. Ensuite, 2, nous introduisons leprobì eme Max k-Hitting Set (Max k-HS en abrégé), qui est simplement un formulation alternative de Max k-Cover. C'est par réduction depuis Max k-HS que nous montrons notre résultat principal concernant

. Mais, quel que soit le réel ? vérifiant 1 ? ? < e e?1 , leprobì eme d'´ evaluation associéassociéà Max k-Cover n'admet pas d'algorithme d'approximation de borne constante ?

. Une-formulation-alternative-de-max-k-cover, Max k-Hitting Set On nomme Maximum k-Hitting Set (Max k-HS en abrégé) leprobì eme suivant Supposons donnés un n-uplet (F 1 F n ) d'ensembles finis et un entier k ? 0. On cherche un ensemble C de cardinal k maximisant le nombre des indices j ? [1, n] tels que C ?F j = ? " . Dans cette section, nous montrons que Max k-HS est exactement aussi difficilè a approximer que Max k-Cover comme suggéré dans

P. Alors and . Np, il n'existe pas d'algorithme d'approximation de borne constante ? pour leprobì eme d'´ evaluation associéassociéà Max k-HS

. Preuve, Supposons (absurde) que Max k-HS E admette un algorithme d'approximation de borne ?. Nous allons montrer, ` a l'aide d'une réduction, que Max k-Cover E admet un algorithme d'approximation de même borne, demanì erè a pouvoir appliquer le résultat de Feige

C. and =. ?}, Correction de notre réduction Résoudre leprobì eme Max k-HS sur l'instance ((F 1, k) consistè a trouver un ensemble C de cardinal k maximisant le cardinal de J C . Pour terminer notre preuve, il suffit de montrer que les deux assertions suivantes sontéquivalentessontéquivalentes

@. Réciproquement, montrons que (ii) ? (i) Supposons (ii) : il existe une partie C ? E de

C. Onécritonécrit, C. La-forme, and . {e-c1, c k sont desélémentsdeséléments de [1, m] (non nécessairement distincts deuxàdeuxà deux) Posant C := {c 1

. Preuve and . Par-hypothèse, Construisons un algorithme d'approximation pour Max k-HS de même borne , F n ), k) de Max k-HS. Posons U := F 1 ? F 2 ? . . . ? F n . Pour chaque u ? U , on calcule l'ensemble E u des indices j ? [1, n] tels que u ? F j . On construit ensuite l'ensemble E := {E u : u ? U } puis l'instance (H, k) de Max k-Cover Cette construction prendévidemment prendévidemment un temps polynomial et on vérifie

?. .. Appliquons-maintenant-l-'algorithme-a-sur, C. Tels-que, }. .. {e-u1, and . Enfin, on obtient, en temps polynomial, une partie C ? E de cardinal au plus k telle que # ( C) soit au moinségaìmoinségaì a 1 / b(k) fois l'optimum

N. Ceprobì-eme-est-trivialement-dans, De plus, lorsqu'on se restreintàrestreintà des tuiles de cardinal 2, d- DFRT se réduit auprobì eme du couplage parfait, donc devient polynomial [31] En revanche, posons H 2 := {(0, 0) 2)} et P 23 := {H 2 , V 3 } : les tuiles H 2 et V 3 sont appelés respectivement domino horizontal et triomino vertical, La restriction de 2-DFRT aux instances (F, P) vérifiant P = P 23 est NP-difficile [9]. Ainsi, décider si 2 tuiles de cardinal au plus 3 pavent une forme finie plane est NP-difficile

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. Mesure, Le cardinal de I, ` a maximiser Page: 18. Notes: La notion d'indépendant est définie page 18. Le cardinal maximum pour un indépendant de l'hypergraphe H est noté ?(H) La restriction de MISH aux hypergraphes r-uniformes est notée r-MISH

. Mesure, Page: 29. Notes: La mesure d'une solution optimale est notée R d (W )

. Nom, Weighted Common Subsequence (-WCS) o` u est une longueur pondérée sur ? , ? désignant un alphabet

. Mesure, Le poids de g, ` a maximiser

. Mesure, Le nombre de similarités appartenantàappartenantà S et détectées par g, ` a maximiser. Page: 95. Notes: ` A ne pas confondre avec sa variante RSOSs définié egalement dans la section 4

. Mesure, Page: 108. Notes: C est l'ensemble des sommets de H couverts par C, c'est-` a-dire l'

. Nom, Maximum k-Hitting Set (Max k-HS)

. Mesure, Le nombre des indices j ? [1, n] pour lesquels C ? F j est non vide