Optimisation combinatoire et environnements dynamiques

N

◦

attribué par labibliothèque

Thèse

Pour l'obtention du titre de

DOCTEUR EN INFORMATIQUE

(Arrêté du 7août 2006)

Optimisation ombinatoire et environnements

dynamiques

Candidat: Ni olas BORIA

JURY

Dire teurde thèse: Vangelis Th. PASCHOS

Professeur àl'UniversitéParisDauphine

Rapporteurs: Christian LAFOREST

Professeur àl'UniversitéBlaise Pas al, Clermont-Ferrand

Denis TRYSTRAM

Professeur àl'Universitéde Grenoble

Surageants: Giorgio AUSIELLO

Professeur àl'UniversitéLa Sapienza, Rome, Italie

Cé ile MURAT

Maîre de onféren e àl'Université ParisDauphine

Peter WIDMAYER

Introdu tion 7

1 Dynami environments: State of the Art 11

Résumé . . . 11

1.1 Introdu tion . . . 14

1.2 Dynami optimization and reoptimization . . . 14

1.2.1 Introdu tion . . . 14

1.2.2 Fully dynami algorithms . . . 16

1.2.3 Reoptimization: Generalproperties . . . 24

1.2.4 Reoptimizing NP-hard optimizationproblems . . . 33

1.2.5 Reoptimizationof vehi le routingproblems . . . 37

1.2.6 Con lusion. . . 43

1.3 Probabilisti CombinatorialOptimization. . . 43

1.3.1 Introdu tion . . . 43

1.3.2 Motivations and Appli ations . . . 45

1.3.3 Post-optimizationand apriori optimization . . . 46

1.3.4 Formalismand obje tive fun tion . . . 49

1.3.5 Main methodologi alissues . . . 52

1.3.6 Con lusion. . . 58

1.4 Con lusion . . . 58

2 Reoptimizationof Min Spanning Tree 61 Résumé . . . 61 2.1 Introdu tion . . . 64 2.2 Node insertions . . . 65 2.2.1 AlgorithmREOPT1+ . . . 65 2.2.2 AlgorithmREOPT2+ . . . 73 2.3 Node deletions. . . 74 2.3.1 AlgorithmREOPT1- . . . 74 2.3.2 AlgorithmREOPT2- . . . 84 2.4 Con lusion . . . 84

3 Reoptimization of hereditary problems 87

Résumé . . . 87

3.1 Introdu tion . . . 90

3.2 Preliminaries . . . 90

3.3 Node insertion. . . 92

3.3.1 Max Independent set . . . 92

3.3.2 Max

k

- olorable Subgraph. . . 94

3.3.3 Max

Pk

-free Subgraph . . . 97

3.3.4 Max Split Subgraph . . . 110

3.3.5 Max PlanarSubgraph . . . 118

3.4 Node deletion . . . 120

3.4.1 Max Independent set . . . 121

3.4.2 Generalnegativeresult . . . 122

3.4.3 Max

k

- olorable Subgraph. . . 123

3.4.4 Restri tion tographs of bounded degree . . . 124

3.5 Con lusion . . . 127

4 Probabilisti Min Spanning Tree 129 Résumé . . . 129

4.1 Introdu tion . . . 132

4.2 Preliminaries . . . 133

4.3 Reoptimization and Min Spanning Tree . . . 134

4.4 Probabilisti Min Spanning Treeunder CLOSEST_ANCESTOR . . . 136

4.4.1 The omplexity of Probabilisti Min SpanningTree . . . 138

4.4.2 Stability of the solutions . . . 143

4.4.3 Probabilisti Metri Min Spanning Tree . . . 146

4.5 Probabilisti Min Spanning Treeunder modi ation strategy ROOT . 154 4.5.1 The omplexity of Probabilisti Min SpanningTree . . . 154

4.5.2 Stability of the solutions . . . 155

4.5.3 ROOTvs. CLOSEST_ANCESTOR . . . 156

4.6 Con lusion . . . 158

5 Probabilisti Dominating Set 159 Résumé . . . 159

5.1 Introdu tion . . . 161

5.2 Modi ation strategy and Fun tional . . . 161

5.2.1 The modi ation strategy M... . . 162

5.2.2 ...andthe omputationof its fun tional . . . 162

5.3 Approximation results . . . 163

5.3.1 Preliminaries . . . 164

5.3.2 Minimaldominating set

S

as anti ipatorysolution. . . 164

5.3.3 Optimaldominatingset

S

∗

asanti ipatory solution . . . 166

5.4 Polynomialsub ases . . . 167

5.4.2 Cy le Probabilisti Dominating Set . . . 174

5.4.3 Bounded Degree Tree Probabilisti Dominating Set . . 177

5.4.4 Equi Tree probabilisti Dominating Set . . . 180

5.5 Con lusion . . . 187

Con lusion & Perspe tives 189

Con lusion. . . 189

Complexités etapproximabilitésde

Π

et

RΠ

. . . 190

Complexités de

Π

et

P Π(

A

)

. . . 190

Réoptimisationet optimisationprobabiliste : deux adres ompleméntaires 191

Perspe tives . . . 192

Problèmes . . . 192

Modèles dynamiques : Généralisationsetvariantes . . . 192

A General Denitions 195

Cette thèseintitulée optimisation ombinatoireetenvironnementsdynamiques s'ins rit

dans le domaine de l'informatique, et plus parti ulièrement dans les thématiques de

l'optimisation ombinatoire et de la théorie de la omplexité algorithmique. Nous nous

intéressons i i aux méthodes de résolution de problèmes mathématiques, plus

ommuné-mentappeléesalgorithmes,quipermettentdetrouver, parmiunensemblenidesolutions,

elle quivaoptimiser(maximiserouminimiserselonles as)un ertain ritère. La

déni-tion detels algorithmesestfondamentale, puisqu'ellevapermettrede résoudreun ertain

nombre de problèmes quiémergent dans les domaines de l'informatique oude l'industrie

(optimisation de réseaux, de haines logistiques, ordonnan ement de tâ hes, et )

Si es problèmes omptent des nombres nis de solution, e n'est pas pour autant

que la puissan e de al ul des ordinateurs modernes permet de les résoudre de manière

e a e. Eneet, omptetenudela omplexitéde ertainsproblèmesposés, ilestsouvent

né essaire d'arbitrer entre laqualité de solutionet letemps de al ul pour trouver ladite

solution. Ainsi,àdéfautde trouverlasolutionoptimaleauregarddu ritère, ommenous

l'avons évoqué plus haut - e qui né essiteraitun temps de al ul beau oup trop long - ,

on préferera dénir une solution dite appro hée, qui quant à elle pourra être trouvée

dans un temps raisonnable. Il est ommunément admis qu'un temps raisonnable peut

être déni ommeun tempsbornépar une fon tion polynomialeen lataillede l'instan e

du problème àtraiter, en termes de nombre d'opérationsélémentaires.

De telles méthodes s'ins rivent dans lathématique de l'approximation polynomiale.

Si le problème du temps de al ul est pour le moins prégnant en optimisation

ombi-natoire,et orrespond àdesproblématiquesréellesdanslesdomainesd'appli ation

indus-triels (où on onsidère souvent qu'uneinformationlégèrementapproximativeaujourd'hui

vaut mieuxqu'uneinformationexa te demain),unautreproblèmedetailleémanedes

ap-pli ationsréelles des algorithmes onçus : elui du ara tère dynamique de l'information

à traiter. Comment modéliser une panne sur un réseau ? un nouveau lient à intégrer

dans une tournée de livraison ? une nouvelle tâ he à traiter dans un ordonnan ement ?

Doit-onobligatoirement onsidérertous es problèmes perturbés ommedes problèmes

omplètementnouveaux, etlestraiter ommetels? Nepeut-onpastirer partidufaitque

es nouveaux problèmessonttrèssimilairesauxproblèmes debase, quisontdéjàtraités?

D'autre part,sil'ondispose, parexemplesur un réseauinformatique,de laprobabilitéde

réseau qui pourra être réoptimisé très e a ement ? Enn, omment traiter une

infor-mationquiest révélée en ontinu : est-onobligéd'attendrequetoutel'informationutile

soitrévélée pour ommen erà réerune solutionauproblème posé ? Ou ne peut-onpas,

pour haque nouvelle information qui est révélée, onstruire en ontinu notre solution,

l'obje tif restant de réduire le temps de al ul, en mettant à prot le temps qui s'é oule

entre deux informationsquiapparaissent?

Le premier type de questionnement évoqué renvoit à la problématique appelée

réop-timisation. Cette problématique fut introduite par C. Ar hetti, L. Bertazzi, et M. G.

Speranzadansun arti lepubliéen 2003[3℄,oùlestroisauteursseproposèrent d'analyser

le problème suivant : onnaissant une solution optimale sur une instan e

I

donnée du problèmedu voyageurde ommer e(TSP),est-ilpossiblede générer fa ilement(i.e.,plus

fa ilement que si on ne onnaissait pas de solutionoptimale sur

I

) une solution sur une instan e

I

′

qui serait une version légèrement perturbée (don très similaire) de

I

? Les perturbationsenvisagées étaient lasuppressionoul'insertiond'un uniquesommet,et des

arêtes in identes. Les auteurs montrèrent que es deux problèmes sont NP- omplets, et

analysèrent don l'approximabilité en temps polynomial de es problèmes, e qui leur

permit d'exhiber un rapportd'approximation pour ha un d'eux.

Depuis etarti lefondateur, denombreuxproblèmesde la lasseNPontétésanalysés

sous l'angle de ette problématique, voire sous des angles légèrement diérents. On

dé-tailleral'ensemble des résultatsgénéraux etspé iques dans lasuite.

Dès que l'on ommen e à analyser les méthodes et résultats de réoptimisation, une

question s'impose d'elle-même : ompte tenu d'un algorithme de réoptimisation donné,

est-il vraiment né essaire et optimal de disposer d'une solutionoptimale sur l'instan e

dedépart? Etpouralleren oreplusloin: est- equed'autrestypesde solutiondedépart

ne seraient pas :

•

plus robustes en moyenne ou aupire des as aux perturbations?

•

plus adaptées aux stratégies de réoptimisation envisagées ?

•

plus fa ilement analysables?

Onentre alorsdansun autretypede problématique,oùl'obje tifne devientplus de

réop-timiser aumieux une fois lesperturbationssurvenues, mais d'anti iperles perturbations

enenvisageantdessolutionsdedépartquipourrontêtrefa ilementadaptéesauxnouvelles

instan es.

On parle alors d'optimisation robuste, etd'optimisation probabiliste.

Enn,ettoujoursdansunsou iderépondreàdesproblématiquesréelles: sil'information

du problème à traiter n'arrive pas en un blo unique, mais par mor eaux, ave un

er-tain temps de laten eentre haque mor eau révélé, peut-on mettre à prot e temps de

pour ommen erà al ulerune solution.

On parle i id'optimisation on-line,et d'optimisation fully dynami .

Toutes es problématiques permettent de modéliser e que nous nommons

environ-nementsdynamiques,etdénissentdesversionsdynamiquesdesproblèmesd'optimisation

lassiques(i.e.statiques),quinediérentdeleurséquivalentsstatiquesquedansl'information

disponible aumoment de la prise de dé ision.

Ainsi, lesproblèmesdynamiquespeuventêtre lassés endeux grandsensembles: eux

où le ara tère dynamique de l'informationreprésente un surplus d'information par

rap-port au problème statique, et eux où le ara tère dynamique représente une aren e

d'information.

Par exemple, la réoptimisation appartient au premier ensemble, dans la mesure où

l'obje tif est de générer une solution sur une instan e

I

′

parfaitement onnue, et l'on

disposeenplusd'unesolution(souventsupposéeoptimale)suruneinstan e

I

trèssimilaire à

I

′

. La pertinen e de ette informationsupplémentaire peut être variable, mais aupire

des as,l'informationn'estd'au uneutilité,etleproblèmedynamiqueestaussi ompliqué

à resoudre que sa version statique. Il en va de même pour l'optimisation fully dynami ,

dont laréoptimisationest un as parti ulier.

A l'inverse,l'optimisationprobabilisteappartientause ondensemble, dans lamesure

où l'obje tif est de générer une solution sur une instan e

I

′

que l'on ne onnaît que de

manière par ellaire : on sait que

I

′

onstitue une sous-instan e d'une instan e générale

I

, parfaitement onnue, et pour haque élément de

I

, on suppose onnue sa probabilité d'appartenir à

I

′

.

Evidemment toutproblèmedu premierensemblepeutêtre onsidéré ommeplus

sim-ple que saversion statique, tandis que tout problème du se ond ensemblepeut être

on-sidéré omme plus omplexes.

Dans e travail de thèse, nous nous proposons de ara tériser plus nement l'impa t

de es surplus et aren es d'informationen termesde omplexitéetd'approximabilité.

Parmitouteslesmanièrespossiblesdemodéliserle ara tèredynamiquedel'environnement,

nous avons retenu la réoptimisationet l'optimisationprobabiliste. Leur parfaite

omplé-mentarité permet d'envisager et de omprendre une grande partie des enjeux liés au

ara tère dynamiques des problèmes posés : la première suppose réglée la question de

la solution de base (on la onsidère optimale sur l'instan e de départ), pour mieux

ap-profondir la question de la stratégie à adopter pour générer une nouvelle solution ; à

l'inverse, la se onde xe la stratégie à adopter (qui fait alors partie intégrante du

prob-lème à traiter),et, ompte tenu de ette stratégie, her he lasolution de base qui sera la

mieux adaptée en moyenne aux nouvelles instan es potentielles.

probléma-tiques, qui nous a permis de mieux erner les enjeux liés à ha une d'elles, ainsi que les

te hniques mises en oeuvres. Puis nous avons nous-mêmesproduit un ertain nombre de

résultats nouveaux en analysant diérents problèmes d'optimisation en environnements

dynamiques.

Le do uments sera don organisé en inq hapitres, le hapitre 1 présente un état de l'art sur les environnements dynamiques, prin ipalement axé sur l'optimisation fully

dynami , la réoptimisation et l'optimisation probabiliste. Les deux hapitres suivants

présentent les résultats obtenus en réoptimisation : le hapitre 2 traite de l'arbre ou-vrant de poids minimum,et le hapitre3 de quatres problèmes héréditairesparti uliers : Max

k

- olorable Subgraph,Max

P

k

-free Subgraph,Max Split Subgraph et Max Planar Subgraph. Les deux derniers hapitres présentent les résultats obtenus

en optimisation probabiliste: le hapitre 4 traite de l'arbre ouvrant de poids minimum probabiliste,etle hapitre5del'ensembledominantminimumprobabiliste. Nousnirons parquelques on lusionset omparaisonsdes adresthéoriquesetdesrésultats

orrespon-dants, et enn par lesperspe tives que nous entendons explorer dans la ontinuité de e

travailde thèse.

Deuxannexesterminent edo umentdethèse,lapremièreprésenteun ertainnombre

de dénitions générales en théorie de la omplexité, et en théorie des graphes (Annexe

A),etlase onde lesdénitions formellesdes diérentsproblèmes d'optimisationévoqués dans lathèse (Annexe B).

Dynami environments: State of the

Art

Résumé

Dans e hapitre,nousproposonsunerevuedelittératuredesdiérentsmodèlesd'optimisation

permettantlapriseen omptedu ara tèredynamiquedes problèmesàtraiter. Les

mod-èles lassiquesd'optimisations'ins riventengénéraldansdes adresstatiquesoùtoutesles

données des problèmessontparfaitement onnues etnesontpas amenéesàêtre modiées

dans le temps, si bien qu'une solution peut être onsidérée omme dénitive. Beau oup

de situation réelles oùl'informationest souvent imparfaite et/ou hangeante ne peuvent

être modélisés dans de tels adres.

Pour pallier ette insusan e liée aux modèles statiques, d'autres types de modèles

ont émergé,où le ara tère dynamique de l'informationdevientun élément déterminant.

La revue de littérature se fo alise en parti ulier sur deux modèles entraux dans e

travailde thèse : d'unepart laréoptimisation(se tion1.2),etd'autrepartl'optimisation probabiliste (se tion 1.3). Ces deux se tions sont onstruites omme deux états de l'art indépendants, et la on lusion (se tion 1.4) apporte quelques pistes de réexions sur les liens possiblesentre es deux modèles,ainsi que sur leur omplémentarité.

La se tion 1.2 ommen e par un historique des modèles dynamiques, et présente en détaillepremiertypedemodèleétudiéàpartirdeannées70: l'optimisationfullydynami ,

où des instan es de problèmes, souvent polynomiaux, sont modiés de manière itérative

(sous-se tion 1.2.2). Dans e adre, l'obje tif devient de déterminer des algorithmes et stru tures de données qui permettent de maintenir une solution optimale après haque

itération,en un temps en moyenne plus faibleque lesalgorithmes statiques.

Mêmesiquelques problèmes NP- ompletsfurentanalysés dans e adre (l'obje tifest

alors de maintenir à haque étape un ertain rapport d'approximation), le modèle fully

dynami resteparti ulièrementadaptéauxproblèmespolynomiaux,etpeuauxproblèmes

NP- omplets, si bien qu'un nouveau modèle, plus restri tif, fut introduit au début des

Là où le modèle fully dynami onsidère un nombre in onnu de modi ations

su - essives, la réoptimisation onsidère une modi ation unique, lo ale, et partant d'une

instan e pour laquelle une solution optimaleest onnue. Quoique plus restri tive en

ter-mes de nombre de modi ations, laréoptimisationpermet d'introduire des modi ations

d'instan es impossibles dans le adre fully dynami : e dernier ne onsidère que des

perturbations relatives à l'ensemble d'arêtes (ave , don , un ensemble de sommets xe),

tandis quela réoptimisationpermet d'envisager tout type de modi ations (arêteset/ou

sommets).

Nousprésentonsd'aborddesrésultatsgénérauxsurla lassedesproblèmeshéréditaires

(sous-se tion 1.2.3), dont la stru ture est parti ulièrement propi e à l'approximation en réoptimisation. Puis, nous présentons un aperçu des prin ipaux résultats obtenus sur

diérents problèmes NP- omplets (sous-se tion 1.2.4), en présentant à part les résultats relatifsàdesproblèmesdetransports,parti ulièrementétudiésdans e adre(sous-se tion

1.2.5).

La se tion 1.3 est onsa rée à la présentation de l'optimisation ombinatoire proba-biliste. Pluttqu'unerevuedelittératureprésentantlesprin ipauxrésultatsobtenusdans

e adre, ette se tion tente d'apporter une ompréhension laire des appli ations et

en-jeux liésaumodèle, en illustrantlesexpli ationspar des exemplesissus de lalittérature.

L'optimisation probabiliste permet de générer des solutions réalisables très rapidement

sur des instan es

I

′

resultantsd'un pro essus probabiliste : une instan e de départ

I

est asso iée à un ve teur de réels entre 0 et 1 sur ses éléments, qui mesurent la probabilité

pour es éléments de faire partie de l'instan e

I

′

qui devra ee tivement être optimisée.

Entout état de ause, pour lesproblèmes modélisés par des graphes oùle graphede

dé-part est noté

G(V, E)

, leproblème devra don êtrerésolu rapidement sur un sous-graphe

G[V

′

_]

induit par un sous-ensemble in onnu de sommets

V

′

, et l'appartenan e de haque

sommet

v

i

ausous ensemble

V

′

est soumis à un épreuve de Bernoulli de probabilité

p

i

. Comme expliqué en sous-se tion 1.3.2, e type de modèle permet de représenter une grande variété de problèmes réels, où les probabilités jouent un rle essentiel. La

sous-se tion1.3.3présentelesdiérentesmanièresdontdetellessituationspeuventêtretraitées sur leplan algorithmique,en mettant l'a entsur le adrethéoriqueretenudans le adre

de ette thèse : l'optimisation a priori. Dans e adre, on suppose xée une ertaine

stratégied'adaptationquipermetd'adapterunesolutionréalisablesurl'instan ededépart

à n'importe quelle sous-instan e potentielle. De ette manière, une espéran e peut être

asso iée a toute solution réalisable sur l'instan e de départ (qui onsiste en la somme

des poids des solutions adaptées à haque sous instan e potentielle, pondérée par les

probabilitésdesinstan esenquestion),appelléealorssolutionapriori,etl'obje tifdevient

de déterminer une solution qui optimise ette espéran e. La sous-se tion 1.3.4 explique en détaille prin ipede l'optimisationapriori, etpropose une dénition formellepourles

problèmes d'optimisationa priori.

Enn,lasous-se tion1.3.5présentelesdiérentsenjeuxliésàl'analysed'unproblème d'optimisationprobabiliste. En parti ulier,il est intéressant de noter que l'appartenan e

de es problèmes à la lasse NPO n'est pas triviale, y ompris quand le problème

de la fon tion obje tif (i.e. de l'espéran e), onsiste en une somme pondérée sur toutes

les sous instan es potentielles, 'est à dire

2

n

termes dans le as général. En première

analyse, la fon tion obje tif n'est don pas dire tement al ulable en temps polynomial,

et l'appartenan e du problème à la lasse NPO n'est pas triviale.

Cependant, une analyse plus poussée de la stratégie d'adaptation peut onduire à

une reformulation ompa te (polynomiale) de l'espéran e. Cette reformulation est un

élément lé dans l'analyse d'un problème probabiliste, puisqu'elle onditionne la plupart

des analyses qui peuvent être menéessur le problème en question, que e soit en termes

de omplexité, d'approximation,ou de ara térisationdes solutions optimales.

La on lusionmeten perspe tivelesdeux problématiquesprésentées dansle hapitre,

en proposant des pistes de réexions sur la mise à prot des résultatsobtenus en

1.1 Introdu tion

It is ommonly known that many optimization problems are intra table, sothat no

e- ient (i.e. polynomial)optimalalgorithmsare known for them. Although ithas notbeen

learly proved that su h algorithms do not exist, it is ommonly believed and a epted

that itis the ase, and the whole omplexity theory relies today onthis assumption.

The mainapproa h tota klethese NP-hardproblems onsists offormulating

approx-imation algorithms whi h run in polynomial time, and whose quality (measured as the

so- alled approximation ratio, i.e. the ratio between the approximate and the optimal

solutions), issomehowgaranteed. This approa h has been a tively appliedand analyzed

forthe pasttwenty years, leadingtothe developmentofmany approximationalgorithms,

aswell as awhole lassi ationof problems basedon their approximability.

At the same time, some new paradigms were investigated, mainlyfor prati tal

appli- ations, where problems are often neither stati nor perfe tly known in advan e. These

un ertainordynami aspe ts ofthe problems thatneed tobesolved an be modeledand

dealt with in various ways. This hapter fo uses on two dierent and omplementary

kinds of dynami models : Se tion 1.2 is fo used on reoptimization (and its generaliza-tion : dynami optimization), while Se tion 1.3 deals with probabilisti ombinatorial optimization.

1.2 Dynami optimization and reoptimization

1.2.1 Introdu tion

We present in this se tion an overview of the results a hieved in the reoptimization

paradigm, as well as main te hniques used in reoptimization algorithms. In the early

1980'snew omputationalframeworkswereappliedtosome polynomialproblemssu has

MinSpanningTree[70℄orShortest Path[67, 139℄,where thegoal wastomaintain

an optimal solution under some more or less slight instan e perturbations su h as edge

insertions, deletions or weight modi ations. These innovative approa hes were soon

fol-lowed by many works improving on their results, mainlyfo used on the data stru tures

allowing omplexityimprovementsinmaintainingoptimalsolutions. Mostoftheseresults

were dealing with polynomial problems, and aimed at maintaining optimal solutions in

fullydynami situations. More pre isely, the paradigmwasa littledierentfrom whatis

known today as reoptimization, onsideringthat the instan es ould besubje t tomany

onse utive or simultaneous perturbations, and the data stru tures aimed at omputing

new optima on perturbed instan es as fast as possible, in parti uler, faster than an ex

nihilo omputation. Rather than reoptimization algorithms, the approa h aimed at

dening whatwas alled fully dynami algorithms. Infa t,the results inthiseld

on- erneddatastru turesratherthan algorithms. Later on,some NP-hardproblemssu has

Bin Pa kingwere alsoanalyzed ina fulldynami setting,where the goalistomaintain

the bound onthe approximation ratio.

It is onlyin the early 2000's that the on ept of reoptimization as it is known today

emerges[3℄. Thisnewparadigm onsidersatwo-steps(initialandperturbed)environment

espe iallywhentheproblemtosolve annotbe ompletelyknowninadvan e,andthetime

window between the moment when the full problem is revealed and the moment where

the solution is needed is so short that an optimalstati  omputation is impossible. In

this ase,whi ho ursveryofteninpra ti alappli ations,itseemsreasonnabletodevote

a tremendous time to ompute an optimal solution on the part of the instan e whi h is

knownforsure,andthenqui klyadaptthissolutiontothefullinstan eon eitisrevealed.

Reoptimizationalgorithms onsist of these qui k adaptation methods.

In this new framework, many new problems and types of perturbations an be

ana-lyzedandsolved. Inparti ular,theappli ationofthereoptimizationparadigmtoNP-hard

problems hasbeen intensivelystudiedinthe pastfewyears. Thesestudiesinvolve

vertex-setperturbationalso,whereas beforetheintrodu tionofthisparadigm,onlyedge-set

per-turbation wherehandled. In many ases, maintainingan optimal solutionin polynomial

time proves tobe impossible unless P=NP. Indeed,a polynomial optimalreoptimization

strategy A ould end up solving any instan e

I

in polynomial time, by starting with a trivialinstan e (for example anemptygraph), and buildingin rementallythe instan e

I

with a polynomial number of steps. Applying the algorithm A at ea h step, one omes

up with an optimal solution on

I

omputed in polynomial, whi h is impossible for NP-hard problems... On the other hand, itis straightforward toarm that areoptimization

problem shouldbe at least aseasy as itsstati version, sin e one an always use a stati

algorithm to solve the reoptimization problem from s rat h, as if no informationon the

initialoptimum were available.

Finally, it is quite lear that, regarding the lassi al P vs. NP di hotomy, a given

reoptimization problem belongs to the same omplexity lass as its stati version. But

this  omplexity heredity between a stati optimization problem and its reoptimization

version goes no further. In parti ular, one annot extend this result to approximability

omplexity lassesinNP(forexampleaproblemhardtoapproximate initsstati version

an be APX in the reoptimization setting), so that the resear h in this eld has been

mainly-and su essfully- fo used onthe denition of approximation algorithms. Indeed,

reoptimizing NP-hard problems an lead to two kindsof advan es:

•

either the reoptimizationalgorithm a hieves the same solution quality (optimality or approximation ratio) as the best stati  algorithm known, but with a better

runningtime,

•

orthereoptimizationalgorithma hievesinpolynomialtimeabetterapproximation than the best approximation ratio known forthe stati version.

In pra ti e, most results a hieved in the reoptimizationsetting onsist of approximation

algorithmwhi hgarantee betterapproximationratiosthan the stati algorithms,oreven

approximationratios that annot be garanteed inthe stati ase unless P=NP.

Themostspe ta ularexampleisprobablythe lassi alMaxIndependentSet

prob-lem, for whi h it has been proved that noalgorithm an a hieve an approximation ratio

better than

n

ǫ−1

(i.e. asymptoti ally returning a trivial solution onsisting of 1 vertex)

unlessP=NP.Inthe reoptimizationsetting( onsideringnodeinsertionsordeletions),the

Thissettinghasalsobeensu essfullyappliedtovarious lassi alNP-hardproblemssu h

as several S heduling problems [14, 15, 141℄, Steiner Tree [34, 42, 43, 65℄ or the

Travel-ing Salesman Problem [3, 8, 40, 45℄. In this se tion we will dis uss some general issues

and properties on erning reoptimizationof NP-hard optimization problems and we will

review some of the most interesting appli ations.

The se tion is organized as follows: subse tion 1.2.2 is devoted to des ribing fully dynami algorithmsand orresponding data stru tures, major results are presented, for

bothPandNP-hardproblems. Subse tion1.2.3dealswithgeneraldenitionsand proper-ties of reoptimization. In subse tion 1.2.4,the omputationale ien y of reoptimization applied to various NP-hard problems is dis ussed, and the latest results of this ongoing

resear h eld are presented, whereas subse tion 1.2.5 is spe i ally devoted to vehi le-routingproblems.

1.2.2 Fully dynami algorithms

As explainedin subse tion 1.2.1, the rst problems studied ina dynami setting- where instan es of optimization problems are onsidered to be modied lo ally, and solutions

are adapted given these input modi ations -, were rather easy (i.e. polynomial)

prob-lems, forwhi hthis additionalfeature an be onsideredas anopportunitytoredu e the

omptutional ost. As stated before, a stati algorithm an always be used to solve a

reoptimizationproblem,sothat the reoptimizationversion of a given optimization

prob-lem is always at least as easy as its stati version. In parti ular, if a stati problem

Π

ispolynomial, itsreoptimization(andfully dynami )version

R(Π)

isalsopolynomial,so that the only kindof result one an expe t is a omputational ost redu tion.

Later on, the on epts of fully- and semidynami graphs and algorithms were also

appliedtoNP-hard problems, wherethe goalis tomaintaina bounded ompetitiveratio

(the ratio between the optimal solution, and the solution omputed iteratively, along

withea h graphupdate)regardless of the natureornumberof graph updates that might

o ur. Inthis ase,the dynami featureofgraphsis learly ageneralizationoftheon-line

model, and is onsidered as a hallenge rather than as an opportunity, in the sense that

designingdynami algorithmswhi h anmaintaingoodsolutionsunderanykindofgraph

perturbations isquite ahard task, somehow harderthan omputing goodsolutionsfrom

s rat h.

Indeed,the maindieren ebetween polynomial,andNP-hardproblemsregarding the

fullydynami setting,isthat,forthepolynomial ase,optimalityismaintainedafterea h

update, and the optimality ata given stage helps omputing an optimalsolution for the

next stage in a more e ient way than omputing this solution from s rat h. In the

opposite,when one has tohandle aNP-hard probleminafully dynami setting,one an

onlyget anapproximatesolutionatalltime,and itishardtoensurethatthe ompetitive

ratio remainsbounded fromone stage to the other.

Toour knowledge, onlythreeNP-hard problems were analyzed infullydynami

environ-ment: Bin Pa king[96℄,Min Vertex Cover[95℄, and Shortest Path in Planar

Denitions

Analgorithmforagivengraphproblemissaidtobedynami ifit anmaintainasolution

fortheproblemasthegraphundergoesmodi ations. Regardingthepolynomialproblems

analyzed inthis setting,the modi ations onsidered are insertions ordeletionsof edges,

or hanges intheweightof someedges (ifrelevant). Atea hstep,a moreorless omplex

data stru ture is updated, and this data stru ture an be used at any time as input to

ompute a solution in a more e ient way than onsidering the raw instan e as input.

The data stru ture is like a pre-pro essed version of the instan e, whi h one an use to

ompute optimalsolutionsfaster. Of ourse,thepro ess isofinterestonlywhenthedata

stru ture iseasier toupdate (i.e. requiresa smallernumberof operationstobe updated)

than the optimum itself.

Update and query pro edures

In fully dynami settings, an update operation denotes a hange in the data stru ture

in response to an in remental hange to the input, and a query is a request for some

information about the urrent solution, using the underlying data stru ture. These two

notions are fundamental in the analysis of dynami algorithms, sin e the omplexity

improvement relies ompletelyonthem.

Indeed, dynami algorithms onsist of maintaining a spe i data stru ture at ea h

step, rather than omputing an optimal solution at ea h step. Whenever one wishes to

have an optimal solutions, one an run the query pro edure. Very often, a tradeo has

to befound between a stru ture easilyupdated, and one easilyqueried. This is pre isely

why omplexitiesof both update and query operations are always analyzed.

Consider, for example, that one wants to maintain a minimum spanning tree in a

dynami graphusingthe lassi alKruskal'salgorithm[114℄. To ompute itfroms rat h,

one needs to:

•

sort all edges inin reasing weight order,whi h takes

O(m log(m))

operations,

•

then, starting from anempty set, insert greedily minimum weight edges in the so-lution, provided they do not reate loops with the urrent solution, whi h, using

Tarjan's method,takesanother

O((m + n)α(m, n))

operations,where

α

isthe fun -tional inverse of A kermann's fun tion.

Now, onsider the same problem in a fully dynami setting, where the set of nodes

is xed, but where edges an be inserted, deleted, or modied at ea h step. To solve

this problem in a more e ient way than re omputing an optimum ex nihilo after ea h

graph modi ation, it su es to maintain the sorted list from one stage to the other.

Unlike building up the sorted list from s rat h (whi h takes

O(m log(m))

operations), maintaining it at ea h stage takes only

O(log(m))

operations for ea h edge insertion, deletion or modi ation: when an edge is deleted from the list, the list remains sorted;

and when an edge is inserted or (resp. modied), one only needs to insert it in (resp.

Wheneveronewantsasolution,itonlytakes

O((m+n)α(m, n))

operationsto ompute itout ofthesortedlist(whi hisstri tlybetterthan

O(m log(m))

,thestati  omplexity of Kruskal's algorithm).

Usingthisverysimpleexample,one understandshowmaintaininganunderlyingdata

stru ture through very e ient update operations (in this ase, maintaining the sorted

list),enablesto omputesolutionsatany stagewithaquery operation,witha omplexity

better than a stati algorithm.

Amortized omplexity analysis

We now formally dene the problems onsidered. We are onsidering a fully dynami

graph

G

over a xed vertex set

V

,

|V | = n

.

m

generally denotes the urrent number of edges, whi h is often assumed to be

0

or

n(n

− 1)/2

(soan independent set, or a lique) atthe beginningof the algorithm.

Mostofthetimeboundspresentedareamortized,meaningthattheyareaveragedover

all operations performed. This ompexity measure is parti ulary adapted to analyzing

onlineorfully-dynami algorithms, sin eit enablesto at h somefeatures that the worst

ase analysis isunableto apprehend. Namely,ina dynami setting, the algorithmmight

be designed to avoid en ountering the worst ase frequently, so that anaverage measure

on all operations allows to dilute the o asional worst ase omplexity in all other

-easilyupdated- ases. Infa t,theanalysisamountstosaving andspending time,justlike

money ona bank a ount. The amortized omplexityis given by the minimalnumberof

operationsone needsatea hstage, operationswhi hwillbeeitherused orsaved. Saving

time whenthe simple ases o ursenables toredu e the number of operationsneeded to

handle hard ases, sin e some operations of the hard ases willbe somehow paid with

allthe operations saved previously.

To givea leareridea of howthese so- alled amortized omplexity analyses are built,

we will present a lassi al example: dynami ally resizable arrays. The pro ess onsists

of, starting from an array of size 1, to add

n

elementsin this array, by doubling its size wheneveritbe omesfull. Theworst ase omplexityo urswhenthearraysizeisdoubled

be ause it takes

O(n)

operations to do this, and onsidering that

n

elements are added, the worst ase omplexity for the overall pro ess is

O(n

2

₎

.

Here onsider that one saves two operations when the simple ase o urs, namely

addinganelementwhenthe arrayisnotfull. Basi ally,this asetakesonlyoneoperation,

plus two operations that are saved for later, so in all: three operations. Also onsider

that one spends all saved operations when the omplex ase o urs, namely adding an

elementwhenthearrayisfull. Basi allythis ase takes

m + 1

operations onsideringthat

m

is thesize of thefull array that hastobedoubled,plus the twosaved operations,so in all:

m + 3

operations. Whenever the omplex ase o urs,

m

operationshavebeen saved (

m/2

new elements have been added sin e the last resize), if one spends them all, one onlyneeds

3

additionnaloperations tosolvethe hard ase.

Thus,theamortized omplexityis

3

,so

O(1)

,farbetterthantheworst ase omplexity

O(n)

. Finally,the amortizedanalysis enables ustoassert that the overall omplexity for inserting

n

elementsin adynami array is

O(n)

and not

O(n

2

₎

A restri tion of the fully dynami setting : online optimization

It would be hard not to mention online optimization while surveying dierent models

of dynami environements in ombinatorialoptimization. However, onsidering the huge

ow of literature ta kling optimization problems in this framework, we will only say a

few words about it, and refer the interested reader to full surveys, su h as [2℄, or [143℄,

fo used onS hedulingproblems.

Onlinemodelsandalgorithmshavemu hin ommonwithfullydynami ones. Indeed,

in the online framework, it is supposed that an instan e

I

of a problem

Π

is revealed gradually,andanonlinealgorithmOAisasked tomaintainafeasible solutionatalltimes.

A naturalrequirementforOA isthat iteventuallyprodu es asolutionas lose to

OPT(I)

aspossible. Tothispoint,anonlineproblemisthesameasafullydynami one,res ri ted

to ases whereno data is allowed tobedeleted.

But the onlinemodelisa tually mu hmore restri tive. The main di ultyregarding

online problems, is that at ea h step, one must de ide if the new data will be added in

the solution ornot. Ifitis not, then itis irrevo ablylost, and annotbetaken ba k ata

later stage of the online pro ess. Thus, it is generallyimpossible to garantee optimality,

and most studies fo us on ompetitiveanalysis.

Competitiveanalysis is amethoddesigned espe iallyfor analyzingonline algorithms,

in whi h the performan e of an online algorithm is ompared to the performan e of an

optimal oine algorithm. An algorithmis said to be ompetitiveif its ompetitiveratio

- the ratio between the value of the solution it eventually returns and the value of an

optimum returned by anoinealgorithm -is bounded.

In ompetitive analysis, one imagines that the instan e is built and revealed by an

adversary that deliberately hooses di ult data, to maximize the ompetitive ratio.

Classi aly,the adversary willtry toprevent the onlinealgorithmfromin ludingelements

of the optimum, by designing aspe i instan e and data-sequen e.

Online models help representing and optimizing numerous pra ti al problems, and

quite naturally, many NP-hard problems were ta kled in this setting. A parti ular lass

of problems whi h online versions were intensively dis ussed are S heduling problems

[1, 62, 132℄, but other problems su h as Min Coloring [57, 82℄, Max Independent

Set [81℄, TSP [38, 102℄, or Min Vertex Cover [60℄ were also dis ussed in online

settings, the listis not exhaustive.

Polynomial problems in dynami graphs

Sin e the rst paper by Frederi kson [70℄ presenting a data stru ture known as topology

trees handlingthefullydynami versionsofConne tivityandof MaximumSpanning

Forest problem in

O(m)

operations per update, many improvements have been made on this result[64, 89, 92℄, and many otherpolynomialproblems were ta kled inthe fully

dynami setting,su hasShortest Path[61,75,109℄,Min Spanning Tree[92,145℄,

2-Edge Conne tivity [64, 71, 90, 92℄,and Bi onne tivity [87, 88,91, 92℄.

Instead of detailingall these results, we willfo us ontwo major problems,

Conne tiv-ity and Shortest Path.

Conne tivity problem Given a graph

G(V, E)

the Conne tivity problem on-sists of de iding whether the graph is onne ted or not. Considering the impli ations it

hasonallrelatedproblems(whileupdatingdatastru turesofvariousfullydynami

prob-lems, one oftenhas toknowwhether the stru tureis onne ted), it isnot surprising that

the Conne tivity problem re eived a spe i attention in the fully dynami setting.

Manysu essiveimprovementswereprovided sin ethe rstresultsbyFrederi ksoninthe

early80's. We willsket h out the te hnique used by Holm etal. [92℄, providingthe best

omplexity known for it,whi h alsoenabledthe authorsto improve omputational osts

for three other onne tivity-related problems.

Theorem 1. ([92℄) Given a graph

G

with

m

edges and

n

verti es, there exists a deter-ministi fully dynami algorithm that answers onne tivity queries in

O(log n/ log log n)

time worst ase, and uses

O(log

2

n)

amortized timeper insert or delete.

Leavingaside theworst ase timefor onne tivity queries,whi hrequiresavery

te h-ni alproof,wewillfo usonthe

O(log

2

_n)

amortizedtime,enhan ingtheprevious

n

1

3

log n

byHenzingerandKing[89℄. Theideaisthefollowing: usingFrederi kson'ste hnique[70℄,

oneonlyhas to(re) omputeaspanningforestatea hstep,whi hwilldrasti allyde rease

the omputational ost of onne tivity queries. Keeping a spanning forest updated on a

dynami graph an be ompli ated only when an edge removal disonne ts a tree of the

spanning forest:

•

Whenanedge

(v, w)

isinserted,either

v

and

w

are onne tedinthe urrentspanning forest

F

(one an he k this with low omputational ost) and

(v, w)

is not added tothe spanningforest, or

v

and

w

are not onne tedin

F

,and one must add

(v, w)

to

F

, tokeep it maximal.

•

When an edge

(v, w)

is deleted, it might dis onne t the graph only if

(v, w)

∈ F

. In this ase, one has to repla eit with another edge, if any. In terms of worst ase

omplexity,this might be very ostly, sin e one has to he k everypossibleedge to

be sure that there isno su h re onne ting edge.

But re all that the measure we are interested inis amortized time. Holm et al. provide

a method to de rease it down to

O(log

2

_n)

. Basi ally, the te hnique used to de rease

the omplexity is to assign a level

l(e) 6 l

max

=

⌊log

2

n

⌋

to ea h edge

e

of the graph. For ea h

i

,

F

i

denotes the subforest of

F

indu ed by edges of level at least

i

. Thus,

F

_{⊆ F}

0

⊆ F

1

⊆ ... ⊆ F

l

max

.

At the beginningof the algorithm,every edge is assignedthe level

0

, and so is every new inserted edge. Edge levels in rease by one whenever they belong in a tree that is

disonne tedduetoanedgedeletion,orea htimetheyarebeing andidateasre onne ting

edges. Withoutgettingtoomu hintodetailregardinghowthealgorithmmaintainsthese

propertiestrue atalltime,the mainpropertiesthat boundtheamortized ostperupdate

• F

isamaximal(withrespe tto

l

)spanningforest of

G

, thatis,if

(v, w)

isanontree edge,

v

and

w

are onne ted in

F

l(v,w)

.

•

The maximal number of verti es in atree in

F

i

is

⌊n/2

i

_⌋

.

The rst property bounds the number of edges tobe onsidered as andidates for

re on-ne ting whenever anedge

e

∈ F

isdeleted, while the se ondensures that

l

max

=

⌊log

2

n

⌋

as stated earlier.

The authors derive also results for Maximum Spanning Forest (result derived

dire tly), Min Spanning Tree (derived from an appli ation of the method, but in

a graph that is onne ted from the beginning, instead of an empty graph, and taking

weights ina ount), 2-Edge Conne tivity and Bi onne tivity.

All Pairs ShortestPathProblem Ina onne tedgraph

G(V, E)

theAllPairs ShortestPathproblem onsistsofndingshortestspathsbetweenallpairofnodesof

G

, thus

n(n

−1)/2

shortestpaths. Thisproblemwasoneoftheveryrsttobeanalyzedinthe dynami setting[115,125,138℄. Although the literature dealingwith this topi has been

rather ri h,it isonly inthe early90's that the rst provably e ient dynami algorithm

was designed. Ausiello et al. [9℄ proposed a de rease-only shortest path algorithm for

dire ted graphs havingpositive integer weights less than

C

: the amortized runningtime oftheir algorithmis

O(Cn log n)

peredgeinsertion(orweightde rease,whi hgeneralizes insertion).

Asanexample,wesket htheideasofthebestfullydynami algorithmfortheproblem

to our knowledge [61℄. The authors enhan e the previous best known results for both

in rease-only and fully dynami ases.

Theorem 2. ([61℄)Inanin rease-onlysequen eof

Ω(m/n)

operations,ifshortest paths are unique and edge weights are non-negative, there exists a fully dynami algorithmthat

answers ea h update operation in

O(n

2

_{log n)}

amortized time, andea hdistan e and path

query in optimal time. The spa e used is

O(mn)

.

The main idea enabling this omplexity is the use of a spe i notion, alled lo ally

Shortest Path, whi h apllies to all paths

Π

xy

, for whi h all proper subpaths are Short-est paths. Let

SP

and

LSP

be the sets of shortest paths, and lo ally shortest paths respe tively. On an dire tly arm that

SP

⊆ LSP

, and it is easy to verify if a path

Π

xy

∈ LSP

or not. A given path islo allyshortest if an onlyif:

•

it isa single edge, or

•

both paths fromthe rst vertex tothe lastbut one vertex, and from the se ond to the last are shortest paths.

The authors prove that the update of both sets

SP

and

LSP

an be done in

O(n

2

_{log n)}

amortizedtime, provingthat nomorethan

n

2

paths anstopbeinglo allyshortestpaths

ompute the modied set

SP

after the update in a e ient way. Moreover, it is proved thatthere annotexistmorethan

mn

lo allyshortest pathsinagiven graph,fromwhi h one derives the spa e bound immediately.

The method and underlying data stru tures are then adapted to the fully dynami

setting(handling both in reases and de reases):

Theorem 3. ([61℄) Inafullydynami sequen eof

Ω(m/n)

operations,ifshortest paths are unique andedge weights are non-negative,there exists a fully dynami algorithm that

answersea hupdateoperationin

O(n

2

_log

3

_n)

amortizedtime,and ea hdistan eandpath

query in optimal time. The spa e used is

O(mn log n)

.

The te hnique used is basi ally the same, though the notion of lo ally shortest path is

extended to the notion of lo ally histori al path, i.e., paths whose proper subpaths are

histori al,meaning that they have been shortest paths at least on e during the pro ess.

Of ourse,the number of lo ally histori alpaths being larger than the numberof lo ally

shortest paths, the amortized time per update goesup to

O(n

2

_log

3

n)

.

These two examples show how proper data stru tures an improve amortized time

om-plexity when dealing with polynomial problems in a fully dynami setting. A whole

dierent pi ture appears when applying this setting toNP-hard problems.

NP-hard problems in dynami graphs

Re all that, in the fully dynami setting, the pro ess always starts with trivial instan es

(for example, graph problems would start with independant sets, or liques), whi h are

modied lo ally atea h step, and the goalis to maintain a given data stru ture atea h

step. This data stru ture helps omputing good solutions in a more e ient way than

when dealingwith a raw instan e.

In the previoussubse tion ,we have presented the kindof datastru tures and results

thatwereproposedintheliteraturewhenapplyingthefullydynami settingtopolynomial

problems. Inthat ase, the data stru tures helped omputing optimalsolutionin amore

e ient way than omputingnew optimafrom s rat h after ea h lo almodi ation. On

the opposite, when applying this setting to NP-hard problems, the data stru tures will

help maintainingapproximate solutions.

Moreover, whenapplying this settingto NP-hardproblems, one annotexpe t better

approximation ratios than inthe stati ase. Indeed, any instan e of a given problem

Π

anbebuiltoutofatrivialinstan emodiedlo allyapolynomialnumberoftimes. Thus,

agiven datastru ture maintaininganapproximationratio

ρ

afterea hlo almodi ation an also be seen as a stati algorithm providing

ρ

-approximate solutions: starting from a trivial instan e, modifying this instan e a polynomial number of times, and updating

the data stru ture after ea h modi ation, one an get a

ρ

-approximate solution on any instan e in polynomialtime.

Thus, in what follows, we will present dynami algorithms whi h aim maintaing

ap-proximate solutions, more e iently than stati approximation algorithms, but not

Only three NP-hard problems were proposed approximation algorithms in this

set-ting: Bin Pa king [96℄, Min Vertex Cover [95℄, and Shortest Path in Planar

Graphs [110℄ whi h are allratherwellapproximatedproblems. In the following, wewill

provide some detailsfor the two rst problems.

Bin Pa king Problem One of the most elementary and easy to approximate

prob-lem, the Bin Pa king problem onsists, given a list of items

L = a

1

, a

2

...a

n

of size

s(a

i

)

∈ (0, 1]

,of nding the minimum

k

so that allthe items t into

k

unit-sizebins. Thisproblemisknowntobehardtoapproximatewithin

3/2

−ǫ

. Ifsu happroximation existed, one ouldpartition

n

non-negativenumbers intotwosets ofminimalsize in poly-nomial time, and this problem is also known to be NP-hard. However, one understands

that this inapproximability o urs only in instan es with rather small optima, so that

approximation algorithmproviding better and better asymptoti al approximation ratios

were proposed for this problem. The problem was shown to be asymptoti allyin PTAS

[106℄, and the best pra ti al algorithm provides an asymptoti al approximation ratio of

71/60

in

O(n log n)

[105℄, generating a solution that uses no more than

71/60OPT + 1

bins, where

OPT

is the optimalnumberof bins.

ThemainproblemwhendealingwithBin Pa kinginafullydynami setting,isthat

lassi al oine methods do not resist insertion of new items, sin e these methods rely

on sorting all the items in de reasing size order, and then using the sorted list to build

the pa king. For instan e, pla ing items in the rst bin where they t provides a

11/9

asymptoti al approximationratio using asorted item list. It is obviousthat this kindof

methodis not adapted to the dynami setting.

Using an adaptation of Johnson's approximation algorithm [103, 104℄, Ivkovi¢ and

Lloyd [96℄ provide a fully dynami algorithm a hieving a ompetitive ratio almost as

goodas the best stati approximation algorithm:

Theorem 4. ([96℄) The fully dynami bin pa king algorithm MMP (Mostly Myopi

Pa king) is asymptoti ally

5/4

- ompetitive and requires

O(log n)

per Insert/Delete oper-ation.

The idea of Johnsons's algorithm is to regroup items in ve size groups: Big, Large,

Small, Tiny and Minus ule. Big ontains items having size in

]1/2, 1]

and Minus ule items having size in

]0, 1/5]

(other thresholds being logi ally

1/3

and

1/4

).

Intuitively,this stru ture is mu hmore adapted tothe dynami settingthan a sorted

list on all items. Ivkovi¢ and Lloyd's algorithm a hieves its ompetitive ratio by using

the te hnique whereby the pa king of an item is done with a total disregard for already

pa ked items of a smallersize (i.e. belongingin smaller-sizegroups).

Ea h of these myopi pa kings may then ause several smaller items to be repa ked

(in a similar fashion). Indeed, an item might be pa ked in a bin where it does not t,

due to the presen e of smaller size items in the bin, whi h need to be repa ked in other

bins. With some additional sophisti ation to avoid ertain bad ases where most of

the smaller size items need to be repa ked in as ading sequen e, the number of items

to be repa ked is bounded by a onstant, thus bounding both the ompetitive ratio and

omplexity.

Min Vertex Cover Problem Avertex overofagraph

G(V, E)

isasetof verti es su h that ea h edge of the graph is in ident to at least one of them. The problem of

ndingaminimumvertex overisparadigmati in omputer s ien eandis approximable

withinratio2. Indeed,Gavril[12℄proved thattakingallendpointsofamaximalmat hing

results ina 2-approximate vertex over.

This method happens to be rather dynami -friendly, but an hardly handle some

pathologi al ase in sparse graphs: while insertion of edges, and deletion of edges not

in the urrent maximal mat hing

M

an be both handled

O(1)

, deletion of edges of

M

might require to s an the whole vertex set in order to re ompute a maximal mat hing,

thus requiring

O(m)

operations, whi h is also the oine ase omplexity. However, this dynami method behavesrather well in dense graph, and will be referred to asClean in

what follows.

Ivkovi¢ and Lloyd's [95℄ propose to maintain at all time an additional bit for ea h

vertex, indi atingifthevertexismat hedorunmat hedinthe urrentmaximalmat hing.

Thisenablestoredu etheworst ase omplexityofre omputingamaximalmat hingfrom

O(m)

to

O(1)

, while

O(m)

operations are needed to maintain the lists of mat hed and unmat hed vertex updated. The key idea of the Dirty method is to update these lists

only on e, thus redu ing the amortized omplexity and keeping a bounded number of

ina ura ies inthe lists. This algorithmbehaves wellin sparse graphs.

But a tually, in the dynami setting, the graph might turn from sparse to dense and

vi eversa,aftera ertainamountofupdates,sothatbothClean(improving omplexityin

densegraphs)andDirty(improving omplexityinsparsegraphs)mightendup onfronted

totheirrespe tivepathologi al ases. TheoverallalgorithmofIvkovi¢andLloyd,denoted

B1, tests frequently the density of the graph (whi h is evolving along with ea h update),

and de ides to use either Clean or Dirty, depending on whether the urrent graph is

dense orsparse, inorder to ensurean overall enha ed omplexity:

Theorem 5. ([95℄) B1 is a 2- ompetitive fully dynami approximation algorithm for

vertex over (thus, it is approximation- ompetitive with the standard o-line algorithm).

Further, both Insert and Delete operations require

O((n + m)

3/4

amortized running time.

It is lear that maintaining a bounded ompetitive ratio in fully dynami setting

is quite a hard task when dealing with NP-hard problems. This di ulty arises from

needingto ompute anapproximatesolutionbasedonanotherapproximatesolution,and

thisseems tobeonlypossibleinveryspe i ases,and withwellapproximableNP-hard

problems. To ope with this resistan e to dynami approximation of some NP-hard

problems, and also to answer to numerous pra ti al appli ations, a restri tion of the

fully-dynami setting was introdu ed: reoptimization.

1.2.3 Reoptimization: General properties

As mentioned earlier, if one wishes to get an approximate solution on the perturbed

modied instan e is always possible. In other words, reoptimizing is at least as easy as

approximating,sin eatworst,theinformationonthe initialoptimum anremainunused,

whi hamountsto the stati problem. Thus, the goalof reoptimizationis todetermine if

it is possible tofruitfully use our knowledge onthe initialinstan e inorder to:

•

either a hieve better approximation ratios,

•

or devise faster algorithms,

•

or both.

In this subse tion, we present some general denitions, and results dealing with

re-optimization properties of some NP-hard problems, many of whi h have already been

presented in [6℄. We rst give some general approximation results on the lass of

hered-itary problems, under node insertion (results regarding the deletion ase an be found

in hapter 3). Then, we dis uss the dieren es between weighted and unweighted ver-sions of lassi al problems, and nally present some ways to a hieve hardness results in

reoptimization,using the example of Min Coloring.

Before presenting properties and results regarding reoptimization problems, we will

rst giveformaldenitions of reoptimizationproblems, instan es, and approximate

reop-timizationalgorithms:

Denition 1. Anoptimizationproblem

Π

isgivenbyaquadruple

(

I

Π

, Sol

Π

, m

Π

, goal(Π))

where:

• I

Π

is the set of instan es of

Π

,

•

given

I

∈ I

Π

,

Sol

Π

(I)

is the set of feasible solutionsof

I

,

•

given

I

∈ I

Π

, and

S

∈ Sol

Π

(I)

,

m

Π

(I, S)

denotes the value of the solution

S

of the instan e I,

m

Π

is alledthe obje tivefun tion,

• goal(Π) ∈ {min, max}

.

Denition 2. A reoptimizationproblem

RΠ

isgiven by a pair

(Π, R

RΠ

)

where:

• Π

is anoptimization problemasdened inDenition 1,

• R

RΠ

is a rule of modi ation on instan es of

Π

, su h asaddition, deletionor alter-ation of a given amount of data. Given

I

∈ I

Π

and

R

RΠ

,

modif

RΠ

(I, R

RΠ

)

denotes the set of instan es resulting from applying modi ation

R

RΠ

to

I

. Noti e that

modif

RΠ

(I, R

RΠ

)

⊂ I

Π

.

Foragiven reoptimizationproblem

RΠ(Π, R

RΠ

)

,areoptimizationinstan e

I

RΠ

of

RΠ

is given by a triplet

(I, S, I

′

₎

, where :

• I

denotes aninstan e of

Π

, refered to asthe initial instan e,

• I

′

denotes an instan e of

Π

in

modif

RΠ

(I, R

RΠ

)

.

I

′

is refered to as the perturbed

instan e.

Fora given instan e

I

RΠ

(I, S, I

′

₎

of

RΠ

, the set of feasible solutions is

Sol

Π

(I

′

₎

.

Denition 3. Foragivenoptimizationproblem

RΠ(Π, R

RΠ

)

,areoptimizationalgorithm A issaid tobe a

ρ

-approximation reoptimizationalgorithmfor

RΠ

if and only if:

•

A returnsa feasible solutionon allinstan es

I

RΠ

(I, S, I

′

₎

,

•

Areturnsa

ρ

-approximatesolutiononallreoptimizationinstan es

I

RΠ

(I, S, I

′

₎

where

S

is anoptimal solutionfor

I

.

Note that denition 3 is the most lassi al denition found in the literature, as well as the one used inthis Chapter. However, analternate (and more restri tive) denition

exists (used for example in [34, 42, 43℄), where a

ρ

1

-approximation reoptimization algo-rithmfor

RΠ

issupposed toensurea

ρ

1

ρ

2

approximationonany reoptimizationinstan e

I

RΠ

(I, S, I

′

)

where

S

is a

ρ

2

approximate solution inthe initialinstan e

I

.

Reoptimizing Hereditary problems under vertex insertion

A property

P

on a graph is hereditary if the following holds: if the graph satises

P

, then

P

is also satised by all its indu ed subgraphs. Following this denition, stability, planarity, bipartiteness are three examples of hereditary properties. On the opposite

hand, onne tivity is no hereditary property sin e there might exist some subsets of

G

whose removal dis onne t the graph. Now, let us dene problems based on hereditary

properties.

Denition 4. Let

G = (V, E, w)

be a vertex-weighted graph. We allHered the lass of problems onsisting,given agraph

G = (V, E)

, ofndingasubset ofverti es

S

su hthat

G[S]

satisies agiven hereditary property and that maximizes

w(S) =

P

v∈S

w(v)

.

Forinstan e, Max Weighted Independent Set, Max Weighted Indu ed

Bi-partiteSubgraph,Max WeightedIndu edPlanarSubgrapharethree lassi al

problems inHered that orrespond tothe three hereditary properties given above.

In whatfollows,

G

willdenotethe initialgraph,

OPT

aknown optimalsolutionon

G

,

G

′

themodiedinstan e(resultingfromalo almodi ationon

G

),and

OPT

′

anoptimal

solution on

G

′

. Under vertex-insertion (where

G

is a subgraph of

G

′

), there are three

powerful propertiesthat one an use when reoptimizing problems inHered :

i- the initialoptimum

OPT

remainsa feasible solutionin the modiedgraph

G

′

; this

derivesdire tly from the denition of anhereditary property.

ii- the part of the new optimum

OPT

′

indu ed by verti es of the initialgraph annot

ex eedthe initialoptimum;otherwise, thispart wouldbeabettersolutionthan the

iii- a single node always veries a hereditary property; this derives also dire tly from

the hara terization of hereditary properties interms of forbiddenminors.

Considering these threeproperties,one an formulate ageneralalgorithmto

approxi-mateanyweightedprobleminHeredwithinratio1/2. LetR1denotethegeneri algorithm

whi h onsists of returning the best solution between the initial optimum

OPT

(noted

S

1

) and the single inserted node

x

(noted

S

2

):

Proposition 1.1. ([6℄) Under vertex insertion, R1 approximates any problem

Π

in Hered within ratio 1/2 in onstant time.

Theresultisquitestraightforwardwhen onsideringthethreepropertiesstatedabove:

while properties (i) and (iii) assert that R1 returns a feasible solution, property (ii) an

be reformulatedwith the following bound:

w(S

1

) > w(OPT

′

)

− w(S

2

)

(1.1)

from whi h one derives dire tly the approximation ratio.

Re all that

S

1

onsistsofasinglenode,sothatone shouldbeableto ompleteitwith some other verti es of the graph. In parti ular, one ould run an approximation

algo-rithmonthe remaininginstan e aftertaking

x

. Consider forinstan e Max Weighted Independent Set, and revisitthe proof of the previous Proposition. If

OPT

′

doesnot

ontain

x

, then the initialsolution

OPT

is optimal. If, on the opposite,

OPT

′

ontains

x

, then onsider the remaininggraphafter havingremoved

x

and its neighbors. Suppose thatoneusesanapproximationalgorithmwhi hgaranteesa

ρ

-approximationratioonthe remaininggraph, and that one adds

x

to this approximate solution. Denote this generi algorithmR2. Then the so-obtained solution

S

′

2

veries:

w(S

₂

′

) > ρ(w(OPT

′

)

_{− w(x)) + w(x) = ρw(OPT}

′

) + (1

_{− ρ)w(x)}

(1.2)

On the other hand,it stillholds:

w(S

1

) > w(OPT

′

)

− w(x)

(1.3)

Denotingby

S

thebestsolutionamong

S

1

and

S

′

2

,adding(1.2) and(1.3)with oe ients

1

and

(1

− ρ)

, one gets:

w(S) >

1

2

_{− ρ}

w(OPT

′

₎

(1.4)

whi his better than

ρ

.

The problem istodene what the remaininginstan e after taking

x

 is. This notion is strongly related to the nature of the hereditary property. To be more pre ise, in

order to run the method that we just des ribed, one must be able to dene a subset of

remaininginstan e after taking

x

is agraph in whi h nonode an violate the hereditary property whenput together with

x

.

If su h a set is easy to dene when dealing - for instan e - with independent sets, it

be omes rather vaguous when dealing with more omplex hereditary properties, su h as

planarity or bipartiteness. The question an nd elements of answer when onsidering

hereditary properties in terms of forbidden minors. These questions are dis ussed in

hapter 3 where spe i approximation algorithms, as well as inapproximability bounds are proposed for various hereditary problems.

Even if not dire tly appli able to problems in Hered (most of them have been proved

to be inapproximable unless P=NP [116℄), the te hnique is rather general, and an nd

appli ations inthe reoptimizationof many problems. We illustrate itin two well-known

problems: Max Weighted Sat (Theorem 6) and Min Vertex Cover (Theorem 7). Givena onjun tionofweighted lausesoverasetofbinaryvariables,MaxWeighted

Sat Problemasks forthe truthassignementof variables maximizingthe sum ofweights

of satised lauses.

Theorem 6. ([6℄)Under theinsertionof a lause,reoptimizing Max WeightedSatis

approximable within ratio 0.81.

Consider a onjun tion of lauses

ϕ

over a set of binary variables, ea h lause being given withaweight,andlet

τ

∗

beaninitialoptimumsolution. Considerthatanew lause

c = (l

1

∨ l

2

∨ . . . ∨ l

k

)

, (where

l

i

is a literal of variable

x

i

, i.e., either

x

i

or

x

¯

i

) has weight

w(c)

. The modied formulaisthus given by

ϕ

c

= ϕ

∪ c

.

k

dierent solutions

τ

i

,

i = 1, . . . k

are omputed. Ea h

τ

i

is built as follows: - set

l

i

to true;

- deletefrom

ϕ

all satised lauses.

- apply a

ρ

-approximation algorithmonthe remaininginstan e (note thatthe lause is already satised); together with

l

i

,this isa parti ular solution

τ

i

.

Finally,return the best solution amongall

τ

i

's and the initialoptimum

τ

∗

.

As previously, if the optimal solution

τ

∗

c

on the modied instan e does not satisfy

c

, then

τ

∗

remainsoptimal forthe new problem. Otherwise, atleast one literal in

c

,say

l

i

, istrue in

τ

∗

c

. Considering that

l

i

is true in

τ

i

, itis easy tosee that:

w(τ

i

) > ρ(w(τ

c

∗

)

− w(c)) + w(c) = ρw(τ

c

∗

) + (1

− ρ)w(c)

(1.5) On e more, asin the generalte hnique des ribed above:

w(τ

∗

) > w(τ

_c

∗

)

− w(c)

(1.6) So, equation(1.4)holdsfor Max WeightedSatalso. Takingintoa ountthat this problemis approximable within ratio

ρ = 0.77

[5℄,the result laimed is on luded.

Letus nowfo us onaminimizationproblem,namely Min Vertex Cover. Given a

vertex-weighted graph

G = (V, E, w)

, the goal is to nd a subset

S

∈ V

su h that every edge

e

∈ E

isin identtoatleastone vertex in

S

,andthat minimizes

w(S) =

P