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Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem

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(1)Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem Omid Amini, Ignasi Sau Valls, Saket Saurabh. To cite this version: Omid Amini, Ignasi Sau Valls, Saket Saurabh. Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem. [Research Report] RR-6237, INRIA. 2007, pp.20. �inria-00157970v4�. HAL Id: inria-00157970 https://hal.inria.fr/inria-00157970v4 Submitted on 3 Jul 2007. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Parameterized Complexity of the Smallest Degree Constraint Subgraph Problem Omid Amini — Ignasi Sau — Saket Saurabh. N° 6237 June 2007. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6237--FR+ENG. Thème COM.

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(45) ÖAÌ,Ô WÖ ÌµÔ¡Õ Œ Ö wTt† Ô ‹ Ö ¤ÓqTz^5qTz‡_†k_*lxw u¡¬¡j ­ I Z»zxw G)p “Tz‡qiqT^5u¡t]“Ou¡t5qT„kl‡u¡pMw ¬kt]_l} qrzBkj

(46) u¡j1l‡[]_ „]z‡qip_Mj‡jq ¤¿„wTpBiu¡t]“djn_r_zxw ¬ijm^wT¬¢¬il‡z‡wGp Hq› ;j u¢tŽl‡q¬¡wTzn“r_z$st]uªlxj Ì  []uvpx[ŸpwTt§¨!_–ln[]_MtŸ„]z‡qkp*_Mj‡jn_M†£wTjjmu¡t]“T¬¡__tŽl‡uªl‡u¢_oj ­¯Ë qTz$_Ò]w ^5„]¬¡_ Ì u¢tŸw t]_*l}  qT2z B¦sjmu¡t]“¨Lq l‡[§lnu¡^5_ × †k#u Žuvjnu¢qrtŸw t†§  w T_M¬¢_Mt]“ l‡[ × †ku iuvjmu¡qTt¦^s¬ªl‡u¢„]¬¡_*Òku¡t]“ Ì Hq› ;j []uvpx[ w z‡_d†]_Mjmlnu¡t]_M†–lnq5wpqT^5^5qTt7t]qk†k_dpMw t7¨!_’„]sklqrt)ln[_’j‡w ^5_d  w r_¬¡_t]“Tln[ w ¬¡¬¡q› u¢t]“ln[]_M^Ælnq ¨!_`†kz‡qT„]„L_M†¨ihw–jmu¡t]“T¬¡_qT„]lnuvpw ¬wT†]† × †kz‡qT„^s¬ªl‡u¢„]¬¡_*Òk_Mz ­ Z []_$qr¨k|}_Mp*ln#u T_oÌ jq ¤H“Tz‡qiqT^5u¡t]“w z‡_ lnq`u¢^5„]z‡q T_€¨w t!†k u¡†kln[sklnu¡¬¢u¡©Mw lnu¡qTtw t†)lnq`^`u¡t]u¡^5u¢©M_€l‡[]_d_ $Žs]u¡„]^5_tŽl;p*qrjml q ¤µl‡[]_€t_*l}  qT2z B ­ ž t ® æ c±qT„]lnuvpw ¬

(47) t]_*l}  qT2z Bkj Ì ln[]_^`qŽj}l’wTpMp*_„]ln_M†Upznu¢ln_Mznu¡qTtUuvjl‡q–^5u¡t]u¢^5u¡©_ln[]_tŽs^¨L_zOqT¤ _¬¡_Mp*lnz‡qTt]uvp$l‡_z‡^5u¢tw lnu¡qTtj Ì twT^5_¬¡h8ln[]_5tis]^¨L_zqT¤ fb &O‚åZ~ æ c¯j ­ fi__ Ô  Ö ¤Óqrz’w7“T_Mt]_zxw ¬ jmszr_hÉqTt “rznqiqr^`u¡t]“ ž l8[wTj7¨!_M_t zn_op*_tŽl‡¬¢hф]z‡q T_M†Ñl‡[w›l ! 12 ? 4 1Cu¡t qT„kl‡u¡pMw ¬ t]_*l}  qT2z Bkj pw t7¨!_z‡_M†]s­ p*_o† ñ^`qk†ks¬¢q5„LqT¬¡hŽ¬¡qT“Žw z‡uªl‡[]^5u¡p;¤ñwTp*lnqTzxj !{l‡q)Å k ¸ Ì qrz _ $Žs]#u WwT¬¢_MtŽln¬¡hlnq °Ñ¸¿°?Å jn__ ž t¤ñwrpl u¢t “Tzxw „][¯ln[]_MqTz‡_*l‡u¡pMw ¬

(48) ln_z‡^j l‡[]_5„]z‡qT¨]¬¡_^/pMw t¯¨L_`lnzxw tjn¬vw›ln_o† lnq„wTzml‡uªl‡u¢Ìqrt]u¡t]“Ô¡Õ*qTÖ ¤»­ l‡[]__M†k“r_MÌ j qT¤w`“T#u T_Mt–z‡_ $Žs]_Mjml“Tzxw „][7u¢ÌtŽl‡q5jns]¨]“rz‡wT„][j j‡w›lnuvjm¤ÓhŽu¡t]“p*_Mzmlxw u¡t ¬¢qŽwT† pqTtjmlnzxw u¡tŽl‡j w t† ln[]_¦qT¨]|}_Mpl‡u r_uvj–l‡q?^5u¡t]u¢^5u¡©_ ln[]_ lnqTl‡wT¬€tis]^¨!_Mz7qT¤ T_Mzml‡u¡p_Mj)qT¤ ln[]_ jms]¨“Tzxw „][j`qT̤€l‡[]_„wTzml‡uªl‡u¢qrt ­  _Mtp*_ Ì u¡tÑln[]uvj–pqTtŽln_Òil Ì „w znlnu¢lnu¡qTt]u¡t]“§w£“Tu#T_MtÑjn_*l)qT¤ _M†k“r_Mj []u¡¬¢_d^5u¢t]u¡^5u¢©Mu¢t“l‡[]_Ol‡q lxw ¬¿tis]^¨L_z q ¤ r_znlnuvp*_oj{uvj [_z‡_ Ì ln[]_Å k ¸7qTz_ $Žs]#u ›w ¬¡_tŽln¬¡h °Ñ¸¿°?Å „]z‡qT¨]¬¡_^.pqT^5_Mj u¡tŽlnq„]¬vwWhkj ­ yHw zxw ^5_ln_z‡u¡©_M†¯aqr^5„]¬¢_Òkuªl}h8uvj€w–zn_op*_tŽl’w „]„]z‡qrwrpx[lnq8†k_MwT¬

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(52) u¢tŽl‡z‡wrpl‡wT¨]u¡¬¢u¢l}hUp*¬vwTj‡j &Oy uvj ®ˆÔ¢ÕÖA× Ì  [u¡px[Ñu¡j5w£jmlnz‡qTt]“£w twT¬¢qr“Ts]_ Ì ¨!_opwTsjm_Uw¯¤Ós]t†]wT^5_tŽl‡wT¬„]znqr¨]¬¡_^±p*qr^5„]¬¢_ln_¤ÓqTz ®ˆÔ¡Õ*Ö uvjln[]_ kÏ ¸LڿǁÀ ¾

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(55) ³ »ÇkÚ,Ç]½,¶€²´³µ²ÓÎÚ¿² ·µ½,²´³

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(57) aqi9q B4 j Z [_qTz‡_^3u¢t§p*¬vwTj‡jnu¡pMw ¬Hp*qr^`„¬¢_Òiu¢l}h ­ ~ÆpqTt T_Mt]u¡_tŽl$jmqrs]z‡p_qT¤ ®ˆÔ¡Õ*Ö¡× [w zx†kt]_ojnjdz‡_M†ksp*lnu¡qTtj’uvj„]z‡q × iuv†k_M†¨ihln[_ z‡_Mjns]¬¢l1ln[w l َ²ÓÜ

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(77) ½ َ²ÓÜ

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(79) ¨wTjn_M†’qrt’ln[]_^5_*ln®ˆ [qi†]qT¬¡qT“Th Bit]q› t$wTj 06  !   2 1  -, -4*

(80)  4 ­ Z []_°?·µÙÚ,² ÏÛ Â

(81) َÂ

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(83) Ùi»½ Û ÙŽ²ÓÜ»·»Ç C t]„skl  w–“rz‡wT„][ G lnqT“r_*l‡[]_zd uªl‡[¦w7„]z‡qT„L_z k× pqT¬¡qTz‡u¢t“)q ¤ln[]_ T_Mzml‡u¡p_Mj q ¤ Z []_„znqr¨]¬¢_M^ uvj„w zxw ^5_ln_z‡u¡©_M†–¨ih s_Mjmlnu¡qTt  ž j l‡[]_z‡_$w k× p*¬¡u#$Žs]_’u¡t G p*qrtjmuvjmlnu¡t]“5q G¤1­ _*Ò]wTp*ln¬¡hqTt]_ r_znln_*Ò)qT¤_owTpx[p*qT¬¡qTz k ­ qT¤µ°?·µÙÚ,² .Ï - »ÙiÂ

(84) ½ Û Ùr²ÓÜ

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(86) َÂ

(87) ½ َ²ÓÜ

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(89) ß ™ ß  œ ß­¡wTÕ t†07 šk˜o­ œ

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(91) ß ™ ß  œ +ß  š],ä  Vß ˜ [c^] w Br_Mj’jns]zn_ln[!w›l’ln[]_qrskl × V“Tqr[cu¢t]“7­ w zxpjOV¤Óz‡[cqT^ ] p*qTz‡z‡_Mjn„!qrt†ku¡t]“’lnqln[]_’jn_¬¡_Mpl‡_M†7w zxp“rwr†k“T_l‡j [w r_€wp*qT^5^5qTt T_znln_Òu¢t Z [w›l uvjwT¬¢¬ V [c ] Ì l‡[]_w zxpj p*qrznz‡_Mjn„LqTt†ku¡t]“lnqln[]_ojm_jn_¬¡_Mpl‡_M†w zxpO“ŽwT†kÌ “r_*lx-j $./01!243-56#% .781/390.:V<[c;!$]#=­ 2*">? V [c ] @ ËwTz‡u¡ptw ¬¡¬¡h Ì 7 ¤Óz‡škqT˜M^ œ

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(93) ¤Óqr¬¢V¬¡q›[c ;j ]Ì  _ Vjm[][cq› ?] []q› £lnq’wT†k[_z‡_ (v, u) ­ l‡q$l‡[]u¡j“T_Mt]_zxw ¬Lj}l‡z‡w ln_M“Th5w t†qr¨kl‡wTu¢t7wzn_o†ksp*lnu¡qTt¤Óz‡qT^.°Ÿ·»ÙÚ,² .Ï - »Ùi»½ Û ÙŽ²ÓÜ»·µÇ5lnq5°É¸¿°ŸÅ ¤ÓqTz Z»qjmu¡^5„]¬¢u¢¤ÓhUl‡[]_„zn_ojm_Mtrlxw›l‡u¢qrt Ì  _¥z‡jml†k_Mj‡p*z‡u¢¨L_qrs]z’zn_o†ksp*lnu¡qTtu¡t †k_l‡wTu¢¬vj€¤ÓqTzdln[]_5pwrjm_ d ≥ 3­. Z []_z‡_ $Žs]u¡z‡_M†)^5qk†ku¢¥!pw lnu¡qTtj ¤Óqrz ln[]_$pMwTjn_ d ≥ 4 u¡jln[]_Mt_Òk„]¬¡wTu¢t_M†7u¢tfis]¨!jm_oplnu¡qTt‹ ­ ‹ ­ d = 3­ . 1. k. i. i. i. j. f. b. f. b. i. j. i. i. i. j. j. ex. i. j. i. i. i. i. i. j. f. j. i. b. d. HJIFK. LAMONQPSRUTWVYX-Z/[(T]\ ^:_`T a2X-ZbMOZ:NdceMf\ N:Z:R [][+ghR []^:i2TkjXS\+T l:RmVYMO[0R. ¤ÓqTz ® _U[w r_ k(k − ƒ1)_M^`p*¬¡_Ms^jmln¨L_M_z‡z€j5lnqT[¤dw “Žl’wTwT†kt¯“r_*w l‡zxj p ­ b€t]_Up*¬¡sjmln_zu¡j€_owT¤Ópxqr[%zn  ¤Ów qTzxz5†l‡uª[]¤ _¯jn_*lq ¤w tw zx†¯pj u¡j€E[c , c ]Ì  ¨ r w  p Ž B.   w x z  † uª¤ 1 ≤ i 6= j ≤ k ­ a ∈ E[c , c ] i < jÌ i > j­ n

(94) 㠙 ⠚]™Wä µß 8 ß œT˜ ñã+ š]ä, ß ˜r¹ž ¤ i < j Ë qTz{_owTpx[w zxp e = (u, v) ∈ E[c , c ]   _[w T_;wphip¬¢_ qT¤¬¡_t]“Tln[ 2 + 2(k − 2) + 3  uªl‡[T_znln_ Ò7jn_*l  C d=3. i. i. j. j. f. i. j. ef. &"  '.

(95) .  

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(99) l‡[]_ `ã

(100)

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(120) ¤Óznqr^ _owTpx[$qT¤]ln[_“ŽSwT†k“r|S|_*l‡j =S k ­ Ì  _*l E ¨L_ål‡[]_ jm_lqT¤_M†k“r_Mj

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(123)

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(128) l‡[]_“Tzxw „[ e ∈ _$E[c w Ž “ w ¡ u 7 t M p w ¬¡¬¿ln[]uvj“rwr†k“T_l S ­ C ­H®  u¢Ò w t ã`

(129)

(130) ß ™ ß  œ +ß  š],ä  ß ˜  㠙 &Oq› ¬¡_*l$p*qrtjnu¡†k_Mzw ¬¡¬1ln[]_jn_¬¡_Mpl‡u¢qrt “Ž wT†k“r_*lxjqT¤»l‡[]_’¤ÓqTz‡^ S Ì p d∈≥{1,4· · · Ë , k} w t!† i Ì p16=≤i ­ i ≤ k ­ Ë qTzUw tih uqTz5∈_rV_[cz‡hŸ] Ìw zxwTp t† w tihÉl}  q?u¡t†kuvp*_oj p 6= quªl‡[ ≤ k Ì p, q 6=   i Ì _7  „_u¡pB§wr†]l‡†[]_Uw%“Tt]zxw _M„ Ð[ _M†k“r_ {u , u } ­ÑË †k_„L_t!†ku¢t“UqTt¦[]eq›  =e(u,u¡j’v)¤Óqrzn  ∈w E[c zx†qr,z’c¨]wTÌ p Bi  w zx† uÌ w ∈t†£V wr[c†]† ] Ì d − 1 _o†k“T_oj’q ¤ ln[]_5¤ÓqrCzn^ Ì x ∈ {f, b} Ì fku¢^5u¡¬¡wTzn¬¡h¤Óqrzw t§_o†k“T_ {e  u¢ln[ u, u∈ },V{e[c ] Ì  , _£u „]},uvp BC. . l‡. [], {e_§“Tzxw „][ C, u Ì }x­ ∈ {f, b} Ì w t!† wr†]† d −e 1=_M(u, †k“r_Mw)jq ∈¤`lnE[c [_ ¤Ó,qrczn^ ] Ì {e , u }, . . . , {e , u }­  qrz;ln[_$l}  q)w zxpj 7 šk˜Mœ

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(132)

(133) ßi㠙 ß  °É¸¿°ŸÅ , ®ˆÔ 1Ö  &5#-  d ≥ 3  ž ¤ _dzn_M„]¬vwTp*_Oln[]_’z‡_ $Žs]u¡zn_M^5_tŽlqT¤»¥t†]u¢t]“5w t7u¢t!†ksp*_o†7jns]¨]“rz‡wT„][)qT¤

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(135) †]_“Tz‡__ w l嬡_MwTjml d u¢t7°É¸¿°ŸÅ  uªl‡[5ln[]_€zn"_ $rsu¢z‡_^5_tŽl{qT¤L¥!t†ku¡t]“w d× z‡_“rs]¬vw z u¡t†ks!p*_M†jns]¨]“rz‡wTk„][ Óz‡_Mjn„ ­ jms¨]“Tzxw „][ ! qT¤ qT¤¿jnu¢©M_;w›l{^5qrjml  _“r_*l °  ¸  Óz‡_Mjn„ ° ¸  !1„]z‡qT¨]¬¡_^ &;qTlnuvp*_ l‡[w›l€ln[]_`^5u¢tu¢H^s]^/Gjms¨]“Tzxw „][UqT¤å†k_“rznk_MÌ _w›l’¬¡_MwTjml dÏ du¡tUÏ  ln[]_`„]z‡qiq ¤ñjO­ qT¤åÏ Z d[]Ï  _qrzn_M^j Õ w t†(­ ‹lns]z‡tj qrskl;l‡q–¨L_w tUu¡t†ks!p*_M†¯jms]¨“Tzxw „][8q ¤Hz‡_“rs]¬¡wTzO†k_M“Tz‡__ d u¢t G ­ ~Ojdw)p*qrtjm"_ $Žs]_tp_ _qT¨klxw u¡tUln[]_ ¤Óqr¬¢¬¡q› u¡t]“`pqTz‡qT¬¡¬¡wTznh  à  °  Ï dÏ  ¸  4° Ï dÏ  ¸   ®ˆÔ 1Ö  &5J#-  d ≥ 3 . 㠙 ã 8 8ñšk™   *0$1x

(136) &8 + #›  18&7 +

(137) 

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(139)  

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