUVG

y.cardinalit maximum

with matching a find ),,(graph bipartite aGiven EVG

Maximum Matching

1

1

Note: Every edge has capacity 1.

2. Can we do those augmentation in the same time?

1. Can we do augmentation directly in bipartite graph?

1. Can we do augmentation directly in bipartite graph?

Yes!!!

Alternative Path

path. e within thedges matched

and unmatchedbetween alternates and vertex,free aat

ends vertex,free aat startspath that a ispath augmentingAn

vertex.free a called is

matching somein edgean ofendpoint not the ishat A vertex t

M

Optimality Condition

path. augmenting no hasit iff maximum is matchingA

. .path w.r.t augmentingan contains *Then

.|*||| with matchings twobe * and Let )(

Trivial. )(

MMM

MMMM

M *M *M

*M

*MM *MM

1)(deg vM 2)(deg * vMM1)(deg * vM

Puzzle

contain? * does w.r.t.

paths augmentingdisjoint many how ,|||*| If

MMM

kMM

why?,exactly :Answer k

Extension to Graph

Matching in Graph

common.

in endpoint no have edges any twoin which edges of

subset a is matching a ),,(graph aGiven EVG

y.cardinalit maximum

with matching a find ),,(graph aGiven EVG

Maximum Matching

Note

• We cannot transform Maximum Matching in Graph into a maximum flow problem.

• However, we can solve it with augmenting path method.

Alternative Path

path. e within thedges matched

and unmatchedbetween alternates and vertex,free aat

ends vertex,free aat startspath that a ispath augmentingAn

vertex.free a called is

matching somein edgean ofendpoint not the ishat A vertex t

M

Optimality Condition

path. augmenting no hasit iff maximum is matchingA

. .path w.r.t augmentingan contains *Then

.|*||| with matchings twobe * and Let )(

Trivial. )(

MMM

MMMM

M *M *M

*M

2. Can we do those augmentation in the same time?

Hopcroft–Karp algorithm

• The Hopcroft–Karp algorithm may therefore be seen as an adaptation of the Edmonds-Karp algorithm for maximum flow.

In Each Phase

. to from paths

shortest disjoint ofset maximal find graph, residualIn

ts

s t

least two.at by increasespath

augmentingshortest theoflength thephase,each In

Running Time

phases. /2

mostat through from obtained becan * Hence,

/2. is *in paths augmenting of # Thus,

1.2length has themofEach

.for paths augmenting |||*|

contains *Then matching. maximum *

and phase after matching thebe Let

time.)(in excuted becan phaseEach

n

MM

nMM

n

MMM

MMM

nM

mO

|||| EV

Reading Material

Max Weighted Matching

Maximum Weight Matching

weight.total

maximumis with matching a find ,:weight

edge positive with ),(graph bipartite aGiven

REw

EVG

1

3

?

It is hard to be transformed to maximum flow!!!

Minimum Weight Matching

weight. toalmaximumis with matching a find ,:

weight edge enonnegativ with ),(graph aGiven

REc

EVG

Augmenting Path

edges. matchedon that edges unmatchedon weight

total with thecycle ealternativan is cycle augmentingAn

edges. matched

on weight total the edges unmatchedon weight total thethat,

propert path with ealternativ maxinal a ispath augmentingAn

vertex.free a called is

matching somein edgean ofendpoint not the ishat A vertex t

M

Optimality Condition

. w.r.t.path/cycle augmentingan contains *Then

*).()( with matchings twobe * and Let )(

Trivial. )(

MMM

McMcMM

M *M *M

*M

cycle. augmenting no andpath

augmenting no hasit iffweight -maximum is matchingA

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Chinese Postman

distance.

possibleleast with theletters,deliver order toin city ain

roadevery along travel toshesPostman wi Chinese The

once.least at traversedis edge

each in which graph theof walk closedshortest a

find weight,edge enonnegativ graph with aGiven

Minimum Weight Perfect Matching

• Minimum Weight Perfect Matching can be transformed to Maximum Weight Matching.

• Chinese Postman Problem is equivalent to Minimum Weight Perfect Matching in graph on odd nodes.