Improved Bounds on the List Decreasing Heuristic for the Vertex Cover Problem

Transcription

1 0 Naoal our Sou Irov Bou o h L crag Hurc for h Vrx ovr Probl Ta-Pao huag of SIE Naoal Tawa Noral Uvr a hg Yu Uvr Eal: chuag@cuuw Shu-Sh L of SIE Naoal Tawa Noral Uvr Eal: l@cuuw Abrac Th l crag hurc for h vrx covr robl a ol vrx covrg algorh A ur bou /3/ ha b roo o h aroxao rao for a a grah ha alo b gv o rach a lowr bou of // I h ar w rf h chqu of rvou rarchr a coruc a w of grah whch ca hac h lowr bou o / a all h grah ca b cagorz o a grou Th w roo a of algorh o oba a ghr bou a rov wh a xal Ix Tr aroxao algorh l crag hurc vrx covr robl I INTROUTION Th u vrx covr robl h ozao robl of fg a u caral vrx covr for a gv grah L G V E b a urc uwgh grah A of vrc V call a vrx covr f for a g E a la o of o coa Th vrx covr robl a faou NP-har ozao robl urrl hr o oloal algorh o olv oall Svral aroxao algorh for h vrx covr robl hav b roo wh varou rforac guara or al [] crb a vr l aroxao algorh ba o axal achg whch gv a aroxao rao of ag al [] roo h ol vrx covrg robl Th u o rl ow a h bgg vrc ar rval o b o a a co of lco u b a for ach rval vrx Th ca vrx lc f a ol f ha a la a olc alra rval ghbor So aroxao algorh ar ba o a ac orrg of vrc r b hr gr a h vrx gr ar o ua h roc A rg ol wha Av al call h l hurc [3] Th of algorh ca h vrc o b o a fx gv orr call a l a a a fv co of lco for h currl ca vrx Av al [3] roo h l crag hurc lbo call LLf [] ha choo vrc orr of crag gr lcg a vrx f ajac o a ucovr g Th rov ha aroxao rao a o whr h axu gr of h grah Th alo how ha a lowr bou o h aroxao rao // Sc l forao avalabl a ach h l crag hurc LLf rfor wor ha h gr algorh whch ral lc a vrx ajac o h larg ubr of ucovr g O h ohr ha lbo al [] rouc a br l hurc algorh LRgh whch ra vrc crag orr of hr gr I h ar w oba a ghr ur bou 3 7 orgh 0 Naoal ha Uvr All rgh rrv

2 0 Naoal our Sou for h l crag hurc [3] Morovr w coruc a grou of grah o how ha h l crag hurc ha a lowr bou of / Th rul br ha ha of Av al [3] whch // Th ar orgaz a follow Sco brfl rvw h rolog Sco 3 roo a ghr ur bou for h l crag hurc algorh Sco coruc a grou of grah rcuro for Lal a brf cocluo a Sco 5 II PRELIMINARIES A Vrx covr robl A vrx covr a of vrc a grah Gv a urc grah G V E a of vrc V call a vrx covr f for a g E a la o of o coa I ohr wor gv G a vrx covr of G a of vrc V uch ha V V u v E u V or v V or boh For a vrx v V w o Nv h of ghbor of v a Nv v h gr of v ubr of ghbor L rr h ubr of vrc h ubr of g a h axu gr of G Au h vrc ar labl uch ha L b a vrx covr a o o h z of h u vrx covr a h /o o h aroxao rao B L hurc algorh ag al [] roo a ol vrx covrg algorh Th ca vrx lc f a ol f ha a la a olc alra rval ghbor L hurc algorh a of ol vrx covrg algorh A ruao of h vrc of V call a l Th l a b or accorg o h vrc gr Th vrc ar rval o b o fro h l Th algorh ca h l ral a a a co of lco or o for h currl ca vrx al Th vrx gr ar o ua urg h roc L crag hurc algorh wa r b Av al [3] 007 whch call LLf [] I ca vrc orr of crag gr fro lf o rgh a lc a vrx f ajac o a ucovr g lbo al [] of h l crag hurc algorh o LRgh hurc algorh Th ca h l fro rgh o lf Th ca vrx lc f a ol f a la a rgh ghbor o lc I h ar w al wh LLf a aalz bou o h aroxao rao Lar rograg A ora ool h aal of aroxao algorh a lar rograg rlaxao of h rla gr rograg robl a wa fr u b Lováz [3] Th raoal aroach how Fgur Igr Lar Prograg Igral Soluo Aroxao Rlaxao Roug Lar Prograg Solvr Fraco Soluo Fgur Th raoal aroach h gr lar rograg robl Th followg fo a lar rograg rlaxao for h vrx covr robl For a vrx covr of a grah G V E w o x v h wgh of vrx v a f v X{x v } v V a follow: x v 0 v 73 orgh 0 Naoal ha Uvr All rgh rrv

3 0 Naoal our Sou whr X a 0 or valu fabl oluo for h lar rograg a covrl vr 0 or fabl oluo corro o a vrx covr for G For a g uv E of a grah G V E w o h wgh of g a f Y{ } E a follow: /u or /v or / g o h g algorh whr Y a fraco valu fabl oluo for h lar rograg Th a ag of ogav wgh o h g of G uch ha h u of h g wgh a a vrx a o o A Y{ } E ha af h coo abov call a fracoal achg of G W f h z of X a X x a clarl X W f h z of Y a Y I h vrx covr robl w r o z h uao of x v whr v V x v v V x x v u v E x 0 v V v u Th ual o axz h uao of whr E ax E δ v v V 0 E For a grah G l o b a u vrx covr of G b a vrx covr of G a Y b a fracoal achg larl X o Y E Ug h abov fac Av al oba o bou o h aroxao rao for h LLf v V v E algorh III AN ANALYSIS FOR THE UPPER BOUN ON THE APPROXIMATION RATIO Gv a grah G V E l o h gr of vrx v a au h vrc ar labl uch ha Th axu gr a h u gr Th l crag hurc algorh [3] how blow Algorh LLf Iu : A grah G a a aoca l L<v v v > or b crag gr : ; // Iall For o - //Sca h l L fro lf o rgh { }; L v b h currl ca vrx; L R o h of g c o v bu o c o a vrx alra ; If R o h : { v }; Rur ; Th abov algorh ouu a vrx covr - b cag h vrc o b o I [3] Av al coruc a ual fabl oluo Y{ } E a follow L b h u x uch ha {v v v } a u caral of vrc Y oba b all g for ach g For ach for whch v lc b LLf h choo a arbrar g fro R a rag a wgh 7 orgh 0 Naoal ha Uvr All rgh rrv

4 Now a fracoal achg Y{ } E of G ca b oba fro I h followg w wll rf h chqu of Av al [3] a Ia Hroh [5 6] o rv h followg hor Thor L b h oluo oba b LLf a o b h z of h oal oluo h * o Proof I h lc ar oba b LLf w choo h u o v v v uch ha h gr u of h o ju grar ha or qual o W h l - Fr w focu o h fr o Th gr u of h o wll b l ha or qual o - a how blow Sco w focu o h followg o Th gr u of h o wll b l ha or qual o - a blow a W h al h auch-schwarz qual u [3] Sc ar ov gr wh u a o - h L o b h z of h oal oluo Now w coruc a ffr fracoal achg Y{ } E a follow L R o h of g c o v bu o c o a vrx alra W lc a arbrar g fro R a ag h wgh Each of h ohr g E ag h wgh Th o Ug h arhc-gorc a qual o * o o * o Th quao or rc ha h rul [5] bu o goo ough So w roo a ghr ur bou for LLf h followg ho Naoal our Sou orgh 0 Naoal ha Uvr All rgh rrv

5 0 Naoal our Sou r Thor L b h oluo oba b LLf b h ubr of vrc wh axu gr a o b h z of h oal oluo If < h o Proof I LLf w v h o o wo ar h lc ar a ulc ar Th lc ar qual o h vrx covr I h lc ar w hav a o - o a ra a la o o o b ulc W ca wr h quao a follow { Now w coruc a fracoal achg Y } Th oal wgh of h g ca b wr a follow Σ Q o Σ o 3 If a < h W u h chqu roo b Av al [3] o ruc h forula a follow: o Σ o Fro a w ca g h aroxao rao a follow: o L u how a xal Fgur No ha h oal ubr of vrc 5 h oal ubr of g 8 h axu gr 6 N h ubr of vrc wh axu gr Th xal ca b gralz o l N b a arbrar valu W hav -- o *8 *6 6 3 o 8 Now w f h lowr bou forula [3] 6 w ca ar o 3 W ca f ha h ur bou o h aroxao rao ug 5 ju qual o h lowr bou / [3] Th a w hav fou a xal o clo h ga bw h wo bou 76 orgh 0 Naoal ha Uvr All rgh rrv

6 0 Naoal our Sou L u al wh h ca ha all vrc h grah hav h axu gr grn Grou A 3 #N grn Grou B #N grn Grou grn Grou 3 5 #N # Vrx gr Wgh of hc g V V V 3 V /6 A B /6 /6 /6 V 5 V 6 V 7 V 8 V 9 V 0 V V V 3 V V 5 V 6 V 7 V 8 V 9 V 0 V V V 3 V V 5 V 6 5 6**6*0*8 6 *6* *6 * /5 /5 /5 /5 /5 /5 /5 /5 /5 /5 /5 /5 /5 /5 /5 / Fgur A xal ug N /5 /5 /5 /5 orollar or a grah GV E wh a L b h oluo oba b LLf al o G a o b h z of h oal oluo h Proof o If a h ach g E ca b ag h wgh / a o Fro w ca g h aroxao rao a follow: o 6 Exal L u how h rul of col grah Fgur 3 I col grah f w u axu gr o xr h aroxao rao h w wll g bggr rao wh bggr So w ca g a ghr bou o / 77 orgh 0 Naoal ha Uvr All rgh rrv

7 our forula ha ha of Av al [3] Th xac aroxao rao h col grah alwa o -/- Fgur 3 Th rul of col grah ug 3 o 00 L u al wh h ca ha h grah ha vrc wh u gr or 0 orollar or a grah GV E wh < a - - or 0 hr ar vrc wh u gr or 0 L b h oluo oba b LLf al o G a o b h z of h oal oluo h o q Proof I LLf w v h o o wo ar h lc ar a ulc ar Th lc ar qual o h vrx covr If hr ar vrc wh u gr or 0 h all h vrc wll b u o h ulc ar So w hav 7 Th wgh of h g ca b ag larl o ho h roof of Thor W coruc a fracoal achg Y { } Th Σ o Σ Q o If - - or 0 a < h 78 0 Naoal our Sou orgh 0 Naoal ha Uvr All rgh rrv

8 0 Naoal our Sou W u h chqu roo b Av al [3] o ruc h forula a follow: o Σ o Fro 8 a 9 w ca g h aroxao rao a follow: o Exal 9 L u u h bar grah a a xal Fgur No ha h oal ubr of vrc 7 h oal ubr of g 8 h axu gr 3 h ubr of vrc wh axu gr 8 h ubr of vrc wh u gr 6 I h bar grah 7 6 Now w ag h valu / /3 a h wgh of h hc g Fgur Th ohr g ar alo ag a wgh of / /3 a : Ug Equao 9 w hav - o /6569 o a : Ug h ur bou [3] o a 3 : Ug h ur bou w roo o W ca f 866 < 75 W hav a ghr bou /ha ho of Av al [3] Fgur A xal ug axu gr 3 79 orgh 0 Naoal ha Uvr All rgh rrv

9 0 Naoal our Sou IV A NEW GROUP OF GRAPHS IN REUR- SIVE FORM TO MATH THE LOWER BOUN Rucg h ga bw h lowr bou a ur bou o of h a challg facg h rarchr I h co w roo a w grou of grah whch ca hac h lowr bou o h aroxao rao for LLf roo b Av al [3] Th grah ca b rr a a rcuro rucur how Fgur 5 Th axu gr of h grah N o N h h lowr bou o h aroxao rao wll b N N N N I a ha h aroxao rao of h lowr bou alo co u o / Av al [3] oba a ur bou /3/ whch ca b rr or r- c for a If w l N h aroxao rao of our grah ca b wr a N N N N N N N N N a l / N N Th ga bw h wo bou N N N / N 05 I a h ga bw N N N N h ur a lowr bou co u o 05 wh f Exal I Fgur 6 h axu gr of h grah N o N h lowr bou o h aroxao rao wll b I a ha h aroxao rao of h lowr bou co u o / N h ur bou [3] wll b N Th N grn Grou A A A #N grn Grou A A N- A N- #N N - grn Grou B B B B 5 B 6 grn #N Grou B #N B N- B N- BN-5 BN-6 N - grn grn Grou # Grou N # grn Grou # N Grou gr Toal No # * S A N N N * N - B N N N * N - N N * N N * N N N N N on N N Rao N / Fgur 5 Th rucur of h xal grou 80 orgh 0 Naoal ha Uvr All rgh rrv

10 0 Naoal our Sou grn Grou A A A A 3 A #N grn #N Grou B B B B 3 B B 5 B 6 B 7 B 8 B 9 B 0 B B B 3 B B 5 B 6 grn Grou grn grn grn # Grou # Grou 3 # Grou # N grn Grou # Grou gr Toal No # * S A N N N * B N N N * N N * N N * NN N N N o N N N Rao N/ Fgur 6 A xal of grah wh axu gr N IV ONLUSION Afr carful corao a rvao w oba h quao Thor Th rul br ha ha of Ia Hroh [5 6] Furhror w hav roo a w grou of grah whch ca co u h lowr bou o / Th rul alo br ha ha of Av al [3] Th ga bw h lowr bou a ur bou co u o 05 wh f W hav roo a ghr bou forula of h l crag hurc for h vrx covr robl W alo gv a xal o how h rul of h ur bou qual o / Th bou of LLf rv b o rarchr a u ar how Fgur 7 I [3] h auhor rov ha a ur bou o h aroxao rao of LLf /3/ a ca b rr or r- c for a Th ffrc bw h wo bou a h xac ur bou / Ia Hroh [5] u h la 8 roo b Av al [3] o oba h followg forula: o * 0 Furhror Ia Hroh [6] go h followg quao: q o * whr q Ia Hroh [6] ca ha wh h axu gr allr ha 9 h abov ur bou br ha ha of Av Fall 8 orgh 0 Naoal ha Uvr All rgh rrv

11 0 Naoal our Sou w uarz our a rvou rul Tabl o * 3 o * 9 o * o * 9 o * o * o * < 9 o * o * < 9 o * o * o * o * Fgur 7 Th bou of LLf 8 orgh 0 Naoal ha Uvr All rgh rrv

12 0 Naoal our Sou Tabl oaro of h rul wh rvou wor Rarchr Ma a Th rul of h bou quao Av al [3] o o * Ia Hroh [5] o o * Ia Hroh [6] o q o * q whr q Our rul Thor o o * Our rul Thor o Σ o * AKNOWLEGEMENT Th rarch wa uor ar b a gra NS99--E MY3 fro Naoal Scc oucl Tawa RO REFERENES [] Thoa H or harl E Lro Roal L Rv a lffor S Irouco o algorh Nw Yor: MIT Pr 00 [] Marc ag Vagl Th Pacho O-l vrx-covrg Thorcal our Scc [3] av Av a Tooazu Iaura A l hurc for vrx covr Orao Rarch Lr [] Fraco lbo a hra Lafor A br l hurc for vrx covr Iforao Procg Lr [5] Ia Hroh 飯田浩志 頂点被覆へのリスト減少法の解析に関する一考察 cuo ar r o 小樽商科大学ビジネス創造センタ 007 [6] Ia Hroh 飯田浩志 頂点被覆に適用されたリスト減少法の解析についての再考 cuo ar r o 小樽商科大学ビジネス創造センタ orgh 0 Naoal ha Uvr All rgh rrv