Mininum Vertex Cover in Generalized Random Graphs with Power Law Degree Distribution

Transcription

1 Mnnum Vrtx Covr n Gnralzd Random Graphs wth Powr Law Dgr Dstrbuton André L Vgnatt a, Murlo V G da Slva b a DINF Fdral Unvrsty of Paraná Curtba, Brazl b DAINF Fdral Unvrsty of Tchnology - Paraná Curtba, Brazl Abstract In ths papr w study th approxmablty of th mnmum vrtx covr problm n powr law graphs In partcular, w nvstgat th bhavor of a standard 2-approxmaton algorthm togthr wth a smpl pr-procssng stp whn th nput s a random sampl from a gnralzd random graph modl wth powr law dgr dstrbuton Mor prcsly, f th xpctd dgr of a vrtx s c β, for, 2,, V, whr β > 2 and c s a normalzng constant, th xpctd approxmaton rato s + L β ρβ, whr s th Rmann Zta functon of β, Lβ s th polylogarthmc spcal functon of β and ρβ L β 2 ζβ Kywords: Chung-Lu random graph modl, Brtton random graph modl, gnralzd random graph modl, approxmaton algorthms, vrtx covr problm, powr-law graphs Introducton Th mprcal study of larg ral world ntworks n th lat 990 s and arly 2000 s [, 2, 3, 4, 5, 6, 7, 8, 9] showd that th vrtx dgr dstrbuton of a varty of tchnologcal, bologcal and socal ntworks can b approxmatd by a powr law,, th numbr of nods of a gvn dgr s proportonal to β, whr β > 0 Thr s som vdnc that som optmzaton problms mght b asr n ntworks wth such dgr dstrbuton [9, 0,, 2, 3] than n gnral ntworks Mor rcntly, som analytcal rsults nvstgatng optmzaton problms on modl of powr law graphs appard n [3, 4, 5, 6, 7] Th problm that w dal n ths papr fts n ths contxt W nvstgat th approxmablty of th mnmum vrtx covr problm n th gnralzd random graph modl wth powr law dgr dstrbuton Random graphs wth arbtrary dgr dstrbutons hav bn studd bfor [8, 9, 20, 2, 22], but n th last 5 yars modls for random graphs wth a powr law dgr dstrbuton, rfrd hr as powr law graphs, hav rcvd a spcal ntrst [23, 24, 25, 26, 27, 28] du to th avalablty of mprcal data Ths modls can b roughly dvdd nto thr groups: gnralzd random graph modl, confguraton modl and prfrntal attachmnt modl Th gnralzd random graph modl, proposd by Brtton t al [27] n 2006, s a gnralzaton of th Erdös-Rény modl [29, 30] whr dgs ar addd ndpndntly, but modratd by vrtx wghts that ar usd to gnrat arbtrary dstrbuton of vrtx dgrs, ncludng th powr law dstrbuton not also that th cas whr vry vrtx hav th sam wght, ths modl s quvalnt to th G n,p modl Th wll known Chung-Lu modl [34] also fts n ths catgory Th rsults n ths papr ar provd n th gnralzd random graph modl, n partcular du to th advantag of dg probablts bng ndpndnt Th confguraton modl uss a dffrnt approach whr dg connctons ar probablstc, but th lst of vrtx dgrs and thrfor thn numbr of dgs s Emal addrsss: vgnatt@nfufprbr André L Vgnatt, murlo@danfctutfprbr Murlo V G da Slva Prprnt submttd to Thortcal Computr Scnc July 27, 205

2 fxd Ths schm orgnatd n th lat 970 s [8, 9], but currntly th most wll know such modl n th spcfc contxt of powr law graphs s th ACLα, β modl, proposd by Allo t al [23] A thrd wll know approach s th prfrntal attachmnt modl Th da s to dscrb th growth of a graph drvn by a random procss n whch nw vrtcs connct to th currnt graph wth probablty proportonal to th dgr of th vrtcs alrady n plac Ths modl was frst dscrbd by Barabas and Albrt [2] and thn mor formally by Bollobás t al [24] In [3], Hofstad gvs a dtald and formal tratmnt of ths thr gnral catgors for random powr law graphs A vrtx covr n a graph G V, E s a st of vrtcs S V such that vry dg E has at last on ndpont n S Fndng a mnmum vrtx covr s a wll known N P-hard problm [32] Th problm s also conjcturd not to admt approxmaton algorthms wth constant factor smallr than 2, unlss P N P [33] Rcntly, Gast and Hauptmann [4] usd ACLα, β modl to show that thr s an approxmaton algorthm wth xpctd factor of approxmaton strctly smallr than 2 for random powr law graphs s Fgur a for a plot of such approxmaton rato for dffrnt valus of β In ths papr w show a smlar rsult, but n th gnralzd random graph modl proposd by Brtton t al Furthrmor, w also show that th sam rsults hold for th Chung-Lu random graph modl W obtand a bttr xpctd approxmaton factor usng a smplr algorthm, vn though ts worth mntonng that ths approxmaton factors cannot b drctly compard, snc th undrlyng random graph modls ar not th sam Consdr th followng algorthm for vrtx covr on an nput graph G: Stp : Insrt n th vrtx covr C vry vrtx that s adjacnt to a vrtx of dgr ; Stp 2: Rmov from G vry vrtx that s thr of dgr or adjacnt to a vrtx of dgr ; Stp 3: Run any 2-approxmaton algorthm g, [35], chaptr 35 on th rmanng graph and lt C b th output of ths algorthm; Stp 4: Output C C as a covr for G W show that th xpctd approxmaton rato of ths algorthm s + L β ρβ, whr β > 2 s a paramtr rlatd to th xponnt of th powr law, ζ s th Rmann Zta functon, L s th polylogarthmc spcal functon and ρβ L β 2 ζβ Ths bound can b bttr undrstood n Fgur b, whr w plot ths functon for 2 < β 4, as wll as th xpctd approxmaton rato of th algorthm of Gast and Hauptman for th sak of comparson a Expctd approxmaton rato of th algorthm of Gast and Hauptman [4] for powr law graphs n th ACL α, β modl Th plottd curv s th approxmaton rato of 2 2 β 2 β ζβ for valus of β from 2 to 4 b Th blu sold ln rfrs to th xpctd approxmaton rato of th algorthm of Gast and Hauptmann [4] and th rd dashd ln rfrs to th xpctd approxmaton rato of algorthm dscrbd n th currnt papr for valus of β from 2 to 4 Fgur : Expctd approxmaton rato of algorthms for powr law graphs 2

3 2 Gnralzd random graph modl In ths scton w dscrb gnralzd random graphs GRG, ntroducd by Brtton [27] For a dtald ntroducton on ths modl and a fw asymptotc quvalnt varants w rfr to [3] In ths modl w start wth a vrtx st V {, 2,, V } and ach vrtx V s assocatd to a wght w Lt w b a vctor wth ntrs w,, w V In th dg st E, vry dg s cratd ndpndntly wth probablty Pr E wwj l n+w w j, whr l n k V w k Naturally, th dgr dstrbuton of th random graph dpnds on th vctor w In ordr to crat a powr law random graph wth a gvn xponnt β > 2 w buld w usng prncpls smlar to Allo t al [23]: Lt α ln V, whr j s th Rmann Zta functon Lt α For ach j β j,,, lt { α f n > y j j β α othrws Now construct w so that y j of ts ntrs ar qual to j, for j,, In ths way thr ar y j vrtcs wth wght j not that, smlarly, n th ACL modl thr ar y j vrtcs of a gvn fxd dgr j From th dfnton of α, obsrv that V α As dscussd n [23], w can gnor roundng and dal wth α and α β β as ral numbrs Furthrmor, not that n th ACLα, β modl n th dfnton of y som xtra car has to b takn, snc th sum vrtx dgrs has to b vn In our modl, no such rstrcton s ncssary snc y rfrs to th vctor of wghts Throughout th papr a graph G V, E s a random sampl n th GRG modl Lmma 2 Lt, j {,, α β } Thn α ζβ + α ζβ Proof W show that Th lmt abov s th sam as But Snc β > 2, th lmt gos to zro α ζβ + lm α α ζβ lm + α α ζβ lm α α ζβ 0 lm α α 0 lm α α lm α α β α β α lm α α 2 β From th dfnton of Pr E and Lmma 2, w hav Pr α ζβ + α ζβ W not that n th Chung-Lu [34] random graph modl, th probablty of an dg s dfnd to b Pr E wwj w l n nstad of w j l n+, as w hav n our dfnton Equaton shows that w w j l n wwj l n+ and thrfor all rsults n ths papr also hold n th Chung-Lu modl as wll Convrsly, snc th dg probablty n Chung-Lu modl s dfnd n way that th xpctd dgr of a vrtx s w, w also hav ths proprty n our modl 3

4 Dfnton 22 St of vrtcs wth sam wght Lt W k th st st of vrtcs wth wght k,, W k {l V w l k} Lt p b th probablty of a random vrtx of W b adjacnt to a random vrtx of W j Snc vrtcs wth a gvn wght ar ntrchangabl w hav p α ζβ In th ltratur p usually rfrs to th probablty of an dg connctng vrtx and vrtx j Th notaton hr s dffrnt snc n th ntr papr t s much clar to argu condtond on th vrtx wght nstad of th vrtx ndx W clos ths scton gvng som furthr graph thortcal notaton and dfntons Dfnton 23 Graphs thortcal dfntons Th dgr of v V s dnotd by dv W dnot V k th st of vrtcs of dgr k Th maxmum dgr of G s dnotd by,, th largst k such that V k Lt V V \ V 0 V For S V, dnot G[S] th graph nducd by S and dnot NS th st o vrtcs contanng at last on nghbor n S 3 Tchncal Lmmas In ths scton our am s to stmat th sz of sts V and NV Lmma 3 Lt q k p k Thn, k q W k k Proof k q W k k k k k α k β α ζβ k k k ζβ ζβ α /k k/ζβ β α k β k β k k β ζβ ζβ ζβ 4

5 Lmma 32 Lt v V Thn Prv W Proof Drctly from th dfnton of W V α β α ζb β ζb Lmma 33 Lt v W Thn Prv V 0 v W Proof Lt v W and lt X b th random varabl for dv Lt X j b th random varabl countng th numbr of nghbors of v n W j By lnarty of xpctaton, X j X j Hnc, Prv V 0 v W PrX 0 Snc all X j s ar mutually ndpndnt, PrX + X X 0 PrX 0 and X 2 0 and and X 0 Prv V 0 v W PrX j 0 Now w calculat PrX j 0 Not that X j s a bnomal random varabl wth paramtrs n W j α and p p j β α ζβ Thrfor whr q p Hnc PrX j 0 Prv V 0 v W whr n th last approxmaton w us Lmma 3 j n p 0 0 p n q Wj j q Wj, Lmma 34 Lt v W Thn Prv V v W Proof Lt v W and lt X b th random varabl for dv Lt X j b th random varabl countng th numbr of nghbors of v n W j Thrfor X j X j Hnc, Prv V v W PrX PrX + X X PrX and X j 0, j + PrX 2 and X j 0, j PrX and X j 0, j 5

6 Snc all varabls X j s ar mutually ndpndnt, w hav Prv V v W PrX j PrX j 0 + PrX 2 j 2 PrX j PrX PrX j 0 Now w calculat PrX j 0 and PrX j Not that X j s a bnomal random varabl wth paramtrs n W j α and p p j β α ζβ Thrfor n PrX j 0 p 0 0 p n q Wj n PrX j p p n W j p q Wj W j p q Wj, q whr q p Thrfor, j Prv V v W W p q W q W2 2 q W3 3 q W q + q W + q W + q W k W 2 p 2 q W2 2 q W3 3 q W q 2 q W2 2 q W2 2 q W k k W 3 p 3 q W3 3 q W q 3 q W3 3 W p q W q W j p Now w bound th product and th sum sparatly Lt s frst look at th sum: j W j p q j α j β α α, j j ζβ j q α ζβ α ζβ j β α ζβ j β α ζβ j j β ζβ ζβ 6

7 whr th frst approxmaton s analogous to Lmma 2 w omt th proof whch s almost dntcal to th proof of Lmma 2 Now, usng Lmma 3, w dal wth th product k q W k k Thrfor, Prv V v W k q W k k W j p j q In th nxt Lmma w nd th spcal functon polylogarthm, dfnd L s z Lmma 35 Lt v V Thn Proof Prv V 0 Prv V 0 L β/ Prv V 0 v W Prv W β L β/, β whr n th last approxmaton w us Lmmas 33 and 32 Th proof of Lmma 36 blow s omtd snc s smlar th proof of Lmma 35 Th ky stp s to us Lmma 34 nstad of Lmma 33 k z k k s Lmma 36 Lt v V Thn Prv V L β / Dfnton 37 Lt W {v V v W and v V } W {v V v W and v V } Lmma 38 Lt v W Thn Prv W v W + 7

8 Proof Lt v W and lt X b th random varabl for dv Lt X j b th random varabl countng th numbr of nghbors of v n W j Thrfor X j X j Hnc Prv V PrX 2 PrX 0 or X PrX 0 + PrX PrX 0 and X Not that PrX 0 and X 0, snc X cannot assum both valus at th sam tm Thrfor, Prv V PrX 0 PrX Prv V 0 v W Prv V v W, whr th approxmaton s obtand from Lmmas 33 and 34 Lmma 39 Pr W k W k k W k Proof Drctly from th fact that W s a bnomal random varabl wth paramtrs n W and p Prv W v W, whr th approxmaton s obtand from Lmma 34 In th nxt Lmma, dnot th vnt v W dos not hav a nghbor n W j by v W j Lmma 30 Prv W j j j β 2 ζβ Proof Usng th Law of Total Probablty and Lmma 39, Prv W j k Prv W j W j k Pr W j k k p k Wj j k j j Wj k j Wj p j k k j j Wj k } {{ j } k k If W j, thn k W j k Othrws, > W j, th vry trm W j k s qual to 0 whn k > W j and hnc, k W j Thrfor, Prv W j W j Wj k k k p j j k j j Wj k 8

9 Usng th Bnomal Thorm, Prv W j p j j + j Wj j Wj j p j j α ζβ j 2 j ζβ α j β 2 j ζβ j j β 2 ζβ α j β α j β Lmma 3 Prv dos not hav a nghbor n V ρβ, whr ρβ L β 2 ζβ Proof Lt ε j b th vnt v W dos not hav a nghbor n W j Thrfor Prv dos not hav a nghbor n V Prε ε 2 ε Prε j j j j j β 2 ζβ, whr th scond stp s a consqunc of th ndpndnc of th vnts ε j and th uppr bound s consqunc of Lmma 30 Thrfor, Prv dos not hav a nghbor n V j j j β 2 ζβ j j j β 2 ζβ ζβ j ζβ L β 2 j j β 2 In th nxt Lmma, dnot th vnt v V dos not hav a nghbor n V by v V 9

10 Lmma 32 whr ρβ L β 2 ζβ Proof Usng th Law of Total Probablty, Prv V Prv V L β ρβ, Prv V and v W Prv V v W Prv W Prv V v W Prv W v W Prv W Usng Lmmas 3, 38 and 32, w hav Prv V ρβ / / β ρβ β L β ρβ In th nxt Thorm w stmat th sz of th st of vrtcs adjacnt to vrtcs of dgr Thorm 33 E[ NV ] V L β/ L β / L β ρβ Proof W dnot th vnt v V has a nghbor n V by v V Lt X v b th ndcator random varabl whr X v whn v V and v V othrws X v 0 Thrfor, by Lmmas 32, 35 and 36, E[X v ] PrX v Prv V v V Prv V v V Prv V Prv V v V Prv V \ {V 0 V } L β ρβ L β/ L β / Lt X b a random varabl such that X v V X v, X NV Thrfor, by th lnarty 0

11 of xpctaton and usng th valu of E[X v ], E[ NV ] v V E[X v ] L β ρβ v V V L β ρβ L β/ L β / L β/ L β / 4 Approxmaton Algorthm For any S V, lt OPTS b th cardnalty of a mnmum vrtx covr n G[S] Lt V V NV In Lmma 4, w gv an auxlary rsult that holds for any graph G V, E, no probablstc argumnt s usd n th proof In ths partcular cas, quantts such as OPTS and NV ar not tratd as xpctd valus of random varabls In th rst of th papr ths quantts ar tratd as such Lmma 4 Th followng thr condtons hold: G contans a mnmum vrtx covr C such that NV C and no vrtx of V s n C; OPTV NV ; OPTV OPTV + OPTV \ V Proof Lt uv E such that u V Not that any mnmum vrtx covr C of G contans xactly on vrtx of {u, v} othrws C s not a covr or C s not mnmum Not that f u C, thn C \ {u} {v} s also a mnmum vrtx covr Thrfor, usng ths sam xchang argumnt, thr s a mnmum vrtx covr contanng vry vrtx of NV Hnc holds Applyng to th graph nducd by V, NV s a mnmum vrtx covr of G[V ], and hnc holds Now lt C C NV b a mnmum vrtx covr rspctng condton Thr s no vrtx covr C n G[V \ V ] such that C < C, othrws C NV contradcts th mnmalty of C Thrfor OPTV \ V C Combnng ths wth, condton holds Lmma 42 OPTV OPTV L ρβ Proof By Lmma 4, OPTV V OPTV V V L β/ L β / Combnng ths wth Lmmas 35 and 36 w hav By Lmma 4 and Thorm 33, OPTV NV V L β/ L β / L β ρβ Combnng ths bounds w hav OPTV OPTV L β ρβ

12 Corollary 43 OPTV \ V OPTV L β ρβ Proof By Lmma 4, OPTV \V +OPTV OPTV Algorthm 44 By Lmma 42, th corollary holds Lt C b a covr obtand by a 2-approxmaton algorthm n G[V \ V ] 2 Output C NV Thorm 45 Th xpctd approxmaton factor of Algorthm 44 s + L ρβ whn appld to a random powr law graph Proof Th sz of a soluton obtand by Algorthm 44 s C NV Snc C and NV ar dsjont, C NV C + NV By Lmma 4, NV OPTV Snc C s obtand by a 2-approxmaton algorthm n V \ V, C 2OPTV \ V Thrfor C NV OPTV + 2OPTV \ V OPTV + OPTV \ V, whr th last stp s obtand by Lmma 4 By Corollary 43, C NV OPTV + OPTV \ V OPTV + L β ρβ OPTV + L β ρβ OPTV 5 Concluson In ths papr w prsnt an approxmaton algorthm for th mnmum vrtx covr problm on powr law graphs W us th th gnralzd random graph [27] modl as wll th Chung-Lu modl to analyz th xpctd approxmaton rato of th algorthm W obsrv that ths graph modl s potntally asr to handl than th modl usd n [5] manly du to th fact that dgs ar addd ndpndntly n th random procss As a futur work w ar ntrstd n nvstgatng th approxmablty of othr optmzaton problms on powr law graphs [] M Faloutsos, P Faloutsos, C Faloutsos, On powr-law rlatonshps of th ntrnt topology, SIGCOMM Comput Commun Rv [2] A-L Barabs, R Albrt, Emrgnc of scalng n random ntworks, Scnc [3] J M Klnbrg, R Kumar, P Raghavan, S Rajagopalan, A S Tomkns, Th wb as a graph: Masurmnts, modls, and mthods, n: Procdngs of th 5th Annual Intrnatonal Confrnc on Computng and Combnatorcs, COCOON 99, Sprngr-Vrlag, Brln, Hdlbrg, 999, pp 7 [4] A Brodr, R Kumar, F Maghoul, P Raghavan, S Rajagopalan, R Stata, A Tomkns, J Wnr, Graph structur n th wb, Comput Ntw

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