Algorithmic GraphTheory:MatchingProblemsin RandomGraphs

MicheleZito

April 22,2001

Chapter 1

TechnicalPreliminaries

We recallsomebasicterminologyandwell-known resultsin thedifferentareasof ComputerScienceandMathematicswhichwill beusedlateron. Thissectioncontainsall thosebackgrounddefinitionsandresultswhichareparticularlyusefulin morethanoneof thefollowing chapters,or simply too longto beput in thespecificchapter, withoutdistractingthereader’sattention.

Someknowledgeof basicsettheoryandelementarycalculusis assumed[Giu83]. If�

is a functiononrealnumbers,for every ����� ������ ����� �� , ��������� � (or simply

��� � whentheindependentvariableis clearfrom thecontext) is a shorthandfor �! #"$&%(' ���)���+* �Also, the readershouldbe familiar with sequentialcomputationalmodelslike Turing machinesor Ran-domAccessMachines[AHU74] andbasiccomplexity theoreticdefinitions[GJ79, BDG88]. Asymptoticnotationslike , ��-/.0� , 1 �320� , 4 �)-5� , 6 �87:9;� and < ��-5� will denotefunctionclassesbut we will normallywrite�=* 4 ��-5� (insteadof

� �>4 ��-5� ) with the intendedmeaningthat thereexists a constant? suchthat����-5��@ ? - for-

sufficiently large. Thereaderis referredto Section2.1 in Cormen,LeisersonandRivest[CLR90], for moreformaldefinitions.In particular, giventwo functions

�and A on integers,wewill write��B A andwe will saythat

�is asymptoticto A if theratio

����-5�DC A ��-5�E�F2(theconceptcanbeextended

to functionson realnumbers).This section’s contentcanbe subdivided into two parts. The first sectionsdescribesconceptsfrom

ComputabilityTheory. Theremainingsectionspresentsomerelevantdefinitionsandresultsfrom differentareasof Mathematics.

More specifically, in Section1.1we recallsomeelementarydefinitionsrelatedto computationalprob-lemsthatwill beusedthroughoutthis thesis.Section1.2introducesthebasicterminologyrelatedto Prob-ability Theory. Section1.3describestherelevantconceptsin graphtheory. Section1.4providesa glimpseinto the beautifulandby now well establishedtheoryof randomgraphs.We describeseveral modelsofrandomgraphs,eachproviding a different framework for the analysisof combinatorialandalgorithmicpropertiesof graphs.

1.1 Problems

Many computationalproblemscanbe viewed asthe seekingof partial informationabouta relation(see[BDG88, Chapter1] or [BC91]). MorespecificallysupposeG isafinitealphabetandthatprobleminstancesandsolutionsareencodedasstringsover G . A relation HEI(JLKLGNM(O�GNM definesanassociationbetweenprobleminstancesandsolutions. For every

� �PGNM theset HEI(J �)��� containsall the ���PGNM thatencodeasolutionassociatedwith

�, with theimplicit conventionthat if

�doesnot encodea probleminstancethenHEI(J �)���Q*SR . For every

� �TG M , let U � U denotethe lengthof (theencoding)�. Noticeincidentallythat ifV

is a set, U V U will beused,in the usualsense,asthecardinality ofV

. In this settinga decisionproblemis describedby therelation HNI�J anda questionwhich hasa yes/noanswerin termsof HEI(J [BDG88, p.11]. If

� �TGNM , a decisionproblem(alsoknown asexistenceproblem) answersthefollowing question:is

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therea �W�XGNM suchthat ���PHEI�J ���Y� ? If� �Z���XGNM aregiven,anotherdecisionproblem(which will be

referredto asthemembershipproblem) answersthequestion:does� belongto HEI�J ���Y� ?Thereis a naturalcorrespondencebetweena decisionproblem [ andtheset \ of instancesthathave

a “yes” answer. Thusno strongdistinctionbetween[ and \ will be kept: the “name” of the decisionproblemwill alsodenotethesetof instanceswith a solution. Notice that \>KLGNM sothewordslanguageor propertywill alsobeusedasqualifiers.

Severalothertypesof problemsaredefinablein this setting.Informally, if� �]GNM is (theencodingof)

a probleminstancethen

1. aconstructionproblem(or search problem) aimsat exhibiting a �^�WGNM suchthat�)� �Z� � �HEI(J ;

2. anoptimisationproblem, givenacostfunction ? ��� �Z� � , aimsatfindingthe ���]GNM with��� �Z� � ��HEI�J

suchthat ? �)� �_� � is maximised(respectively minimised);

3. auniformgenerationproblem, aimsat generatinga word ��]GNM satisfying��� �Z� � �HEI(J suchthat

all � in HEI(J ����� comeupequallyoften;

4. acountingproblem, aimsatfinding thenumberof elementsin HEI�J ���Y� .Example. In whatfollowswerecall,in a ratherinformal way, a numberof definitionsrelatedto booleanalgebra. The readeris referredto [BDG88, Chapter1] or [Dun88] for a more formal treatmentof thesubject.

A booleanformulais anexpressionlike` ���ba � �Yc . � � .Dd � ��aZe � ��fZgh�i*kj�lm�)� .Dd+n �Yc . �/o�pE�qpr�bf_gsoP�8pr��a n �ba_e0�_�built up from the elements

��tof a countableset u of propositionalvariables, a finite setof connectives

(usuallyincluding n ,o

andp

) andthebrackets.If variablesare assignedvaluesover a binary set of truth-valuesdenotedby v * 0wx� 2 � and con-

nectives are interpretedin the usualway as operationson v then eachformula representsa functionon v . It is evident that under this interpretationthe formula

` �)�ba � �Yc . � � .Dd � �ba_e � �bf_g0� is equivalent to` ���yc � � . � �ba � �be � � d � obtainedby replacing�ba

with�Yc

,�Yc . with

� . andsoon. Thus,without lossof gen-erality, a formulaon

-differentvariablescanberegardedascontainingexactly thevariables

� c �{z|z{z|� � 9 .Let uT}} 9 * � c �|z|z{z~� � 9 � . Sometimesnotation

` �{��/�will beusedinsteadof

` ��� c � � . �|z{z|z~� � 9 � .n1 1 11 0 00 0 10 0 0

o1 1 11 1 00 1 10 0 0

p1 00 1

Any function ���/u }} 9 � v is calledan-

variable-truth-assignment(or simply a truth-assignment).Notation

` 0�i� will beusedfor thetruth-valueof`

aftereachvariable� t

hasbeenreplacedby � ��� t � andthetablesabove(calledtruth-tables) havebeenusedto find thetruth-valueof conjunctions,disjunctionsornegationsof truth-values.Sincetruth-valuesarenothingbut binarydigits, thesetof all

--variable-truth-

assignmentswill normallybedenotedby &w�� 2 � 9 .Theformulais saidto bein conjunctivenormalform (CNF for short)if it is in theform�Nc n � . z{z|z n �s�

whereeachclause� t

is thedisjunctionof somevariablesor negationof variables(expressionslike� t

orpr� tarecalled literals). Every booleanformulacanbe transformedby purelyalgebraicrulesinto a CNF

formula(seefor example[BDG88, p. 17–18]).A CNF formula`

is in � -CNF if themaximumnumberofliteralsforminga clauseis � .

Every � -CNF formulaecanbeencodedover thefinite alphabetG * n � o � p � � � � �Dwx� 2 � (e.g. variable�btis encodedby the binary representationof � ). In this setting, � -SAT is the well known NP-complete

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problem[GJ79] of decidingwhethera � -CNF`

is satisfiable,i.e. whetherthereexistsanassignment� ofvaluesin v to all variablesin

`suchthatthevalueof

`underthisassignmentis one.

Combinatorialstructuresassociatedwith computationalproblemscanbecharacterisedby anumberofparameters, describingspecificfeaturesof the probleminstances.For examplea � -CNF formula is builtonsome

-variables,� clausesandeachclausehasatmost � literals.A graph(seeSection1.3for relevant

definitions)mighthave-

vertices,� edges,maximumdegree� . In mostcasesit will bepossibleto definetwo functionson naturalnumbers,theorderandthesize. � 9 K�GNM is thesetof probleminstancesof order-

.The relationshipsbetweendifferentparameterscharacterisingthe instancesof two specificproblems

will betheobjectof thework describedin Chapter2. Weconcludethissectionwith a remarkandacoupleof usefuldefinitions.

It is worth noticingthat theremight beno relationshipbetweentheseparametersandthelengthof theencodingof the instancesof a particularproblem.For instancethenaturalencodingof a � -CNF formulaof order(i.e. numberof variables)

-andsize(i.e. numberof clauses)� describedin theexampleabove

haslengthatmost ��� �!��� - .A set \�K>GNM is a monotoneincreasingproperty (respectively monotonedecreasingproperty) with

respectto a partialorder �Q� if for everyfixed-� ��� 9�� \��_�^��� 9 � � � � � � respectively ��� � �Y�+� ����\

A set \�K�GNM is a convex propertyif for every� �Z�Y�D���k� 9 � �Q����Q�T� and

� �_���]\ imply �^�]\ .

1.2 Probability

Many of theresultsin this thesisareprobabilistic.In thissectionsometerminologyandgeneralresultsaregiven.

Following [GS92], a probability spaceis a triple� 4���G��_�+� � , where 4 is a setcalleda samplespace,G * 0����� K�4�� is thesetof eventsand �i� is a non-negativerealvaluedmeasureon G with �i�&¡ 4�¢ *S2 .

The elementsof 4 areparticulareventscalledelementaryevents. Unlessotherwisestated 4 will be afinite set and G will be the set of all subsetsof 4 . For every �£�¤G the probability of the event � ,�i�&¡ ��¢ * j�l¦¥�§b¨�© �+�{¡ 6ª¢ .Theorem1.2.1 Theprobabilitiesassignedto theelementsof a samplespace4 satisfythefollowingprop-erties(for every �m�D«S�]G ):

1. �i�{¡ ��¢ @ w .2. (Monotonicity)If �¬K�« then �i�&¡ ��¢®­¯�+�{¡ «�¢ .3. �i�{¡ ����«�¢ * �+�{¡ ��¢��X�i�&¡ «�¢��P�+�{¡ � � «�¢ .4. �i�{¡:°��¢ *L2 �P�+�{¡ ��¢ .

Theorem1.2.2 (Totalprobability)If � c �|z{z|z~�D� 9 is a partition of 4 with � t �]G for all � and � ��G then�i�&¡ ��¢ * ¥ 9t!± c �i�&¡ � � � t ¢ .Proof. Immediatefrom Theorem1.2.1.3 sincetheevents� � � t areall disjoint. ²�i�&¡ �U «�¢ will denotetheprobabilityof theevent � giventhattheevent « hashappened.If �+�{¡ «�¢5³�w ,we define�+�{¡ ��U «�¢ * j�l �i�&¡ � � «�¢ C �i�|¡ «�¢ . A sequenceof events� t aremutuallyindependentif �+�{¡ � c �z|z{z_� 9 ¢ *µ´ 9t#± c �i�&¡ � t ¢ . Mutual independencebetweenpairsof eventsis calledpairwiseindependence.

3

Example. Let 4 * &¶��¸·0�_?h�D¹x� with �i�&¡ ¶�¢ *»ºand �i�&¡ ·¸¢ * �i�&¡ ?¸¢ * �i�{¡ ¹�¢ *½¼ . Let � c * &¶��¸·&� ,� . * &¶��_?h� and � a * &¶��D¹x� . �+�{¡ � t ¢ *¾¼ � º and �i�{¡ � c � � . � � a ¢ * �+�&¡¿&¶b�{¢ *�º . Solving

º�*��À¼ � º:� awith theconstraintÁ ¼ � º�*L2 weget�i�&¡ � c � � .+� � a ¢ * �i�&¡ � c ¢�Â&�i�&¡ � . ¢bÂ|�+�{¡ � a ¢for¼*S� ÁN�Wà Á �_ChÄ and

º�*S� Á�à ÁE��Å �_ChÄ . On theotherhand�+�{¡ � t U �sÆ|¢ *�º�C;�¿¼ � º:�ÈÇ* �+�{¡ � t ¢ . Similarlysamplespacescanbebuilt in which it is possibleto constructeventsthatarepairwiseindependentbut notmutuallyindependent.

A realvaluedrandomvariable É on a probabilityspace� 4���G��_�+� � is a function from 4 to thesetof

real numberssuchthat for every real number�

the set |6Ê��4»�iÉ � 6 � ­ � �X�SG . The distributionfunctionof a randomvariableÉ is thefunction «S�:� � � ¡ w�� 2 ¢ with « ������* �+�{¡ ÉË­ � ¢ .Moments. If Ì is any real-valuedfunctionon thesetof realnumbers� � thentheexpectationof Ì ���Y� isÍ � Ì � É �Z�i* j�l¦Î¸Ï Ì ����� �i�{¡ É *�� ¢In particularthemeanof a randomvariable É , usuallydenotedby Ð , is

Í � É � andthe � -th momentof ÉisÍ � É�Ñ � (of course,if 4 is not finite, thesequantitiesmight not exist). The � -th binomialmomentof É

isÍ �{Ò8Ó ÑbÔ � whereasthe � -th factorial momentof É is

Í Ñ � É ��*�Õ×Ö Í � ɻ � ÉØ� 20� z|z{z� � ÉØ�Ù�k� 2&�_� . Itfollows that

Í Ñ � É �i* �YÚ Í � Ò Ó Ñ�Ô � .Theorem1.2.3 If É *µ¥ É t then

Í � É �i*µ¥ Í � É tq� .This result,known aslinearity of expectation, (theproof followsimmediatelyfrom thedefinitionof expec-tation)is averyusefultool for computingthemeanof a randomvariable.For exampleif thevalueof É isthesumof a numberof very simplerandomvariablesÉ t thenthemeanof É is easilydefinedin termsofthemeansof the É t .

Thevarianceof É , usuallydenotedby Û . , is definedby ÜiÝ:� � É �i*kj�l Í �Z� ÉÊ��Ð � . �+* Í � É . � �PÐ . .Theorem1.2.4 If É * ¥ É t andthe É t are pairwiseindependentthen ÜiÝ�� � É �i* ¥ ÜiÝ�� � É t8� .Theorem1.2.5 If É is a positiverandomvariable then �i�&¡ É @ Þ ¢È­ Í � É�Ñ �_C:Þ Ñ for every

Þ ³ w andinteger �³¯w .Proof. By definition

Í � É�Ñ ��@ ¥ $�ßYà � Ѫ�+�{¡ É *�� ¢ . If�á@�Þ

thenthesumabove is lower boundedbyÞ Ñ��i�{¡ É @¯Þ ¢ andtheresultfollows. ²Theorem1.2.5hasmany usefulspecialcases,dependingon thechoicesof

Þand � .

Theorem1.2.6 (Markov inequality) �i�&¡ É�³Ùw:¢5­ Í � É � .Theorem1.2.7 (Chebyshev inequality) �i�&¡!U ÉÊ� Í � É � U @¯Þ Â{Û�¢5­ Þ/â . .

An importantuseof Chebyshev inequalityis in proving thata positive randomvariabletakesa valuelargerthanzerowith “high” probability.

Corollary 1 If É @ w then �+�{¡ É * wh¢/­¯ÜiÝ:� � É �DC Í � É � . .Proof. �+�{¡ É * wh¢®­Ù�i�&¡!U ÉÊ� Í � É � U @¯Þ Â{Û�¢ if Þ�* Ð C Û . ²

Soassumingthata naturalnumber-

canbeassociatedasa parameterwith theelementsof thesamplespaceunderconsideration,andthat

Í � É � and ÜiÝ:� � É � arethusfunctionsof-

, if�! !" 9 %(' Í � É �ã*

and ÜiÝ�� � É �i* 1 � Í � É � . � thenthelastcorollaryimpliesthat �i�{¡ É * w:¢ becomessmallerandsmallerasafunctionof

-.

4

Distrib utions. Wenow briefly review thediscreteprobabilitydistributionsthatwill beusedin laterchap-ters.Thediscreteuniformdistribution on afinite samplespace4 containing

-elementsis definedby�+�{¡ 6 t ¢ * 2- ä �ª�P 2 �|z{z|z�� - �

andin thiscasewesaythat 6 t is generateduniformly atrandom.TherandomvariableÉ»��4 � 2 �|z{z|z~� - �definedby É � 6 t)��* � hasdiscreteuniform distribution (or equivalently that its valuesare distributeduniformly over 4 ). Usingthefollowing simpleidentities,whichcanbeeasilyprovedby inductionon

-,9Î t!± c � * -i��- � 2&�7 9Î t#± c � . * -i��- � 20�~�87:- � 2&�å

it is possibleto derive Í � É �i* 2- 9Î t!± c � * 2-  -i��- � 20�7 * - � 27ÜiÝ:� � É �æ* 2- 9Î t!± c ç �5� - � 27éè .* 2-�ê 9Î t#± c � . � 9Î t#± c � �)- � 2&� � 9Î t!± c ��- � 20�Ä .~ë

* 2- 9Î t!± c � . � 9Î t!± c �5� - � 27 � ��- � 20�Ä .* 2- 9Î t!± c � . � -i��- � 2&�7 � - . � 2Ä* 2- 9Î t!± c � . � ç - � 27 è .* ��- � 2&�|�87h- � 20�å � ç - � 27 è . * - . � 22&7

If Éì��4 � 0wx� 2 � and �i�&¡ É *�2 ¢ *�¼ then É is calleda 0–1-randomvariableor randomindicator.0–1-randomvariablesmodelanimportantclassof randomprocessescalledBernoulli trials. Duringoneofthesetrials anexperimentis performedwhich succeedswith a certainpositive probability

¼. In particular

from now on wewill alwaysabbreviate �i�&¡ É *�2 ¢ by �+�{¡ É�¢ and �i�{¡ É * w:¢ by �i�&¡ É]¢ . We haveÍ � É �ª* w� �32 � ¼�� � 2  ¼*¾¼ÜiÝ:� � É ��*S� w�� ¼�� .  �Z2 � ¼Y� � �Z2 � ¼�� .  ¼*L�Z2 � ¼Y�~�¿¼ . � ¼ � ¼ . �i*á¼®�Z2 � ¼Y�Randomindicatorshave many applicationsin probability. For examplethey canbe usedto estimatethevarianceof a randomvariable.

Theorem1.2.8 If É canbedecomposedin thesumof-

not necessarilyindependentrandomindicatorsthen

1. ÜiÝ:� � É � ­ 7 ¥îí t�ï Ƹð �+�{¡ É t n ÉãÆ~¢;� Í � É � where thesumis over all 2-setson 2 �|z|z{z~� - � .2. ÜiÝ:� � É � ­ Í . � É � �Ù, � Í � É �Z� .

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Proof. It follows from the definition that ÜiÝ�� � É � ­ Í � É . � . For every real number� ³Ëw we can

write� . *�� � 7 Ò $ . Ô . Hence

Í � É . �]* Í � É � � 7 Í � Ò Ó . Ô � . If É *ì¥ t É t thenÒ Ó . Ô is the number

of ways in which two different É t canassumethe valueone,disregardingthe ordering. SoÍ � ÒqÓ . Ô �m*¥ í t)ï Ƹð Í � &É t �ZÉãÆ:� �i* ¥ í t)ï Ƹð �+�{¡ É t n ɦÆ~¢ overall {�¸�òñx��óî 2 �{z|z{z~� - � .

Thesecondinequalityis trivial sinceÍ . � É �i*�7 Í � Ò8Ó . Ô � . ²

Theproofof Theorem1.2.8givesacombinatorialmeaningtoÍ �|ÒqÓ . Ô � in termsof therandomindicatorsÉ t . If É *µ¥ 9t!± c É t whereÉ t arerandomindicatorsalsothe � -momentandthe � -th factorialmomentofÉ have aninterpretationin termsof the É t . Í � É . � is thesumoverall pairsof (not necessarilydistinct) �

and ñ of �i�&¡ É t n ÉãÆ~¢ whereÍ . � É � is thesumoverall orderedpairsof distinct � andñ of �i�&¡ É t n ÉãÆ�¢ .

If É t are-

independentrandomindicatorswith commonsuccessprobability equalto¼

then É *¥ 9t#± c É t hasbinomial distribution with parameters-

and¼. Simplecalculations(usingTheorem1.2.3

and1.2.4)imply Í � É �ª*�-;¼ ÜiÝ:� � É �i*�-;¼®�32 � ¼��If a sequenceof identicalindependentrandomexperimentsis performedwith commonsuccessprob-

ability equalto¼

then the randomvariable ô c countingthe numberof trials up to the first successhasgeometricdistributionwith parameter

¼. �i�0¡ ô c * ��¢ *T¼r�32 � ¼�� Ñ â c henceusingthebinomialtheoremand

someeasypropertiesof powerseriesÍ � ô c~�i*�20C�¼ ÜiÝ�� � ô c|�+* 2 � ¼¼ .In the samesettingasabove ô Ñ countingthe numberof trial up to the � -th successhasthe Pascal

distribution (or negativebinomial distribution). �i�&¡ ô Ñ *½- ¢ *õÒ 9 â cÑ â c Ô ¼ Ñ �32 � ¼��39 â Ñ . Sinceeachtrial is

independentô Ñ * ¥ ÑÆ ± c ô Æc wherethe ô Æc have a geometricdistribution. Henceby Theorem1.2.3andTheorem1.2.4andtheresultsfor thegeometricdistributionwe haveÍ � ô Ñ �i* � C¸¼ ÜiÝ:� � ô Ñ �i* � �32 � ¼��¼ .Impr oved Tail Inequalities. Thebeautyof Theorem1.2.5residesin thefacttheonly assumptionmadeon É is on the existenceof

Í � É�Ñ � . If moreaccurateinformation is available it is possibleto improveconsiderablythe quality of the results. The following Theoremstatesa coupleof inequalitiesproved in[Hm90].

Theorem1.2.9 Let- �P� ö andlet

¼ c �|z{z|z{� ¼ 9 �P� � with w�­ ¼�t ­ 2 , � *¤2 �|z{z|z~� - . Put¼W*½�320Ch-5� ¥ ¼�t

and � *L-;¼andlet É c �|z{z|z{�ZÉ 9 beindependent0-1 randomvariableswith �i� � É tq�Q*�¼bt �ª� *¤2 �|z{z|z{� - .

LetV¾*µ¥ É t . Then �i� �qVÙ@µ�32 �á÷ � � � ­¯ø âYù8ú �NûDa �rw�­Ù÷È­ 2

and �+� �8V ­ �32 �T÷ � � � ­Ùø âYù8ú �Nû . ��wã­�÷Q­ 2 zIn mostcasesthe Chernoff boundsstatedabove will be usedon a sequenceof

-independentidentically

distributed0-1randomvariables.Undertheseassumptions,V

hasbinomialdistributionandsomeimprovedboundsarepossible(see[Bol85, Ch. I]).

1.3 Graphs

Most of the graph-theoreticterminologywill be taken from [Har69] and[Bol79]. A (simpleundirected)graph ü * �8ý �_� � is a pair consistingof a finite nonemptyset

ý¬*þý^� ü � of vertices(or nodesor points)andacollection � * � � ü � of distinctsubsetsof

ýeachconsistingof two elementscallededges(or lines).

If ÿ * ��®�����^�á� thenthevertices� and � areadjacent, vertex � andthewholeedgeÿ are incident(or

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elsewesaythat � belongsto ÿ , sometimesusingtheset-theoreticnotation�]��ÿ ). Also if��* ��������(���

then ÿ and�

are incident. If «�K�� � ü � thený^� « � is the setof verticesincidentto some ÿP��« . For

every� K ý^� ü � , � � � � will denotethesetof verticesadjacentto some��� �

andnot belongingto�

. If� * ��b� we write � � � � insteadof � � ���� � . If� � K ý thencut

� � � ý¦� is thesetof edgeshaving oneendpointin

�andtheotherin .

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v1 2

34

v

vv

v1 2

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v

vv

v1 2

34

v

vv

v1 2

34

v

vv

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v

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Figure1.1: The64 distinctlabelledgraphson 4 vertices.

Thedegreeof a vertex � is definedas �;ø � � � *�Õ×Ö U � � � � U . Theminimum(resp.maximum) degreeofü is � * � � ü �È* "� ���� ¨ � �;ø ��� � (resp. � * � � ü �È* " Ý�� � ¨ � �;ø � � � ). For all ���¯&wx�{z|z{z�� - � 2 � letýbt_� ü �Q* ���� ý ���;ø � � � * �_� . A multisetis a collectionof objectsin which a singleobjectcanappearseveral time. A multigraph is a pair � * � � �_� � in which

�is thesetof verticesand � is a multisetof

edges.If ÿ appears��� ³ 2 timesin � theneachof its occurrencesis a parallel edge. The skeletonof a

multigraph� *�� � �_� � is agraphü withý^� ü �i* �

and � � ü � containingasinglecopy of everyparalleledgein � plus all the ÿT�þ� with

����*Ë2. A graphis directedif the edgesareorderedpairs. Round

bracketswill encloseverticesbelongingto a directededge.A graphis labelledif its verticesaredistinguishedfrom oneanotherby names.Figure1.1showsthe64

differentlabelledgraphsonfour vertices.Someof thesegraphsonly differ for thelabellingof theirvertices,their topologicalstructureis thesame.More formally, two graphsü c and ü . areisomorphicif thereis aone-to-onecorrespondencebetweentheir labelswhich preservesadjacencies.A graphis unlabelledif itis considereddisregardingall possiblelabellingof its verticesthatpreserveadjacencies.Figure1.2shows

7

theelevenunlabelledgraphson four vertices.A graphis completelydeterminedby eitherits adjacenciesor its incidences.This informationcanbe

convenientlystatedin matrix form. Theadjacencymatrix of a labelledundirected(resp. directed)graphü *��qý �D� � with-

vertices,is an- O - matrix � suchthat,for all � t ���&Æ(� ý , � t)ï Æ *�2 if � t is adjacentto�&Æ (resp.if

� � t ���&Æ � ��� ) and � t�ï Æ * w otherwise.A subgraph of ü *ì�8ý �_� � is a graph � * � T�_« � with K ý

and «õK � . � is a spanningsubgraphif *µý

andit is aninducedsubgraphif whenever �®���^�� with ��®��������� then ��®��������« .If K ý^� ü � we will denoteby üm¡ µ¢ theinducedsubgraphof ü with vertex set . � 9 is thecompletesimplegraphon

-vertices.It has

-i��- � 20�_C:7 edges.Everygraphon-

verticesis a subgraphof � 9 .A graphü *S�8ý �_� � is bipartite if

ýcanbepartitionedin two sets

ýYcand

ý . suchthatevery line of üjoinsavertex in

ý�cwith avertex in

ý . . � 9 � ï 9 ú is thecompletebipartitegraphon-]*�-®c � - . vertices.A

graphis planar if it canbedrawn on theplanesothatno two edgesintersect.If ü is a graphand ��� ý then ü��!� is the graphobtainedfrom ü by removing � andall edges

incidentto it; if � Ç� ý then üþ�!� * �8ý �"�b�D� � . If ÿ * ��®�������¯� then ü��¯ÿ * �qý �D�$#�&ÿ�� � andü��áÿ *��qý �W��r���b���_���ÿ � . Theseoperationsextendnaturallyto setsof verticesandedges.

Figure1.2: The11 distinctunlabelledgraphson 4 vertices.

A pathin agraphü *��qý �D� � is anorderedsequenceof verticesformedby astartingvertex � followedby a pathwhosestartingvertex belongsto � � � � . The path is simple if all verticesin the sequencearedistinct. The lengthof a path % * � � c �{z|z{z~��� Ñ � is ��� 2 . A cycle is a simplepath % * � � c �{z|z{z|��� Ñ �suchthat � c * � Ñ . A singlevertex is a cycle of lengthzero. Since � Ç�&� � � � thereis no cycle of lengthone.An edge��®��������� belongsto a path % *�� � c �{z|z|z{��� Ñ � if thereexists �Q�á 2 �{z|z|z|�D�k� 2 � suchthat��r���b� * �� t ��� t(' c � . Two vertices� and � in a graphareconnectedif thereis a path % * � � c �{z|z{z|��� Ñ �suchthat ��®����� * �� c ��� Ñ � . Thedistance¹ )�* � � �®��� � betweenthemis thelengthof ashortestpathbetweenthem.Thesubscriptü will beomittedwhenclearfrom thecontext. A connectedcomponentis a subgraphwhosevertex setis

� K ý , suchthatall �®����� �areconnectedandno ��� ý # � is connectedto some�¯� �

. A graphis connectedif all its verticesareconnected.A tree is a connectedgraphcontainingnocycles.Any graphwith nocyclesis a forest. A treein whichonevertex, theroot, is distinguished,is calleda rooted-tree. In a rooted-treeany vertex of degreeone,excepttheroot, is calleda leaf. Thereis preciselyonepathbetweenany two verticesof a tree.Thedepthor level of a vertex in a rooted-treeis thelengthofthepathfrom theroot to thatvertex. If ��®����� is anedgeof a rooted-treesuchthat � lies on thepathfromtheroot to � , then � is the fatherof � and � is a child of � . An ancestorof � is any vertex of thepathfrom� to therootof thetree.Similarly, if � is anancestorof � , then � is adescendantof � . Finally abinary tree(resp. � -ary tree) is a rooted-treein which everyvertex, unlessit is a leaf,hastwo (reps. � ) children.

1.4 RandomGraphs

Let + 9 ï � bethesetof all (labelledandundirected)graphswith-

verticesand � edges.If � *ØÒ 9 . Ô and+ 9 *-,/.� ± g + 9 ï � then U + 9 ï � U * Ò .� Ô and U + 9 U *¤7 . . Informally, a randomgraph is a pair formedbyan elementü of + 9 alongwith a non-negative real value

¼ �suchthat

¥ � ¨ 021 ¼ � *Ê2. In otherwords

randomgraphsareelementsof a probability spaceassociatedwith + 9 , calledthe randomgraph model.Thereareseveralrandomgraphmodels.In mostcasesthesetof eventsis thesetof all subsetsof + 9 andthedefinitionis completedby giving a probabilityto eachü=�3+ 9 . If 4 is a randomgraphmodelwe willwrite ü �54 to meanthat �+�{¡ ü�¢ is definedaccordingto thegivenmodel.

Theprobabilityspace+ ��- �_� � is obtainedby assigningthesameprobabilityto all graphson-

verticesand � edgesandassigningto all othergraphsprobabilityzero. For each� * wx� 2 �|z{z|z~�� , + ��- �_� � hasÒ .� Ô elementsthatoccurwith thesameprobability

Ò .� Ô â c . Sometimesthealternativenotation+ � � 9 �Z� � isusedinsteadof + �)- �Z� � , where � 9 is calledthebasegraphsincetheelementsof thesamplespaceareallsubgraphsof thecompletegraph.Variantsof + �)- �Z� � arethusobtainedby changingthebasegraph.For

8

examplethesamplespaceof + � � 9 ï 9 �_� � is thesetof all bipartitegraphson- � - verticesand � edges.

This is madeinto a probabilityspaceby giving thesameprobabilityto all suchgraphs.In the model + �)- � ¼Y� (sometimesdenotedby + � � 9 � ¼�� ) we have wT� ¼ � 2

andthe modelconsistsof all graphswith

-labelledverticesin which edgesarechosenindependentlyandwith probability

¼. In

otherwords if ü �6+ ��- � ¼Y� and U � � ü � U * � then �i�{¡ ü�¢ *½¼ � �32 � ¼�� . â � . A variantof + �)- � ¼Y� is+ � � 9 � �À¼�t�ï Æ �_� in which edge &�D�òñ�� is selectedto bepartof thegraphor not with probability¼�t�ï Æ . So for

example + � � 9 ï 9 � ¼Y� , whosesamplespaceis the setof bipartitegraphson- � - verticesin which each

edgeis presentwith probability¼, is indeedaninstanceof + � � ._9 � �À¼bt)ï Æ �_� .

To avoid undesiredinconsistenciesit is importantthatunderfairly generalassumptionsresultsobtainedon onemodeltranslateto resultsin anothermodel.A property \ holdsalmostalways(or a.a.),for almostall graphsor almosteverywhere (a.e.) if

�! #" 9 %¦' �i�{¡ ü»��\�¢ * 2. The following theorem,reportedin

[Bol85, Ch.II], relates+ �)- � ¼�� and + ��- �_� � .Theorem1.4.1 (i) Let \ beanypropertyandsupposethat

�! !" 9 %(' ¼®�Z2 � ¼Y� � * �� . Thenthefollowingtwo assertionsare equivalent.

1. Almosteverygraphin + ��- � ¼�� has \ .

2. Given� ³½w and ÷�³>w , if

-is sufficiently large, there are 7 @»�32 ��÷ �Z7:�98 ¼®�32 � ¼�� � integers: c �|z{z|z{� :<;

with¼ � � � 8 ¼®�32 � ¼�� �Ë� : c � : . �þz|z|z�� :<; � ¼ �=� � 8 ¼r�32 � ¼�� �such that �i�>=@?~¡ \�¢5³ 2 �P÷ for every � *�2 �|z{z|z~�7 .

(ii) If \ is a convex propertyand�! #" 9 %(' ¼®�32 � ¼Y� � * �� , thenalmosteverygraphin + ��- � ¼�� has\ , where

: *BA ¼ �=� �98 ¼®�Z2 � ¼Y� �5C .(iii) If \ is a propertyand w�� ¼�* : C �Ë� 2 then�+� = ¡ \�¢5­Ù�+�ED�¡ \�¢!ø cZûF = 8 7�G;¼r�32 � ¼�� �»­�Á à : �i�HD;¡ \�¢The successof a randomgraphmodeldependson many factors. From a practicalpoint of view the

modelmustbereasonablein termsof realworld problemsandit mustbecomputationallyeasyto generategraphsaccordingto the specificdistribution assignedby the model. From the theoreticalpoint of viewthechoiceof onemodelover anotherdependson thespecificproblemat handandit is oftena matteroftrading-off thesimplicity of combinatorialcalculationsperformedundertheassumptionthatagivengraphwassampledaccordingto a certainmodel, for the tightnessof the desiredresults. + �)- �Z� � often givessharperresultsbut it is sometimesmoredifficult to handlethan + �)- � ¼Y� . In Chapter2 a slightly differentmodel will be usedwhich keepsthe good featuresof + �)- �Z� � and is easierto analyse. Let I 9 ï �

bethesetof all (labelledandundirected)multigraphson

-verticesand � edges;let I 9W* , '� ± g I 9 ï �

.I ��- �_� � is theprobabilityspacewhosesamplespaceis thesetof pairs� : �_Û � where

: �&I 9 ï �andÛ is a permutationof � objectsgiving an orderingon the � edgesof

:. The probability measureon

the samplespaceassignsthe sameprobability � â � to all elementsof I 9 ï � O V � . Strictly speaking,I ��- �_� � is a randommultigraph model. Figure1.3 shows a graphon 7 verticesand13 edgesanda

1

2

3

45

6 7

1

2

3

45

76

Figure1.3: Examplesof randomgraphs

multigraphwith the samenumberof verticesandedges.In particularthe multigraphshown on the rightcorrespondsto �PÚ elementsof thesamplespaceof I �)- �Z� � , onefor eachpossibleordering.Themodelis somehow intermediatebetween+ �)- �Z� � andtheuniformmodelor multigraphprocessmodelasdefinedin [JKŁP93] in which � orderedpairsof (not necessarilydistinct) elementsof ¡ - ¢ * 2 � 7 �{z|z|z�� - � aresampled.

9

The practicalsignificanceof I ��- �Z� � is supportedby the very simpleprocesswhich enablesus togenerateanelementin this space:for � timesselectuniformly at randomanelementin ¡ - ¢KJ .�L , thesetofunorderedpairsof integersin ¡ - ¢ * 2 �|z|z{z�� - � .

Againaresultthatrelatespropertiesof I ��- �_� � to thoseof + �)- �Z� � is needed.Thefollowing sufficesfor thepurposesof Chapter2.

Theorem1.4.2 Let É and ô betwo randomvariablesdefinedrespectivelyon + ��- �_� � and I ��- �_� � . IfÉ � ü �i* ô � ü � for every ü¬��+ 9 ï � � I 9 ï �and � * ? - then

Í � É � ­�, � Í � ô �_� .Proof. Í � É � * Î� ¨ 0 1�M N É � ü � ��Ú � � �P� � Ú�¾Ú­ Î� ¨ 0 1�M N É � ü � ��Ú� � ��� � �* Î� ¨ 0 1�M N É � ü � ��Ú� �PO 2 � ��RQ â �If � * ? - since � *�-i��- � 20�_C�7 we haveO 2 � ��RQ â � ­¯øTS ú 11VU �which is asymptoticto ø�W ú . Hence Í � É � ­�, �320� Î� ¨ 0 1�M N É � ü � �PÚ� �For every simplegraph ü with � edgesthereareexactly ��Ú elementsof

� ü��DÛ � �XI 9 ï � O V � SinceÉ � ü �i* ô � ü � for every ü ��+ 9 ï � � I 9 ï �wecanwrite

Í � É � ­�, �Z2&� Â Í � ô � . ²1.5 Concluding Remarks

This chapterhasprovided both the algorithmiccontext andthe necessarytechnicalbackgroundfor thisthesis.A few morespecificconceptswill bedefinedin therelevantchapters.We arenow in a positiontoapplya numberof randomisedtechniquesto severalcombinatorialproblemareas.

10

Chapter 2

Hard Matchings

This chapterwill investigateanotherapplicationof randomnessto ComputerScience,sometimescalledinput randomisation.This is basedon the simpleprinciple that sometimesthesetof instancesfor whicha particularproblemis difficult to solve forms a rathersparseor well structuredsubsetof the setof allinstances.In this context theassumptionthatnot all inputsoccurwith thesameprobabilitycanbeusedtoderive improvedperformancesfor specificalgorithmicsolutions.

The settingwill be that of optimisationproblems.Many of theseproblemsseemto be quite hardtosolve exactly. For examplethe problemof finding the minimumlengthtour arounda setof towns is thewell known NP-completeTravelling SalesmanProblem[GJ79]; thatof finding the maximumnumberofnon-adjacentverticesin a graphis equivalentto solvingthesocalledMaximumIndependentSetproblem[GJ79]. The hardnessresultstell us that it is possibleto constructa set of instanceson which, underreasonableassumptions,theoptimumcannotbefoundin polynomialtime.

Oneway to copewith theseresultsis to relax the optimality requirementandbe happy with an ap-proximatesolution. In Section2.1 the notion of approximationwill be madeprecise: the conceptsofapproximationalgorithmandapproximationcomplexity classeswill bedefined.

A matchingin a graphis a setof disjoint edges.Severaloptimisationproblemsaredefinablein termsof matchingsby just changingthe cost function or the optimisationcriterion. For instance,if for anygiven graph ü andmatching

:in ü , the cost function returnsthe numberof edgesin

:andthe goal

is to maximisethe valueof this function, thenthe correspondingproblemis that of finding a maximum(cardinality)matchingin ü . Theproblemof findingamaximummatchingin agraphhasaglorioushistoryandhasan importantplaceamongcombinatorialproblems.TheclassNP canbecharacterisedasthe setof all decisionproblemsfor which finding a solutionamongall possiblecandidatescantake exponentialtimebut checkingwhetheracandidateis a solutiononly takespolynomialtime (seefor example[BDG88,Ch. 8]). Maximum matchingis a nice exampleof a problemfor which, despiteof the existenceof anexponentialnumberof candidates,a solutioncanbefoundquickly. This fact,discoveredby [Edm65], ledto a numberof algorithmicapplications(seefor example[HK73, MV80]).

Few othermatchingproblemssharethe nice propertiesof maximummatching. In this chaptertwoproblemswill beconsideredwhich arenot known to besolvablein polynomialtime. Section2.2providesthereaderwith therelevantdefinitionsandgivesanoverview of theknown resultsfor theseproblems.Thefollowing four sectionspresenta numberof worstcaseresults.

Theresultsin Section2.3and2.6 imply thatbothproblemsareNP-hardfor graphswith givenboundson the minimumandmaximumdegree. In Section2.3 we introducetheconceptof almostregulargraphandwe prove that thefirst problemis NP-hardfor almostregularbipartitegraphsof maximumdegree Á )for every integer )^³µw . For thesecondone,we show thatevenfinding a solutionwhosesizeis at leastaconstantfraction ÷ from theoptimumis NP-hardfor somefixed ÷�� 2 , even if the input graphis almostregularof maximumdegree

Ä ) for every integer )�³�w .Section2.3.2describesa slightly unrelatedhardnessresult. A relationshipis establishedbetweenone

of thematchingproblemsunderconsiderationandtheproblemof finding a particularsubgraphof a givengraphcalleda2-spanner(thereaderis referredto Section2.3.2for furtherdetailsaboutthisproblem).Thereductionhasbeenrecentlyused[DWZ99] to prove approximationresultsfor an optimisationproblem

11

relatedto 2-spannersin a classof planargraphs.The resultwas includedin this work asan interestingapplicationof thematchingproblemconsidered.

A numberof simplepositive approximationresultsaregivenin Section2.4 and2.5. In thefirst of thetwo a generaltechniqueis describedfor obtainingsimpleguaranteeson thequality of anyapproximationheuristicfor the given problemsif the input graphshave known boundson the minimum andmaximumdegrees.Also, familiesof regulargraphsareconstructedthatmatchtheseguarantees.Secondly, Section2.5describesalineartimealgorithmwhichsolvesoptimallythesecondproblemif theinputgraphis atree.Theproblemwasknown to besolvableexactlyontreesbut thepreviouslyknown algorithmicsolutioninvolvedmatrixmultiplicationandanintricatealgorithmfor computingalargestindependentsetin achordal graph.Our solutionis greedyin flavour in that it builds theoptimalmatchingby traversingtheedgesin the treeonly once.

Thesimplepositiveresultspresentedin Section2.4arefinally matchedby anumberof negativeresults,presentedin Section2.6.

2.1 Approximation Algorithms: General Concepts

The definition of optimisationproblemwasgiven informally in Section1. This chapterwill be entirelyconcernedwith a particularclassof optimisationproblems. All the definitionsin this Sectionare from[Cre97].

Definition 1 AnNP optimisationproblem(NPO) % is a tuple� �N�3HEI(J��_?h�D1 ¼ * � where:

(1) � is thesetof the instancesof % andthemembershipin � canbedecidedin polynomialtime.(2) For each

� �¯� , HEI(J �)��� is the setof feasiblesolutionsof�. Membership in HEI(J �)��� can be

decidedin polynomialtimeandfor each ���HEI�J �)��� , thelengthof � , U �/U is polynomialin thelengthof�.

(3) For each� ��� and each ����HNI�J ���Y� , ? ��� �_� � is an integer, non-negativefunction,called the

objectiveor costfunction.(4) 1 ¼ *(�� " ÝV��� "� �� � is theoptimisationcriterionand tells if theproblem % is a maximisationor a

minimisationproblem.

For exampleif � is thesetof undirectedgraphs,HEI�J � ü � is, for every ü¬�k� , thecollectionof all setsof vertices

� K ý�� ü � suchthatevery edgein ü hasat leastoneendpoint in�

, ? � ü�� � �s* U � U for every� �]HEI�J � ü � , and 1 ¼ * * "� �� thentheproblemunderconsiderationis thatof finding a, socalled,vertexcover of theedgesof minimumcardinality(denotedby M INVC). Thenumberof verticesof this optimalcover is agraphparameternormallydenotedby Y � ü � [LP86].

Definition 2 Let % be an NPOproblem. Givenan instance�

anda feasiblesolution � of�, the perfor-

manceratioof � with respectto�

isZ ��� �Z� �i* " ÝV�\[ ? ��� �Z� �1 ¼ * �)��� � 1 ¼ * �����? ��� �Z� ��]where 1 ¼ * ����� is theobjectivefunctionevaluatedat anoptimalpoint.

U

Figure2.1: PossibleVertex Covers

For examplethevertex cover�

in Figure2.1hascardinalitya.2Y � ü � (anoptimalcover is shown on the

left). Hence

Z � ü�� � �i* Á C�7 .12

Definition 3 Let % be an NPO problemand let ^ be an algorithm that, for any given instance�

of % ,returnsa feasiblesolution ^ ����� of

�. Givenan arbitrary function _��y� ö � �32 �¸ � we saythat ^ is an_ ��-5� -approximationalgorithmfor % if, for everyinstance

�of order

-,Z ��� ��^ �)���_� ­`_ �)-5� z

Alsowesaythat % canbeapproximatedwith ratio _^³ 2 if there existsan _ -approximationalgorithmfor% .

For thevertex coverproblema verysimpleandelegantargumentprovesthatthenumberof end-pointsof theedgesin amaximalmatchingin ü alwaysapproximatesY � ü � within afactorof two (seefor instance[CLR90, Sect.37.1]).

Optimisationproblemscan be groupedinto classesdependingon the quality of the approximationalgorithmsthat they have. The classAPX containsall NPO problemswhich admit a polynomial time� -approximationalgorithmfor someconstant�á³ 2 . TheclassPTAS containsall NPOproblemswhichadmit a polynomial time � -approximationalgorithmfor any constant�î³ 2

. The classPTAS takes itsnameandis characterisedin termsof aparticularfamily of approximationalgorithms.

Definition 4 A polynomialtimeapproximationscheme(or ptas)for anNPOproblemP is analgorithm �which takesasinputan

� �k� andan errorbound ÷ andhasa performanceratioZ ù ��� ��� �����_� ­ 2 �á÷Thealgorithm � runsin timepolynomialin theinputorder andin ÷ â c .2.2 ProblemDefinitions

In this sectionthe relevantoptimisationproblemswill bedefined.All graphsin the following discussionwill beundirectedandlabelled.If ü *¬�8ý �_� � is a graph,a set

: K�� is a matching in ü if ÿ c � ÿ . *þRfor all ÿ c �_ÿ . � :

. Letý^� : �

bethesetof verticesbelongingto edgesin thematching.A matching:

is maximalif for every ÿ��¾�a# : , thereexists� � :

suchthat ÿ � �îÇ*¬R (we saythat�

covers ÿ ). Amatching

:is inducedif for every edgeÿ * ��®����� , ÿ�� :

if andonly if �®����� ý�� : � and ÿ��á� . Anumberof parameterscanbedefinedto characterisematchingsin graphs:

Definition 5 If ü *��qý �D� � is a graphthen

1. b � ü � denotestheminimumcardinality of a maximalmatching in ü ;

2. c � ü � denotesthemaximumcardinality of a matching in ü ;

3. c�d � ü � denotesthemaximumcardinality of an inducedmatching in ü .

In thefollowing sectionssomeof thecombinatorialandcomputationalpropertiesof parametersb � ü �and c d � ü � will be described.The appelationsM INMAXLMATCH andMAX INDMATCH will be usedtoidentify thecorrespondingoptimisationproblems.

A vastliteratureis concernedwith theparameterc � ü � (see[LP86] for a beautifulandcompletetreat-ment)but muchlessseemsto beknownaboutb � ü � and c d � ü � . Thefollowingparagraphsgiveanoverviewof theknown results.

Small Maximal Matchings. The oldestresulton b � ü � is by Forcade[For73] who proved an approx-imation result on hypercubes.The first negative result is by YannakakisandGavril [YG80]. An edgedominatingsetin agraphis asetof edges« suchthatfor all ÿ(��� � ü � thereexists

� ��« with ÿ � �áÇ*µR .All maximalmatchingsareedgedominatingsets.YannakakisandGavril provedthateveryedgedominat-ing setcanbetranslatedinto a possiblysmallermaximalmatching.Thenthey provedthatfinding anedgedominatingsetwith at most � edgesis NP-completeeven for planaror bipartitegraphswith maximumdegreethree. In the samepaperthey give a polynomial time algorithmfor trees. SubsequentlyHorton

13

andKilakos[HK93] extendedtheNP-completenessto planarbipartitegraphs,planarcubicgraphsandfewotherclassesof graphs.A graphis chordal if for every simplecircuit of lengthat leastfour thereexistsanedgenot in thecircuit connectingtwo verticesbelongingto thecircuit. HortonandKilakosalsogavea, ��- a � algorithmfor classesof chordalgraphsanda few otherclasses.

Induced Matchings. Thefirst proof of NP-completenessis in [SV82]. Theauthorspresenttheir resultsin termsof � -separatedmatchings.The notion of distancebetweentwo vertices,definedin Section1.3,canbe extendedin the obvious way to pairsof edges.Given a graph ü *»�8ý �_� � , for all ÿ�� � ��� thedistance¹ )�* � � ÿ�� �y� is thelengthof theshortestpathamongall pathsconnectingtheverticesin ÿ and

�. A

matching:

is � -separatedif theminimumdistancebetweentwo edgesin:

is � . Obviouslya2-separatedmatchingis an inducedmatching.Stockmeyer andVaziraniprove thatfinding a � -separatedmatchingofsizeat least � is NP-complete.Their reductionis from vertex cover andholdseven for bipartitegraphsof maximumdegreefour. An alternative proof of NP-completeness,againevenfor bipartitegraphs,is in[Cam89]. Inducedmatchingshave attractedattentionfollowing two questionsby ErdosandNesetril(see[Erd88]):

1. Whatis themaximumnumberof edgesin agraphof maximumdegree� andsuchthat c d � ü � ­¯� ?2. Whatis theminimum * for which theedgesetof ü canbepartitionedinto * inducedmatchings.

A seriesof improvedanswershavecomein anumberof papers[FGST89, HQT93, SY93, LZ97]

Not muchis knownabouttheapproximabilityof b � ü � and c d � ü � . Theonly nontrivial resultis claimedin [Bak94] whereapolynomialtime approximationschemefor b � ü � is describedfor planargraphs.

2.3 NP-hardnessResults

In this Sectionwe startdescribingthe new resultsin this chapter. The classof graphsfor which M IN-MAXLMATCH andMAX INDMATCH areNP-hardto find is extended.The first resultdealsthe hardnessof M INMAXLMATCH for graphswhich show someregularity whereasthesecondonerelatesMAX IND-MATCH on bipartite graphsto anotherinterestingcombinatorialproblem. Other hardnessresultswillfollow asby-productsof theresultsin Section2.6.

A� �:�D� � -graphis agraphwith minimumdegree� andmaximumdegree� . A

� ¹b�_¹ � -graphis a regulargraphof degree¹ (or a ¹ -regulargraph).If % is anNPOgraphproblem,then

� �h�D� � -P (resp. ¹ -P)denotesthesameproblemwhentheinput is restrictedto thebeinga

� �h�¸� � -graph(resp.a ¹ -regulargraph).Finally,a� �h�¸� � -graphü is almostregular if � C � is boundedby a constant.

2.3.1 M INMAXLMATCH in Almost Regular Bipartite Graphs

TheNP-completenessproofsin [HK73] and[YG80] show that it is NP-hardto find a maximalmatchingof minimumcardinalityin a planarcubicgraphandin planarbipartite

�32 �_Á � -graphs.This last resultcanbeextendedto bipartite

� ��)��_Á ) � -graphsfor every integer )k³îw and � * 2 � 7 . Thefollowing resultshowshow to removeverticesof degreeonefrom a

�32 �¸� � -graph.

Lemma 1 There is a polynomialtime reductionfrom�Z2 �D� � -M INMAXLMATCH to

�87 �D� � -M INMAXL-MATCH.

v

v

1

2

v0

v3

vG

Figure2.2: Gadgetreplacinga vertex of degreeonein a bipartitegraph.

14

c1

c2

a

b

cd

a

a

b b

d

d

1

2

1

2

1 2

Figure2.3: A�32 �DÁ � -graphandits 2-padding.

Proof. Givena�Z2 �DÁ � -graph ü , thegraph üfe is obtainedby replacingeachvertex � of degreeonein ü by

thegadgetü � shown in Figure2.2.Theedge��b����� incidentto � is attachedto thevertex � g . Theresultinggraphhasminimumdegreetwo andmaximumdegreethree. If

:is a maximalmatchingin ü it is easy

to build a maximalmatchingin üfe of size U : U&� 7 U ý c � ü � U���U ý�� : � � ý c � ü � U . For every �� ý c � ü � add�� c ��� . � to: e moreover if ��®����� Ç� :

then: e will containalsothe edge �� g ��� a � . Converselyevery

matching: e in üfe canbe transformedinto a matching

: e e * : e ec � : e e. , with U : e eòUy­=U : eqU , suchthat: e ec is a maximalmatchingin ü and: e e is asetof edgesentirelycontainedin thegadgetsü � . ²

To provethemainhardnessresultin this section,thefollowing graphoperationwill beuseful.

Definition 6 The ) -paddingof a graph ü , ühg , is obtainedby replacingeveryvertex � by a distinctsetoftwin vertices� c �|z{z|z~���2g with �� t ��� Æ �(��� � ühg � if andonly if ��r���b����� � ü � .i Theverticesof ü g are partitionedinto ) layers. Verticesin each layer alongwith edgesconnecting

pairsof verticesin thesamelayer forma copyof theoriginal graph ü .i For each ÿ * ��®��������� � ü � edges ÿ tÈ* �� t ��� t � for � * 2 �|z{z|z~�>) are called twin edges. Edgesÿ t Æ * �� t ���&Æ:� for each � Ç* ñ are calledcrossedges.i Each copyof � g ï g obtainedby replacingtwo verticesandan edge in ü is calleda paddingcopy of� g ï g andsometimesdenotedby � �g ï g , to showdependencyon theedge in theoriginal graph ü .

Thefollowing resultis a simpleconsequenceof Definition6.

Lemma 2 If ü is a� �:�D� � -graphwith

-verticesand � edgesthen ü g is a

� )�Â2�h�)�Â�� � -graph with ) -verticesand ) . � edges.

To proveTheorem2.3.1it is importantto relate b � ühg � to b � ü � .Lemma 3 b � ühg � ­!)jb � ü � , for all graphsü and ) @þ2 .Proof. If

:is a maximalmatchingin ü , a maximalmatching

: g in ühg is obtainedby takingtheunionof U : U perfectmatchingsonein eachcopy of � �g ï g with ÿ¦� :

. ²Lemma 4 b � ü g �E@ )jb � ü � , for all bipartitegraphs ü and ) @µ2 .Proof. Let ü g be the paddedversionof a bipartite graph ü , and

: g be a maximal matchingin ü g .The ) -weightingof the edgesof ü is a function ���+� � ü �^� &wx�{z|z{z~�)�� . For each ÿ¾��� � ü � define� � ÿ ��*�Õ!Ö U : g � � � � �g ï g � U . Thefollowing propertieshold:

1. If � � � �+*�Õ×Ö ¥ í ��k � ¨ � ð � � ÿ � then � � � � �W0wx�|z{z|z~�>)�� for all �� ý�� ü � .Thesumof theweightsof theedgesincidentto � cannotbe largerthan ) otherwisetherewould bemorethan ) edgesin

: g incidentto � c �|z|z{z~��� g ; thereforeoneof theseverticeswould beincidenttomorethanoneedgein

: g .2. Let � � � �s* 0ÿk��� � ü � ��� � ÿ �E* �_� for �N�á0wx�|z{z|z��>):� . Then

, gt!± c � � � � is anedgedominatingsetof ü (asdefinedin Section2.2).

This is truebecauseif therewasanedgeÿ in ü not adjacentto any edgewith positive weight then� �g ï g would not becoveredby: g in ü g .

15

3. Let ü � � � � �Z� be the subgraphof ü inducedbyý^� � � � �Z� . Then � � ) � is a maximal matchinginü � � � ) �_� .

Theedgesin � � ) � mustbeindependentbecauseeachof themcorrespondsto a perfectmatchinginapaddingcopy of � g ï g .

4. Let üî�Tü � � � � �Z� bethegraphobtainedby removing fromý�� ü � all verticesin

ý^� ü � � � � �Z�Z� andalledgesadjacentto them.Then

, g â ct#± c � � � � is anedgedominatingsetin üî�Tü � � � ) �Z� .5. Let �^� ý^� üµ�Tü � � � ) �Z�_� ; if � � � � �!) then � � � �+* ) for every ����� � � � .

Let � t beoneof thetwin verticesassociatedwith � that is not iný^� : g � . For each�X�l� � � � each

of theedges�� t ���xÆh� (for ñ *Ê2 �|z{z|z~�>) ) mustbe adjacentto a differentedgein: g otherwisethey

wouldnot becovered.

Usingthe ) -weightingdefinedabove theedgesin: g canbepartitionedinto ) matchingseachcorre-

spondingto amaximalmatchingin ü .Firstof all, eachedgeÿ in � � ) � correspondsto ) distinctedgesÿ c �{z|z|z|�_ÿ�g in

: g . Define: � ñ �i* &ÿ Æ �

for eachÿ�Ù� � ) � � . We prove, reasoningby inductionon ) , that the setof remainingedgesin: g , : eg ,

canbepartitionedinto ) sets: Æ suchthat,

: Æ � : � ñ � correspondsto amaximalmatchingin ü , for eachñ *�2 �{z|z{z~�) .BASE. If ) *µ7 by property4 above,theset � �32&� is formedby a numberof pathsandevenlengthcycles,� c �|z{z|z~�D� Ñ . Eachcycle of length

7 � (for someinteger � ³ 2 ) canbedecomposedinto two matchings: cand

: .of size � by takingalternatingedges.If �sÆ is a paththen,by property5 above,neitherof its

end-pointscanbeadjacentto a vertex � with � � � �+* w . Thereforeagaintwo setof edgesareaddedto: c

and: .

.STEP. Let � g bethegraphinducedby theedgesof positiveweightlessthan ) . If

: g isamaximalmatchingin � g , then �@e � ÿ ��* � � ÿ � � 2 (resp. �@e � ÿ ��* � � ÿ � ) if ÿ�� : g (resp. ÿ Ç� : g ) is an

� )�� 20� -weightingof � � üî�Wü � � � ) �Z�_� correspondingto theamaximalmatchingin the )s� 2 -paddingof ü��Wü � � � ) �Z� . Theinductivehypothesisapplies. ²Theorem2.3.1 M INMAXLMATCH is NP-hard for almostregular bipartitegraphs.

Proof. We will prove hardnessfor� ��)��_Á ) � -graphs,with � *æ2

or � *F7. Yannakakisand Gavril

[YG80] proved that�32 �_Á � -M INMAXLMATCH is NP-hardfor bipartite graphs. The hardnessof

�87 �DÁ � -M INMAXLMATCH followsfrom Lemma1. Then, � *�2 � 7 the ) -paddingcanbeusedto obtainaninstanceof� ��)��DÁ ) � -M INMAXLMATCH restrictedto bipartitegraphs.The resultthenfollows from Lemma3 and

Lemma4. ²2.3.2 MAX INDMATCH and Graph Spanners

Given a graph ü * �8ý �_� � a7-spanneris a spanningsubgraphüfe with the propertythat ¹�)m* �9n � �r��� � ­7 Â:¹ )�* � � �®��� � for every �r���T� ý . Let ) . � ü � be the numberof edgesof a sparsest2-spannerof ü . The

problemhasmany applicationsin areaslike distributedcomputing,computationalgeometryandbiology[PU89, ADJS93]. Peleg and Ullman [PU89] introducedthe conceptof graphspannersas a meansofconstructingsynchronisersfor Hypercubicnetworks(their resultshavebeenrecentlyimprovedin [DZ00]).Theproblemof findinga2-spannerwith theminimumnumberof edgesis NP-hard[Pm89]. In thisSectionwe presenta reductionfrom the problemof finding a sparsest2-spannerin a graphto that of finding alargestinducedmatchingin a bipartitegraphwithout smallcycles.

For every ü on-

verticesand � edges,let o � ü � bea bipartitegraphwith vertex sets� * �� � �xÿ��� � ü � � and * ��qpþ� � is a cycleof length3 in ük� . Two vertices� � � �

and �qpþ�r in o � ü � areadjacentif theedgeÿ belongsto thecycle

�in ü .

Lemma 5 ) . � ü � ­¯�=�<c d � o � ü �_� .Proof. Let

:bean inducedmatchingin o � ü � . Define

Vî* &ÿ��X� � ü � �5�� � ��� p � Ç� : � . We claimthat

Vis a spannerin ü . This is sobecausefor every �� � ��� p �¦� :

theedges� �3A��]� � ü � thatform the

cycle�

alongwith ÿ aresuchthat �� Ö ���spN���~��ut����qpN� cannotbein:

andthereforeare� �3A^� V . ²

16

Lemma 6 ) . � ü �E@ �=�<c d � o � ü �_� .Proof. Let üfe be a 2-spannerin ü . We prove that we canconstructan inducedmatchingin o � ü � . Ifÿ(��� � ü � #s� � üfe � thenthereexist two edges

� �ZA���üfe suchthat��* 0ÿ�� � �3A�� is a trianglein ü . We add�� � ��� p � to thematchingin o � ü � andwe saythat thetriangle�

covers ÿ . Let:

bethesetof edgesin� � o � ü �_� constructedin this way. Sincea trianglecanonly coveroneedge,thereareno two edgesin:

sharingavertex �qp . Also by ourconstructioneveryedgein � � ü � #�� � üfe � is consideredonly oncesothatthereareno two edgesin

:sharinganedge-vertex. We claim that

:is an inducedmatchingin o � ü � .

Let �� � ���qpN�W� :, assumeedge ÿ belongsto triangles

� c �|z|z{z|� �wv J � L , andlet�Ø* 0ÿ�� � �3A�� . No edge��yx;���qp�z0� canbein

:because,by thedefinitionof

:, ��yx;���qp�z0�(� :

would imply that ÿ¦��� � üfe � andtherefore�� � ���qpE� Ç� :

. Similarly neither� Ö , nor ��t canbeiný^� : �

. ²Thegirth of agraphis thelengthof its shortestcycles.

Theorem2.3.2 MAX INDMATCH is NP-hard on bipartite graphswith girth at leastsix.

Proof. For everygraphü , thegirth of o � ü � is at leastsix, sinceno two vertices� � ��� Ö � �cansharetwo

neighboursin . Theresultfollowsfrom Lemma5 and6. ²2.4 Combinatorial Bounds

The NP-hardnessproofs in the previous section(and thosein Section2.6) imply that underreasonableassumptions(see[GJ79]) therecanbeno polynomialtime algorithmwhich findsa maximalmatchingofminimumcardinalityor oneof thelargestinducedmatchingsin thegivengraph.Thenext bestoptionis tolook for algorithmsreturningapproximatesolutions.In this sectionwe describesomeeasycombinatorialresultswhich imply someguaranteeson the quality of the solutionsfor the matchingproblemsunderconsiderationproducedby abroadfamily of algorithms.

The setof all matchingsin a graph ü , which we denoteby I � ü � , definesan instanceof a type ofcombinatorialobjectcalledanindependencesystem.Thefollowing definitionis from [KH78].

Definition 7 An independencesystemis a pair� �^��{ � where � is a finitesetand { a collectionof subsets

of � with thepropertythat whenever «¬ó�ü �|{ then « �5{ . Theelementsof { are called independentsets. A maximalindependentsetis anelementof { that is not a subsetof anyotherelementof { .

KorteandHausmann[KH78] analysedtheindependencesystem� � � ü � ��I � ü �_� andprovedanupper

boundof 2 on theratiobetweenthecardinalitiesof any two maximalmatchings.Theresultthey proveis

Theorem2.4.1 c � ü � ­ 7 b � ü � for anygraph ü .

The theoremimmediatelyimplies the existenceof a simpleapproximationheuristicfor b � ü � : anal-gorithm returningany maximal matchingin ü is a 2-approximationalgorithm. We thereforehave thefollowing theorem.

Theorem2.4.2 M INMAXLMATCH canbeapproximatedwith ratio 2.

In thenext resulta similar argumentis appliedto�32 �D� � -MAX INDMATCH. Let ü bea

� �h�D� � -graph.Let I}d � ü � be the set of all inducedmatchingsin ü . The pair

� � � ü � ��I}d � ü �_� is an independencesystem.For every

V K=� the lower rank ofV

, ~ �qVi� is the numberof edgesof the smallest(maximal)inducedmatchingincludedin

V; theupperrank °~ is definedsymmetrically. By atheoremin [KH78], if

:is a maximalinducedmatching,then c d � ü �U : U ­ " ÝV���� © ~ �8Vi�~ �8Vi�Theorem2.4.3 Let ü bea

� �h�¸� � -graphand� � � ü � ��I}d � ü �_� begiven.Then" Ý����� © ~ �qVi�~ �qVi� ­ 7x� ��� 20� z

17

Proof. Let: c

and: . betwo maximalinducedmatchingsin ü andlet ÿ¦� : . # : c . By themaximality

condition, the set: c ��&ÿ�� is not independent(i.e. it is not an inducedmatchinganymore). Hence

thereexists` � ÿ � � : c

at distanceless than two from ÿ and since: . is maximal and independent,` � ÿ � � : c # : . . Indeed definesa function from

: . # : c to: c # : . . Let

�beoneof theedgesin

therangeof`. A boundon thenumberof edgesÿ(� : . # : c thatcanbethepre-imageof

� � : c # : .is needed.In theworstcase(occurringwhen ÿ is not adjacentto

�) therecanbeat most

7�� �S� 20� such ÿ .Theresultfollows. ²

Thelastresultprovesthatany algorithmthatreturnsamaximalinducedmatchingin thegivengraphisa7x� ��� 20� -approximationalgorithm.

Theorem2.4.4�32 �D� � -MAX INDMATCH canbeapproximatedwith ratio

7x� �L� 2&� .Thenext setof resultsdescribesa techniqueto obtainimprovedboundson b � ü � and c d � ü � on

� �h�D� � -graphs.Theideacanbereadilyintroducedin thecontext of vertex cover. Let ü *��8ý �_� � bea

� �h�D� � -graphandlet

�be a vertex cover of cardinality � in ü . If

�is a vertex cover then

Z *Øý # �mustbe an

independentsetotherwisetherewould beanedgenot coveredby�

. This impliesthatall edgesgoingoutfrom

Zmustendin

�. Let ¹ c � � � bethenumberof edgesadjacentto �^� �

andto some� e/� Z. ThenÎ� ¨2� �;ø � � � �+* Î� ¨ � ¹ c � � � (2.1)

If ü hasminimumdegree�¦³�w andmaximumdegree ��)- �T� � �¦­ Î� ¨2� �;ø � � � �+* Î� ¨ � ¹ c � � � ­��x�from which � @ - ��þ�&� z

Similarargumentcanbeappliedto boundb � ü � and c d � ü � .Theorem2.4.5 If ü is a

� �h�D� � -graph

1. b � ü �E@ �m� � J � L �. J(� ' � â c L ;2. c d � ü � ­ � � � J � L �. J�� ' � â c L .

Proof. If:

is a maximalmatchingthen

Z *�ý # ý�� : � is an independentset. Otherwise:

couldbeextendedwith any edgeconnectingtwo differentverticesin

Z. This impliesthatall verticesadjacentto a

vertex in

Zmustbelongto

ý^� : �. Theanalogueof (2.1) isÎ� ¨2� �;ø � � � �+* Î� ¨ � J = L ¹ c � � �

If ü hasminimumdegree�¦³�w andmaximumdegree � @þ2 ,��- � 7 � � �¦­ Î� ¨2� �;ø � � � �+* Î� ¨ � J = L ¹ c � � � ­ 7 � � �S� 20�Thus � @ �xU ý�� ü � U7�� �µ�&��� 20�

If:

is amaximalinducedmatchingthenfor each� suchthat ��®������� :,� � � ���� � ����®� � � ý^� : �i*R

. On theotherhandit is not trueanymorethat

Zis an independentset(anedgebetweentwo verticesin

18

Zis atdistanceonefrom edgesin

:soit cannotbechosen,but its presenceis legal in amaximalinduced

matching).In this casewe canwrite (where ¹ c now refersto

Z)Î� ¨2� ¹ c � � �i* Î� ¨ � J = L � �;ø � � � � � 2&�

If ü hasminimumdegree� @þ2 andmaximumdegree � @ ���- � 7 � � � @�7 � � ��� 2&� (2.2)

from which wegettheupperbound �­ ��U ý�� ü � U7�� �µ�&��� 20� ²Noticethatif ü is ¹ -regularthelowerboundon b � ü � andtheupperboundon c�d � ü � coincide.Indeed

it is not difficult to constructa family of graphsmatchingthesebounds.

z1

u4z2

v4

u1

u2

v1

v2

v3 u3

Figure2.4: A cubicgraphwith smallmaximalmatchingandlargeinducedmatching.

Theorem2.4.6 Let ¹��� ö ' . If ü is a ¹ -regular graphon-

verticesthenc�d � ü � ­ ¹�U ý�� ü � U7x�q7 ¹�� 2&� ­�b � ü �Moreover for every ¹ *þ7 �y� 2 with ����� ö ' there existsa graph ü t on

7x�q7 ¹�� 2&� verticeswith c d � ü tò�+*b � ü tò�i* ¹ .Proof. The first part is an immediateconsequenceof Theorem2.4.5. The secondpart canbe provedbygiving a recursivedescriptionof ü t for all �i��� ö ' . To simplify thedescriptionit is convenientto draw ü tso thatall its verticesareon five differentlayers,calledfar-left, mid-left, central, mid-right andfar-rightlayer. Figure2.4shows ü c . Vertices� c and � . (respectively � c and � . ) arein thefar-left (respectively far-right) layer. Vertices� a and � e (respectively � a and � e ) arein themid-left (respectively mid-right) layer.Vertices � c and � . are in the centrallayer. Moreover an horizontalaxis separatesodd-indexed vertices(whicharebelow it) from even-indexedones(whichareabove),with smallerindexesbelow higherones.

Let ü t â c , for � @þ7 , begiven. Thegraph ü t is obtainedby addingfour centralvertices,two mid-leftandtwo mid-right vertices. Since ü c hastwo centralandtwo pairsof mid verticesandeasyinductionsprovesthat ü t â c has

7 ¡ 7�� �h� 20� � 2 ¢ centralverticesand7x� �h� 2&� mid-left andmid-rightones.Let � e J t â c L â c ,� e J t â c L , � e t â a , � e t â . bethefour “new” centralvertices,� . t�' c and � . J t�' c L , � . t(' c and � . J t�' c L themid-left

andmid-rightones.ü t hasall edgesof ü t â c plusthefollowing groups:

1. two edgesconnectingeachof � . t(' c , � . J t(' c L , (respectively � . t(' c and � . J t(' c L ) to � c and � . (respec-tively � c and � . ), plusedges �� . t�' c �_� e J t â c L â c ���~�� . t(' c �D� e J t â c L ����� . J t(' c L �_� e t â a ���|�� . J t(' c L �D� e t â . ����� . t�' c �_� e J t â c L â c ���~�� . t�' c �_� e J t â c L ����� . J t�' c L �_� e t â a ���~�� . J t�' c L �_� e t â . �All theseedgesarethecontinuousblacklinesin Figure2.5.(c).

19

v2

v1 u1

u2

(a)

z4(i-1)-1

z4i-2

z4(i-1)

z4i-3

u2i+1

u2(i+1)

v2(i+1)

v2i+1

(b) (c) (d)

Figure2.5: A ¹ -regulargraphwith smallmaximalmatchingandlargeinducedmatching.

Figure2.6: A cubicgraphwith smallmaximalinducedmatching.

2. A final setof edgesconnectseachof theeven index mid verticeswith thecentralverticesof ü t â cwith indices

Ä ñ�� 7 andÄ ñ^��Á for ñ * w�� 2 �|z|z{z��Z�s� 2 . Eachof the odd index mid verticesare

connectedwith thecentralverticesof ü t â c with indicesÄb� ñ�� 20� and

Ä�� ñ�� 2&� � 2 for ñ *�2 �{z|z{z~�Z� .Thesquaresin Figure2.5 representall mid verticesin ü t â c . Thebold solid lines in Figure2.5.(d)representthiskind of edges. ²

Theorem2.4.5.2 is complementedby thefollowing result,giving a lowerboundon thesizeof apartic-ular family of inducedmatchingsin thegivengraph.

Theorem2.4.7 Let��� �h�¸� �s* e � ú�â e � ' .� . If ü is a

� �h�D� � -graph,then c d � ü �È@ U ý^� ü � U C���� �h�¸� � . More-overfor every ¹ @�7 thereexistsa regular graphof degree¹ with ¹� ��� ¹��D¹ � verticesanda maximalinducedmatchingof size ¹ .Proof. Let ü bea

� �h�¸� � -graphon-

verticesand:

amaximalinducedmatchingin ü . An edgeÿ(��� � ü �is coveredby anedge

� � :if thereexistsapathof lengthlessthantwo betweenoneof theend-pointsofÿ andoneof theend-pointsof�

. Eachedgein ü mustbecoveredby at leastoneedgein:

. Converselyeveryedgein

:cancoveratmost

7�� ��� 20�3. � 7 ��� 2 edges.ThusU : U @ U ��U7x� �L� 2&� . � 7 �L� 2 @ �xU ý�� ü � U7x�q7 � . � 7 ��� 20� zA ¹ -ary depthtwo tree ^ Õ is formedby connectingwith an edgethe rootsof two identicalcopiesof acomplete¹ -ary treeon ¹ . �Ù¹¦� 2 vertices.Thegraphobtainedby taking ¹ copiesof ^ Õ all sharingthesamesetof

� ¹ª� 20� . leavesis regularof degree¹ , it has¹ªÂ ��� ¹��D¹ � verticesandamaximalinducedmatchingof size ¹ . Figure2.6shows theappropriatecubicgraph. ²

20

2.5 Linear Time Solution for Trees

Although NP-hardfor several classesof graphsincludingplanaror bipartitegraphsof maximumdegreefour andalmostregular graphsof maximumdegreefour (seeTheorem2.6.4), the problemof finding alargestinducedmatchingadmitsa polynomialtime solutionon trees[Cam89]. Thealgorithmicapproachof Cameronreducesthe problemto that of finding a largestindependentset in a graph � that can bedefinedstartingfrom thegiventree. If ü *=�8ý �_� � is a tree,thegraph � *�� T�_« � has U ý U�� 2 vertices,onefor eachedgein ü andthereis an edgebetweentwo membersof if andonly if the two originaledgesin ü areeitherincidentor connectedby a singleedge.Notice that U «^U * , � U ý U .0� . Moreovereachinducedmatchingin ü is an independentsetof the samecardinality in � . Gavril’ s algorithm[Gav72]findsa largestindependentsetin a chordalgraphwith

-verticesand � edgesin , ��- �¯� � time. Since

thegraph� is chordal,a largestinducedmatchingin thetreecanbefoundin , � U ý U . � time. In thissectionwe describea simplerandmoreefficient way of finding a maximuminducedmatchingin a treebasedondynamicprogramming.If ü *=�qý �D� � is a treewe choosea particularvertex _�� ý to bethe root of thetree(andwe saythat ü is rootedat _ ). If �]� ý #��_�� thenparent

� � � is theuniqueneighbourof � in thepathfrom � to _ ; if parent

� � �¦Ç* _ thengrandparent� � �s* parent(parent

� � � ). In all othercasesparentandgrandparentarenotdefined.If � * parent

� � � then � is � ’schild. All childrenof thesamenodearesiblingsof eachother. Let ? � � � bethenumberof childrenof node � . Theupperneighbourhoodof � (in symbolsUN� � � ) is emptyif � * _ , it includes_ andall � ’s siblingsif � is a child of _ andit includes� ’s siblings,� ’s parentand � ’s grandparentotherwise. � ��� ö � � �_� is the setof edgesin ü connectingthe verticesin

UN� � � .

Claim 1 If ü *��qý �D� � is a treeand:

is an inducedmatching in ü then U : � � �K� ö � � �Z� U;­ 2 , for every�^� ý .

To believe theclaimnoticethatif:

is aninducedmatchingin ü , any node� in thetreebelongsto oneofthefollowing typeswith respectto thesetof edges� ��� ö � � �_� :Type 1. theedge�������Ý��_ø � � � � � � is partof thematching,

Type 2. either ���Ý:�Dø � � � � � � � �¸Ý � ����Ý:�Dø � � � � � � or ���Ý:�Dø � � � � � ����� (where � is somesiblingsof � ) belongsto thematching,

Type 3. NeitherType 1. norType2. applies.

Thealgorithmfor findinga largestinducedmatchingin a tree ü on-

verticeshandlesan- OÁ matrix

ValuesuchthatValue¡ �¸��*ò¢ is thecardinalityof thematchingin thesubtreerootedat � if vertex � is of type * .Lemma 7 If ü is a treeon

-vertices,Value¡ �¸��*ò¢ canbecomputedin , ��-5� timefor every �N�á 2 �|z{z|z~� - �

and * *L2 � 7 �DÁ .Proof. Let ü bea treeon

-verticesandlet _ beits root. We assumeü is in adjacency list representation

andthat somelinear time preprocessingis performedto order(usingstandardtopologicalsort [CLR90,Ch. 23]) theverticesin decreasingdistancefrom theroot.

Thematrix Valuecanbefilled in a bottom-upfashionstartingfrom thedeepestverticesof ü . If � is aleaf of ü thenValue¡ �¸��*ò¢ * w for * *S2 � 7 �_Á . In filling theentrycorrespondingto node ��� ý of type * weonly needto considertheentriesfor all childrenof � , ñ c �|z|z{z|�òñ�� .

1. Value¡ �¸� 2 ¢ * ¥ W�J t LÑ ± c ÜiÝ ��� ø�¡ ñ Ñ � 7 ¢ .Since {�¸����Ý��_ø � � � � � � will be part of the matching,we cannotpick any edgefrom � to one of itschildren. The matchingfor the treerootedat � is just the union of the matchingsof the subtreesrootedat eachof � ’s children.

2. Value¡ �¸� 7 ¢ *î¥ W�J t LÑ ± c ÜiÝ ��� ø�¡ ñ Ñ �_Áh¢ .We cannotpick any edgefrom � to oneof its childrenhereeither.

21

3. If � has ? � � � childrenthenValue¡ �D�DÁh¢ if definedasthemaximumbetween¥ W�J t LÑ ± c ÜiÝ ��� ø�¡ ñ Ñ �DÁh¢ anda

numberor terms )~ÆE� *L2 �áÜiÝ �(� ø�¡ ñ Ñ � 2 ¢�� Î ;��± Ñ ÜiÝ ��� ø�¡ ñ ; � 7 ¢If theupperneighbourhoodof � is unmatchedwe caneithercombinethematchingsin thesubtreesrootedat eachof � ’s children(assumingthesechildrenareof type3) or addto thematchinganedgefrom � to one of its children ñ Ñ (the one that maximises)~ÆE� ) andcompletethe matchingfor thesubtreerootedat � with thematchingfor thesubtreerootedat ñ Ñ (assumingñ Ñ is of type1) andthatof thesubtreesrootedat eachof � ’s otherchildren(assumingthesechildrenareof type3).

Option threeabove is the most expensive involving the maximumover a numberof sumsequalto thedegreeof thevertex underconsideration.Sincethesumof thedegreesin a treeis linear in thenumberofverticesthewholetablecanbecomputedin lineartime. ²Theorem2.5.1 MAX INDMATCH canbesolvedoptimallyin polynomialtimeif ü is a tree.

Proof. The largestbetweenValue¡ _h� 2 ¢ , Value¡ _h� 7 ¢ andValue¡ _h�_Á:¢ is the cardinalityof a largestinducedmatchingin ü . By usingappropriatedatestructuresit is alsopossibleto storetheactualmatching.Thecomplexity of thewholeprocessis , ��-5� . ²2.6 Hardnessof Approximation

Sofar two particulargraphtheoreticparametersbothdefinedon theset I � ü � of thematchingsin ü havebeenconsidered.After proving someNP-hardnessresults,their algorithmicapproximabilitywasstudied.In this sectionwepresentsomenegativeresults.

Polynomialtime approximationschemes(asdefinedin Section2.1) for NPOproblemsareconsideredthenext bestthing to a polynomialtime exactalgorithm.For many NPOproblemsanimportantquestionis whethersuchaschemeexists.Theapproachtakenparallelsthedevelopmentof NP-completeness:someuseful notionsof reducibility are definedand thenproblemsare groupedtogetheron the basisof theirnon-approximabilityproperties.

Althoughseveralnotionsof approximationpreservingreductionshavebeenproposed(seefor example[Cre97]) the L-reductiondefinedin [PY91] is perhapsthe easiestoneto use. Let % be an optimisationproblem.For every instance

�of % , andevery solution � of

�, let ?�� �)� �Z� � bethecostof thesolution � .

Let 1 ¼ * � ���Y� bethecostof anoptimalsolution.

Definition 8 Let % and \ be two optimisationproblems. An L-reductionfrom % to \ is a four-tuple� * c ��* . �_����b � where * c and * . arepolynomialtimecomputablefunctionsand � and b arepositiveconstantswith thefollowingproperties:

(1) * c mapsinstancesof % to instancesof \ andfor everyinstance�

of % , 1 ¼ * � � * ch���Y�Z� ­Ù��Â31 ¼ * � �)��� .(2) for everyinstance

�of % , * . mapspairs

� * ch���Y� �_��e � (where ��e is a solutionof * c:����� ) to a solution �of�

sothat U 1 ¼ * � ���Y� �T? � ��� ��* . � * c������ �Z� e �Z� Ux­`bsU 1 ¼ * � � * c:���Y�Z� �T? � � * c:�)��� �Z� e � U¿zWe write %=­��W\ if thereexistsanL-reductionfrom % to \ . An importantpropertyof L-reductions

is givenby thefollowing result.

Theorem2.6.1 Let % and \ betwo NPOproblemssuch that % ­ � \ with parameters � and b , andit isNP-hard to approximate% with ratio ? .

1. If they arebothmaximisationproblems,thenit is NP-hard to approximate\ with ratio �V� WJ �V� â c L W ' c .2. If they arebothminimisationproblems,thenit is NP-hard to approximate\ with ratio J ��� ' c L W â c�V� W .

22

Proof. The result is essentiallyderived from [Pap94, Proposition13.2]. To prove the first statement,supposeby contradictionthatthereis analgorithmwhichapproximates\ with ratio �V� WJ �V� â c L W ' c . For everyinstance

�of % let ��e betheresultof applyingthis algorithmto * c ����� . Then,by definitionof L-reduction,1 ¼ * � �)��� �P? � �)� ��* . � * ch���Y� �_��e �_�1 ¼ *H� �)��� ­¯�9b 1 ¼ * � � * c:���Y�Z� �T? � � * c:�)��� �Z��e �1 ¼ *3� � * c ���Y�Z�

By definitionof performanceratio, � D v�� J v � J $ L�LW � J v � J $ L ï � n L ­ �V� WJ ��� â c L W ' c , therefore1 ¼ *3� � * c �����_� �T?~� � * c �)��� �_� e �1 ¼ * � � * c����Y�Z� ­ 2 � � �9b�� 20� ?i� 2��b5? * 2��b ç 2 � 2? èandtheresultfollows.

Similarly if % and \ areminimisationproblemsandthereexistsanalgorithmwhich approximates\with ratio J �V� ' c L W â c�V� W then? � �)� ��* . � * c����Y� �_��e �Z� �P1 ¼ * � ���Y�? � �)� ��* . � * ch���Y� �_� e �_� ­¯�9b ? � � * c:����� �Z��e � �T1 ¼ * � � * c:�)���Z�1 ¼ * � � * c����Y�Z�By definitionof performanceratio W � J v � J $ L ï � n L� D v � J v � J $ L�L ­ J �V� ' c L W â c�V� W andtheresultfollows. ²2.6.1 M INMAXLMATCH

YannakakisandGavril [YG80] provetheNP-hardnessof�32 �DÁ � -M INMAXLMATCH usingareductionfrom

3-M INVC restrictedto planargraphs.Their reductioncanbeusedasa building block for a numberof L-reductions.

Theorem2.6.2 Á -M INVC ­�� � �Y�DÁ � -M INMAXLMATCH with parameters � Ñ *>2 � 7���� � � and b Ñ *½2where

��� � �i* Á Ò Ñ . Ô � 7 for � *L2 � 7 �_Á .Proof. Let ü * �8ý �_� � be a planarcubic graph. ReduceÁ -M INVC to

� �Y�_Á � -M INMAXLMATCH, byreplacingevery vertex � t � ý by the gadget� tÑ shown in Figure2.7. The threeedgesincidentto � t areattachedto � t , � t and

¼ t(thereare3! possibilitiesto do this). Theresultinggraph * c:� ü � Ñ is planarand� � * c�� ü � Ñ ��* Á for all � . We claim that b � * c:� ü � Ñ ��*���� � � U ý^� ü � U:��Y � ü � . If

�is a vertex cover of ü

define : c * ��� t ��� t � ���~&� t ��� t�� ���| ¼ t ��� t�� ���V� t � � �+�W��� t � ��� t ú ���~�� t�� ��� t�� ���V� t Ç� � �: . * : c ���&� t � �_� t ú ���|&� t(� �_� t�� ���|&� t�� �D� t�  ��� � t � ý �: a * : c ����* t � ��* t ú ���~�* t�� ��* t�� ���|�* t(� ��* t(  �(�V� t � ý ����&� t � �_� t¢¡ ���|&� t ú �_� t(£ ���|&� t�� �_� t�¤ ���~0� t�� �D� t � g ���|&� t(� �_� t � c ���~0� t�  �D� t �Z. ���2� t � ý �: Ñ is clearly a matchingfor all � . It is maximal becauseall edgesin each � tÑ are adjacentto someÿ¯� :and if ÿ¯�S� was incident to some � t � �

then ÿ���� � * c�� ü � Ñ � and it is incident to oneof�� t ��� t � ���~&� t ��� t(� ���| ¼ t ��� t�� ��� :. We have U : Ñ U *��q7 �áÁ Ò Ñ . Ô �3- �µU � U .Converselyto prove b � * c � ü � c �s@�7:- �lY � ü � YannakakisandGavril show thatany maximalmatching:

in * c � ü � c canbetransformed(in polynomialtime) into another: e containingno edgeof theoriginal

graph ü andsuchthatevery � tc containseithertwo or threeedgesin:

. Everyedgeof theoriginalgraphmustbeadjacentto at leastone � tc containingthreeedgesin

: e ; vertices� t � ý associatedwith such� tcdefineavertex cover in ü .

For ��³ 2, any maximal matching

:in * c � ü � Ñ can be transformedin another(possiblysmaller)

matching: e that usesthe edgesin

: Ñ # : cto cover the sets � � � tÑ � #�� � � tc � . ThenYannakakisand

Gavril’ sappliesto: e # � : Ñ # : c � andtheedgesin � � * c � ü � Ñ � # � � � � tÑ � #s� � � tc �Z� . ²

3-M INVC is APX-complete[BK99]. Thereforethereexistsa ? g ³ 2 suchthatit is NP-hardto approx-imate Y � ü � with ratio ? evenif ü is a cubicgraph.Usingthis constantandTheorem2.6.1we have

Corollary 2 Let ? g be a constantsuch that Á -M INVC is NP-hard to approximatewith ratio ? g . Then� �y�_Á � -M INMAXLMATCH is hard to approximatewith ratio. J Ö J!Ñ L ' c L W¦¥ â cJ . Ö J!Ñ L ' c L W¦¥ .

23

H i1

H i2

zz z z z z

z zz zz z

H i3

z z z z z z

t

t t

t

t

ti1 i3 i4u i p imi

i1 i2 i3 i4

i2 i3 i4

i1s

i1s i3si2s

i2s i3s

i6i1 i2 i3 i4 5i

i2s

i3 5ii4 i6i1 i2

i2

1i

i4

i3 i5

i6

i3si1s

w w w

mw w w pu i i i

mw w w w pu i i i

w

i1

i i8 i9 i10 i11 i127

Figure2.7: Gadgetsfor avertex � t � ý^� ü � .2.6.2 MAX INDMATCH

Let MIS denotethe problemof finding a largestindependentset in a graph(problemGT20 in [GJ79]).Appellations

� �h�D� � -MIS and ¹ -MIS are definedin the obvious way. Thereis [KM] a very simple L-reductionfrom MIS to MAX INDMATCH with parameters� * b *�2 . Givena graph ü *��qý �D� � , define* ch� ü �+*S�8ý e8�_�\e � asfollows:ý e *�ý �]�� e � �^� ý ���¤� e * ���W������� e ��� �^� ý ��zIf�

is anindependentsetin ü then « * ��������e��k� ��� � � is aninducedmatchingin * c:� ü � . Converselyif « is an inducedmatchingin * ch� ü � the set * . � * ch� ü � �_« � obtainedby picking oneendpointfrom everyedgein « is anindependentsetin ü . Thereforethesizeof a largestindependentsetin ü is c�d � * c:� ü �_� .

The ) -paddingof a graphwasintroducedin Section2.3.1. The key propertyof the ) -paddingswithrespectto the inducedmatchingsin the original graphis that they preserve the distancesbetweentwovertices.If ü *��8ý �_� � is agraphthenfor every �r����� ý�� ü � , ¹ )�* � � �®��� � is thedistancebetween� and � ,definedasthenumberof edgesin ashortestpathbetween� and � .

Lemma 8 For all graphs ü and for every ) @ 7 , ¹�)m* � � �r��� ��* ¹ )�* �¨§ � � t ���&Æ � for all �r���¾� ým� ü � with� Ç* � andall �¸�qñm�� 2 � 7 �|z{z|z��>):� .Lemma 9 For all graphsü andfor every ) @�7 , cVd � ü �+* c�d � ühg � .Proof. Let

:be an inducedmatchingin ü . Define

: g * ��� c ��� c �á�þ� � ü g � �E��r���b�¾� : � . ByLemma8 all edgesin

: g areat distanceat leasttwo. Converselyif: g is an inducedmatchingin ü g

define: * ���r���b�ã�P� � ü � ���� t ���&Æ:�k� : g for some�¸�qñ�á 2 �|z{z|z~�>)���� . : is aninducedmatchinginü . ²

Thefollowing Lemmasshow how to removeverticesof degreeoneandtwo from a�32 �D� � -graph.

Lemma 10 Any�32 �D� � -graph ü canbetransformedin polynomialtimeinto a

�q7 �¸� � -graph üfe such thatU ý c � ü � U * c d � üfe � �<c d � ü � .24

0

v1

v2

vGv

Figure2.8: Gadgetreplacinga vertex of degreeone.

Proof. Givena�Z2 �D� � -graphü , thegraph üfe is obtainedby replacingeachvertex � of degreeonein ü by

thegadgetü � shown in Figure2.8. Theedge ������(� incidentto � is attachedto � g . Theresultinggraphhasminimumdegreetwo andmaximumdegree � . If

:is aninducedmatchingin ü it is easyto build an

inducedmatchingin üfe of size U : U|�µU ý�c�� ü � U . Converselyevery inducedmatching: e in üfe will contain

exactly oneedgefrom every gadgetü t . Replacing(if necessary)eachof theseedgesby theedge�� c ��� . �couldonly resultin a largermatching.Thematchingobtainedby forgettingthegadget-edgesis aninducedmatchingin ü andits sizeis (at least) U : eqUh��U ý�c�� ü � U . ²Lemma 11 Any

�87 �D� � -graph ü canbetransformedin polynomialtimeinto a� Áx�¸� � -graph ü e such thatU ý . � ü � U * c d � üfe � �<c d � ü � .

Proof. Let ü bea�87 �D� � -graph.Every vertex � of degreetwo is replacedby thegraph üf© in Figure2.9.

Thetwo edges��r���(� and ��b����� adjacentto � arereplacedby edges��r��� c � and ������ . � . Let üfe bethe

Gw

w w1 2

Figure2.9: Gadgetreplacinga vertex of degreetwo.

resulting� Áx�¸� � -graph. If

:is a maximal inducedmatchingin ü , a matching

: e in ü e is obtainedbytakingall edgesin

:andaddingoneedgefrom eachof thegraphsü © . Figure2.10showsall therelevant

cases.If � � ý^� : � thenwithout lossof generalitywe canassumethat � c � ý^� : e � andoneof thetwoedgesadjacentto � . canbeaddedto

: e . If � Ç� ý�� : � thenany of thefour centraledgesin ü © canbeaddedto

: e . After thesereplacementsno vertex in the original graphgetsany closerto an edgein thematching.Inequality c d � üfe �E@ c d � ü � �îU ý . � ü � U follows from theargumentaboveappliedto a maximuminducedmatchingin ü .

Converselyfor any inducedmatching: e in üfe atmostoneedgefrom eachcopy of üf© belongsto

: e .Thecopiesof üf© with

: e � � � üf© �Q*�R arecalledempty, all othersarecalledfull. Inequality c�d � ü ��@c�d � üfe � ��U ý . � ü � U is provedby thefollowing claimsappliedto amaximuminducedmatchingin üfe .Claim 2 Anymaximalinducedmatching

: e in üfe canbetransformedinto anotherinducedmatching: e e

in üfe with U : eòU;­�U : e eòU andsuch thatall gadgetsin: e e are full.

Claim 3: * Õ!Ö : e e � � � ü � is an inducedmatching in ü .

To provethefirst claim,analgorithmis describedwhich,givenaninducedmatching: eyK�� � üfe � , fills all

emptygadgetsin: e . Thealgorithmvisits in turnall gadgetsin üfe thathavebeencreatedby thereduction

andperformsthefollowing steps:

w

w

any of these edges can be chosen

w

Figure2.10:Possiblewaysto definethematchingin üfe giventheonein ü .

25

u u v

ww1 1

u v

w1

Figure2.11:Filling anemptygadget,normalcases.

(1) If thegadgetüf© underconsiderationis emptysomelocal replacementsareperformedthatfill üf© .

(2) Thegadgetü © is thenmarkedas“checked”.

(3) A maximalityrestoration phaseis performedin which, asa consequenceof the local replacementsin Step(1), someedgesmight beaddedto theinducedmatching.

Initially all gadgetsare“unchecked”. Let ü © beanuncheckedgadget.If ü © is full thealgorithmsimplymarksit ascheckedandcarrieson to thenext gadget.Otherwise,since

: e is maximal,at leastoneof thetwo edgesadjacentto vertices� c and � . mustbein

: e for otherwiseit would bepossibleto extend: e

by picking any of thefour centraledgesin ü © . Without lossof generalitylet ��r��� c ��� : e . Figure2.11shows (up to reflectionsymmetries)all possiblecases.If vertex � doesnot belongto anothergadgettheneitherof the configurationson the left of Figure2.11 is replacedby the oneshown on the right. If � ispartof anothergadgetfew subcasesneedto be considered.Figure2.12shows all possiblecasesandthereplacementrule. In all casesafterthereplacementtheneighbouringgadgetis markedaschecked.Noticethatall replacementrulesdo not decreasethesizeof the inducedmatching.Also astheprocessgoesby,new edgesin � � ü � canonly be addedto the currentmatchingduring the maximality restorationphase.To prove thesecondclaim, assumeby contradictionthat two edgesÿ * ��r���b� and

��* ��(�_��� in:

areat distanceone. Notice that ¹ )�* � n � ÿ�� �y��* ¹ )�* � � ÿ�� �y� unlessall theshortestpathsbetweenthemcontaina vertex of degreetwo. Theexistenceof ÿ and

�is contradictedby thefact that

: e and: e e areinduced

matchingsin üfe andall gadgetsin üfe arefilled by: e e . ²

w1 w1

u v u v

w1 w1

u v u v

w1 w1

u v u v

w1 w1

vu u v

w1 w1

u v u v

Figure2.12:Filling anemptygadget,specialcases.

Lemma 12 Any� Á��D� � -graph ü canbetransformedin polynomialtimeinto a

�)Ä �¸� � -graph üfe such thatU ý a � ü � U * c. c d � üfe � �rc d � ü � .Proof. Let ü bea

� Áx�¸� � -graph.Thegraph üfe is definedby takingtwo copiesof ü andconnectingpairsof correspondingverticesof degreethreewith thegadgetshown in Figure2.13. Thegivengadgethasthefollowing importantproperties:

1. All maximalinducedmatchingsin ü � containexactly two edges.

2. Thereexistsan inducedmatching: M� in ü � suchthatneither � c nor � . areadjacentto a vertex iným� : �

.

26

Gv v1 2

vv

v v

v3

4

5

6

Figure2.13:Gadgetconnectingpairsof verticesof degreethree.

If:

is aninducedmatchingin ü thentheunionof two copiesof:

(onein eachcopy of ü ) anda copyof

: M�� for each �¯� ý a � ü � is an inducedmatchingin üfe . Converselygiven a matching: e in üfe the

replacementof: e � � � ü � � by ��� a ��� e ���~�� d ��� F ��� for every ��� ý a � ü � canonly leadto apossiblylarger

inducedmatching. ²The non-approximabilityof

� ��)�� � �¤� 2&� ) � -MAX INDMATCH (for � * 2 � 7 �DÁx� Ä ) andÄ ) -MAX IND-

MATCH follows from Theorem2.6.1appliedto known resultson independentset[AFWZ95, BK99].

Theorem2.6.3 Let Ì � �h�¸��_? �i* J c 'sª � ûDa�« L¦¬ Ö J � ï � L ' c¦­ WÖ J � ï � L W ' c . DefineA � wx�D��D? � * ?A � �¸�D��D? � * Ì � �¸�D��ZA � �/� 2 �D��_? �Z� � @µ2For every � @ Á , let ? � be a constantsuch that it is NP-hard to approximate

�Z2 �¸� � -MIS with ratio? � . Then for � * 2 � 7 �_Áx� Ä and every integer )�³õw it is NP-hard to approximate� ��)�� � �Ê� 20� ) � -

MAX INDMATCH with ratio A � �ã� 2 �D�þ� 2 �_? � � .Proof. The result for � *Ê2

follows from the L-reductionat the beginningof the sectionfor ) * 2and

a furtherL-reductionbasedon ) -paddingsfor ) @L7 . For �P�Ù 7 �DÁx� Ä � If ü hasminimumdegree ��� 2 ,Theorem2.4.7implies c�d � ü �E@ U ý Ñ â c:� ü � U C:��� �s� 2 �D� � . Theresultfollowsusingtheseboundsalongwiththereductionsin Lemma10,11,and12. ²Theorem2.6.4 Let ? g bea constantsuch that Á -MIS is NP-hard to approximatewith ratio ? g . Thenforeveryinteger )(³¯w it is NP-hard to approximate

Ä ) -MAX INDMATCH with ratiof_f_eDg W¦¥fF�® d W ¥ ' dDd .

Proof. The reductionat the beginning of Section2.6.2 andTheorem2.6.1 imply that it is NP-hardtoapproximate

�32 � Ä�� -MAX INDMATCH with ratio ? g . If theoriginalcubicgraphü has-

vertices,then * c � ü �has U ý c � * c � ü �_� U * U ý e � * c � ü �Z� U *Ø-

, no vertex of degreetwo or three, Å -®C�7 edgesand the maximumnumberof edgesat distanceat mostonefrom a givenedgeis 19. We call onesuchgrapha special

�32 � �� -graph.

Claim 4 Thereis anL-reductionfrom�32 � �� -MAX INDMATCH restrictedto special

�32 � Ä�� -graphsto� Áx� Ä�� -

MAX INDMATCH with parameters � * e¸ad and b *�2 .If ü is a

�32 � Ä�� -graphwith U ý . � ü � U * U ý a � ü � U * w , thenreplacingeachvertex � of degreeonewith thegadgetin Figure2.8,givesa

� Áx� Ä�� -graphü e . Thepropertiesof special�32 � Ä�� -graphsandthesameargument

usedto proveTheorem2.4.7imply c d � ü �E@ da® U ý c � ü � U . ThereforecVd � ü e �i* c�d � ü � �îU ý�c:� ü � U;­Xc�d � ü � � Á2¯Å c�d � ü ��* Ä ÁÅ c�d � ü �Also, for every matching

: e in üfe , define* . � üfe)� : e � asdescribedin Lemma10. It follows that c d � ü � �U * . � üfeq� : e � U�­!c d � üfe � ��U : eqU andtheclaim is proved.Therefore,by Theorem2.6.1,

� Áx� Ä�� -MAX INDMATCH is hardto approximatewith ratio ? c�* eDa W ¥a® W¦¥ ' d .The special� Áx� Ä�� -graphs� generatedby the last reductionhave againa lot of structure. In particular,U ý�a�� � � U * U ýxe�� � � U , U � � � � U *6° U ýba�� � � U C:7 andagainthemaximumnumberof edgesat distanceat most

onefrom agivenedgeis 23. ²Claim 5 There is an L-reductionfrom

� Á�� Ä�� -MAX INDMATCH restrictedto special� Áx� Ä�� -graphsto

Ä-

MAX INDMATCH with parameters � *²± ccDc and b *�2 .27

Thereductionwasdescribedin Lemma12. Theorem2.4.7andthepropertiesof special� Á�� Ä�� -graphsimplyc d � � �s@ cDcF ± U ý a � � � U . Therefore c�d � � e � ­ 2 ¯�w2�2 cVd � � �

andthus,by Theorem2.6.1,4-MAX INDMATCH is hardto approximatewith ratiocH®_g W �c�F ± W � ' c_c . Finally by

Lemma9 thereis an L-reductionfrom 4-MAX INDMATCH toÄ ) -MAX INDMATCH (for ) @¤7 ) with pa-

rameters� * b *�2 .

28

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