The Price of Being Near-Sighted

Transcription

1 The Prce of Beng Near-Sghte Faban Kuhn, Thomas Moscbroa, Roger Wattenhofer Comuter Engneerng an Networks Laboratory, ETH Zurch, 8092 Zurch, Swtzerlan Abstract Achevng a global goal base on local nformaton s challengng, esecally n comlex an large-scale networks such as the Internet or even the human bran In ths aer, we rove an almost tght classfcaton of the ossble traeoff between the amount of local nformaton an the qualty of the global soluton for general coverng an ackng roblems Secfcally, we gve a strbute algorthm usng only small messages whch obtans an (ρ ) 1/k -aroxmaton for general coverng an ackng roblems n tme O(k 2 ), where ρ eens on the LP s coeffcents If message sze s unboune, we resent a secon algorthm that acheves an O(n 1/k ) aroxmaton n O(k) rouns Fnally, we rove that these algorthms are close to otmal by gvng a lower boun on the aroxmablty of ackng roblems gven that each noe has to base ts ecson on nformaton from ts k-neghborhoo 1 Introucton Many of the most fascnatng an funamental systems n the worl are large an comlex networks, such as the human socety, the Internet, or the human bran Such systems have n common that ther entrety s comose of a multlcty of nvual enttes; human bengs n socety, hosts n the Internet, or neurons n the bran As verse as these systems may be, they share the key characterstc that the caablty of rect communcaton of each nvual entty s restrcte to only a small subset of neghborng enttes Most human communcaton, for nstance, s between acquantances or wthn the famly, an neurons are rectly lnke wth merely a relatvely small number of other neurons for neurotransmsson On the other han, n ste of each noe thus beng nherently near-sghte, e, restrcte to local communcaton, the entrety of the system s suose to come u wth some kn of global soluton, or to kee an equlbrum Achevng a global goal base on local nformaton s challengng Many of the systems whch are the focus of comuter scence fall exactly nto the above mentone category of networks In the Internet, largescale eer-to-eer systems, or moble a hoc an sensor networks, no noe n the network s caable of keeng global nformaton on the network Instea, these noes have to erform ther ntene (global) task base on local nformaton only In other wors, all comutaton n these systems s nherently local! Not surrsngly, stuyng the funamental ossbltes an lmtatons of local comutaton s therefore of nterest to theoretcans n aroxmaton theory, strbute comutng, an grah theory The stuy of local comutaton has been ntate by the oneerng work of Lnal [11], an Naor an Stockmeyer [15] more than a ecae ago Also, the work of Peleg [17] has resulte n numerous nterestng an ee results But nonetheless, there remans a great number of mortant oen roblems relate to questons such as what kn of global tasks can be erforme by nvual enttes that have to base ther ecsons on local nformaton, or how much local nformaton s requre n orer to come u wth a globally otmal soluton For nstance, the oen queston from [16], that s, characterzng the trae-off between communcaton among agents to exchange nformaton an the global utlty acheve, has been unanswere It s the goal of ths aer to make a ste towars answerng ths oen roblem an, more generally, to brng forwar some of the unerlyng rncles an trae-offs governng local comutaton Not surrsngly, many global crtera such as countng the total number of noes n the network or obtanng a mnmum sannng tree cannot be met f every noe s ecson s base solely on local knowlege On the other han, many funamental coornaton tasks an alcatons n large-scale networks aear to be easer to hanle from a local-global ersectve Secfcally, classc grah theory roblems such as omnatng set or matchng can be formulate as stanar coverng an ackng roblems The nature of smle coverng an ackng roblems lke mnmum vertex cover or maxmum matchng aears to be local an ntutvely, one may thnk that each noe s (ege s) ecson s not affecte by very stant noes (eges) Interestngly, we rove n ths aer that ths ntuton s msleang even for the most basc ackng roblems

2 On the ostve se, we show that there exst strbute aroxmaton algorthms that almost acheve the otmal trae-off, even n the ractcally mortant case n whch the amount of nformaton exchange n each message s lmte Secfcally, we gve the followng results: Conser a network wth n noes an maxmum egree Assume that each noe n a network grah has to base ts ecson on ts k-ho neghborhoo We resent an effcent etermnstc strbute algorthm that oerates wth small messages of sze O(log n) bts The algorthm acheves a (ρ ) 1/k -aroxmaton for general coverng an ackng roblems n O(k 2 ) communcaton rouns, where ρ eens on the coeffcents of the unerlyng LP When message sze s unboune, each network noe can easly gather the entre nformaton from ts O(k)-neghborhoo n O(k) communcaton rouns Hence, n ths case, the achevable trae-off s a true consequence of localty restrcton only We resent an algorthm roucng an O(n 1/k )-aroxmaton f each noe knows ts O(k)-neghborhoo In combnaton wth (strbute) ranomze rounng, the above algorthms can be transforme nto constant-tme strbute algorthms havng non-trval aroxmaton ratos for varous combnatoral roblems Fnally, we show that the trae-off acheve by our algorthms s almost tght Secfcally, we rove that even the most smle ackng roblem, (fractonal) maxmum matchng, cannot be aroxmate wthn Ω(n c/k2 /k) an Ω( 1/k /k), resectvely Ths mles Ω( log n/ log log n) an Ω(log / log log ) tme lower bouns for (ossbly ranomze) strbute algorthms n orer to obtan a constant or olylogarthmc aroxmaton for maxmum matchng an ackng roblems, even f message sze s unboune Ths lower boun extens a smlar result that has recently been acheve for the mnmum vertex cover roblem n [9] Note that by gvng uer an lower bouns for general coverng an ackng LPs, we show that many fferent natural roblems behave smlarly wth regar to local aroxmablty Ths s of great theoretcal nterest snce such a classfcaton of roblems may rove a comletely new nsght nto the mact of localty on algorthms Relate work an the moel of comutaton are escrbe n Sectons 2 an 3 In subsequent Sectons 4 an 5, we gve strbute algorthm for general coverng an ackng LPs n the boune an unboune message moel, resectvely The ackng lower boun s erve n subsequent Secton 6 Secton 7 conclues the aer Due to lack of sace, some roofs are omtte from ths extene abstract 1 2 Relate Work Lttle s known about the funamental lmtatons of localty-base aroaches Fch an Ruert, for nstance, escrbe a numerous lower bouns an mossblty results n strbute comutng [5] But most of them aly to other comutatonal moels where localty s no ssue or there are atonal, more restrctve lmtng factors, such as boune message sze [4] There have been vrtually no nontrval lower bouns for local comutaton, beses Lnal s semnal Ω(log n) tme lower boun for constructng a maxmal neenent set on a rng [11] In aton, we have shown that mnmum vertex cover an thus coverng roblems cannot be aroxmate better than Ω(n c/k2 /k) an Ω( 1/k /k) f each noe s nformaton s restrcte to ts k-neghborhoo [9] On the ostve se, t was shown by Naor an Stockmeyer [15] that there exst locally checkable labelngs whch can be comute n strbute constant tme, e, wth comletely local nformaton only The focus of ths aer s to unerstan localty n roblems that can be formulate as ackng an coverng LPs There are a number of (arallel) algorthms for solvng such LPs whch are faster than nteror-ont methos that can be ale to general LPs (eg [6, 14, 18, 22]) All these algorthms nee at least some global nformaton to work The roblem of aroxmatng ostve LPs usng only local nformaton has been ntrouce n [16] The frst algorthm achevng a constant aroxmaton n olylogarthmc tme s escrbe n [2] Dstrbute algorthms targete for secfc coverng an ackng roblems nclue algorthms for the mnmum omnatng set roblem [3, 8, 19] as well as algorthms for maxmal matchngs an maxmal neenent sets [1, 7, 13] Ths mles a constant aroxmaton for maxmum matchng All escrbe strbute algorthms have a tme comlexty whch s at least logarthmc n n That s, each noe may gather nformaton whch s as far away as O(log n) hos Hence, whle these algorthms rove solutons to artcular roblems, they o not fully exlore the trae-off between local knowlege an soluton qualty A strbute algorthm for mnmum omnatng set runnng n an arbtrary, ossbly constant number of rouns s foun n [10] 1 A verson contanng all roofs can be foun as TIK techncal reort 229 at ft://fttkeeethzch/ub/ublcatons/tik- Reort229f

3 3 Moel We escrbe the network as an unrecte grah G = (V, E) The vertces V = {v 1,, v n reresent the network enttes or noes (eg rocessors) an the eges reresent brectonal communcaton channels We stngush two rototycal an classc message assng moels [17], LOCAL an CON GEST, eenng on how much nformaton can be sent n each message In the LOCAL moel (eg, [11, 15, 17]), knowng your k-neghborhoo an erformng k communcaton rouns are equvalent It s assume that n every communcaton roun, each noe n the network can sen an arbtrarly long message to each of ts neghbors Local comutatons are for free Each noe has a unque entfer an ntally, noes have no knowlege about the network grah In k communcaton rouns, a noe v may collect the IDs an nterconnectons of all noes u to stance k from v, because messages are unboune Hence, each noe has a artal (local) vew of the grah; t knows ts entre vcnty u to stance k Let T v,k be the toology seen by v after k rouns T v,k s the grah nuce by the k-neghborhoo of v wthout all eges between noes at stance exactly k The labelng (e, the assgnment of IDs) of T v,k s enote by L(T v,k ) The vew of a noe v s the ar, V v,k := (T v,k, L(T v,k )) The vew of an ege e = (u, v) s the unon of vews of ts ncent noes The best a local algorthm can o n tme k, s to have every noe v collect ts k-neghborhoo an base ts ecson on V v,k Snce the LOCAL moel abstracts away other asects arsng n the esgn of strbute algorthms (congeston, fast local comutaton, ), t s the most funamental moel when stuyng the henomenon of localty; artcularly for lower bouns In ractce, the amount of nformaton exchange between two neghbors n one communcaton ste s lmte The CON GEST moel [4, 17] takes nto account the volume of communcaton Ths moel lmts the nformaton that can be sent n one message to O(log n) bts Gven ths atonal restrcton, even roblems on the comlete network grah, whch coul be solve otmally n a sngle communcaton roun n the LOCAL moel, become nontrval A fractonal coverng roblem (PP) an ts ual fractonal ackng roblem (DP), are lnear rograms of the canoncal forms mn st c T x A x b x 0 max st b T y A T y c y 0 where all a, b, an c are non-negatve We wll use the term rmal LP (PP) for the mnmzaton an ual LP (DP) for the maxmzaton roblem The number of rmal an ual varables are enote by m an n, resectvely Let a max := max, {a, b, c be the maxmum coeffcent an a mn := mn, {a, b, c \{0 be the mnmum non-zero coeffcent of (PP) an (DP) ρ := a max /a mn s the maxmum rato between any two coeffcents Analogously to [2, 16], we conser the followng strbute settng The lnear rogram s boun to a network grah G = (V, E) Each rmal varable x an each ual varable y s assocate wth a noe v () V an v () V, resectvely There are communcaton lnks between rmal an ual noes wherever the resectve varables occur n the corresonng nequalty Thus, (v (), v () ) E f an only f x occurs n the th nequalty of (PP), e, v () an v () are connecte ff a > 0 2 The egrees of v () an v () are calle δ () an δ (), resectvely := max δ () an := max δ () are calle the rmal an ual egree, resectvely The set of ual neghbors of v () s enote by N (), the set of rmal neghbors of v () by N () Where convenent, N () an N () also enote the sets of the nces of the resectve noes 4 Boune Messages In ths secton, we escrbe an effcent strbute algorthm to aroxmate coverng an ackng lnear rograms n the CON GEST moel For our algorthm, we nee the LPs (PP) an (DP) to be of the followng secal form: (41), : b = 1, a = 0 or a 1 The transformaton to (41) s one n two stes Frst, every a s relace by â := a /b an b s relace by 1 In the secon ste, the c an â are ve by λ := mn {â \ {0 The otmal obectve values of the transforme LPs are the same A feasble soluton for the transforme LP (41) can be converte to a feasble soluton of the orgnal LP by vng all x-values by the corresonng λ an by vng the y-values by the corresonng b Ths conserves the values of the obectve functons Note that the escrbe transformaton can be comute locally n a constant number of rouns For the rest of ths secton, 2 Note that n orer to solve such a roblem n a real network settng where only some varables correson to noes, the other varables may be smulate by the noes as well Varables assocate to eges (lke n vertex cover or maxmum matchng) can be smulate by ncent noes

4 LP Aroxmaton LP Aroxmaton Algorthm for Prmal Noe v () : Algorthm for Dual Noe v () : 1: x := 0; 1: y := y + := w := f := 0; r := 1; 2: for e := k 2 to f 1 by 1 o 2: for e := k 2 to f 1 by 1 o 3: for 1 to h o 3: for 1 to h o 4: ( γ := c max c a r ) 4: r := r ; 5: for e := k 1 to 0 by 1 o 5: for e := k 1 to 0 by 1 o 6: γ := cmax c a r ; 6: 7: f γ Γ e /k then 7: 8: x + := 1/Γ e /k ; x := x + x + ; 8: 9: f; 9: 10: sen x +, γ to ual neghbors; 10: receve x +, γ from rmal neghbors; 11: 11: y + := y + + r a x + / γ ; 12: 12: w + := a x + ; 13: 13: w := w + w + ; f := f + w + ; 14: 14: f w 1 then r := 0 f; 15: receve r from ual neghbors 15: sen r to rmal neghbors 16: o; 16: o; 17: 17: ncrease uals(); 18: receve r from ual neghbors 18: sen r to rmal neghbors 19: o 19: o 20: o; 20: o; 21: x := x / mn N () l a lx l 1 c l a ly l 21: y := y / max () N Algorthm 1: Dstrbute LP Aroxmaton Algorthm we assume that the coeffcents of the LP are gven accorng to (41) We start the escrton of the algorthm wth a general outlne As our algorthm borrows from the greey omnatng set/set cover algorthm, t s useful to vew the strbute LP algorthm n ths context The greey mnmum omnatng set (MDS) algorthm starts wth an emty set an sequentally as the noe whch covers the most not yet covere noes The LP relaxaton of MDS asks for varables x for the noes v such that the sum of the x n the 1-neghborhoo of every noe s at least 1 Analogous to the sequental greey aroach, we also start wth all x set to 0 an we gve rorty to noes wth many uncovere neghbors when ncreasng the x In artcular, we always ncrease the x of all the noes whose number of uncovere neghbors s maxmum u to a certan factor (actve noes) In orer not to over-cover a noe wth many actve neghbors, we have to carefully choose the ncrement of the x at actve noes As we rocee, we smultaneously comute a soluton for the ual LP such that the obectve values of the solutons stay the same In the en, each noe s covere at least f tmes an each ual constrant s fulflle u to a factor αf Hence by vng by f an αf, we obtan feasble, α- aroxmate rmal an ual solutons, resectvely In orer to acheve that every rmal nequalty s fulflle f tmes, each ual noe v () nees a requrement r 1 whch s ecrease every tme the corresonng rmal constrant s acheve an a varable f whch counts how many tmes the rmal constrant has been fulflle (cf [21]) The ecson whether a rmal noe v () s actve an can ncrease x s base on the effcency er cost rato γ whch s efne as follows: γ := c max c a r For smlcty, we assume that all noes know c max := max{c as well as two other global quanttes Γ an Γ whch are efne as Γ := max c max c n =1 a an Γ := max m a =1 At the rce of a conserably more comlcate (an less reaable) algorthm, t s ossble to get r of ths assumton For etals, we refer to the full aer The etale algorthm s gven by Algorthm 1 along wth the roceure ncrease uals() whch s

5 roceure ncrease uals(): 1: f w 1 then 2: f f f then 3: y := y + y + ; y+ := 0; 4: r := 0; w := 0 5: else f w 2 then 6: y := y + y + ; y+ := 0; 7: r := r /Γ w /k 8: else 9: λ := max{γ 1/k, Γ 1/k ; 10: y := y + mn{y +, r λ/γ e/k ; 11: y + := y + mn{y +, r λ/γ e /k ; 12: r := r /Γ 1/k 13: f; 14: w := w w 15: f use by the ual noes The algorthm has two arameters k 1 an k 1 whch etermne the traeoff between tme comlexty an aroxmaton qualty The bgger k an k, the better the aroxmaton rato of the algorthm On the other han, smaller k an k lea to a faster algorthm Algorthm 1 also makes use of two values f an h whch are efne as follows: k + 1 k f := an h := 1 + Γ 1/k 1 Γ 1/k ln Γ In the followng, we resent lemmas whch establsh all the necessary etals to analyze Algorthm 1 The goal of the outer e -loo s to reuce the maxmum weghte rmal egree γ Ths s reflecte by the followng lemma Lemma 41 For each rmal noe v (), at all tmes urng Algorthm 1, γ Γ (e +2)/k One comlete run (k teratons) of the nnermost e -loo can be seen as one arallel greey ste Prmal noes wth large γ ncrease ther x such that the corresonng ncreases y + of the ual varables are almost feasble Lemma 42 Each tme a ual noe enters ncrease uals() n Algorthm 1, (42) y + r w Γ e/k an y + r Γ1/k + 1 Γ e/k As shown n Lemma 44, all the ncreases of the ual varables together rener the ual constrants feasble u to a small factor tmes (k + f + 1) We frst nee the followng heler lemma Lemma 43 Let v () be a rmal noe an let Y := a y be the weghte sum of the y-values of ts ual neghbors Further, let Y + be the ncrease of Y an γ be the ecrease of γ urng an executon of ncrease uals() We have Y + Γ3/k max{γ 1/k, Γ 1/k γ (Γ 1/k 1) c c max γ Proof We rove the lemma by showng that the nequalty hols for every ual neghbor v () of v () Let β be the ncrease of y an let r be the ecrease of r We show that (43) β Γ1/k max{γ 1/k, Γ 1/k r Γ e /k (Γ 1/k 1) The lemma then follows because γ Γ (e +2)/k (Lemma 41) an because Y + = a β an γ = c max c a r To rove Inequalty (43), we agan conser the cases where w 2 an where 1 w < 2 If w 2, by Lemma 42, β = y + r (1 + Γ 1/k )/Γ e /k The requrement r s ve by at least Γ 2/k an therefore r (Γ 2/k 1)/Γ 2/k Together, we get r β 1 + Γ1/k Γ e /k ( ) 1 + Γ 1/k Γ e/k Γ 2/k Γ 2/k Γ 1/k ( Γ 1/k r max{γ 1/k, Γ 1/k ) ( ) r Γ 1/k 1 For 1 w < 2, the roof s along the same lnes Here, β r max{γ 1/k, Γ 1/k /Γ e /k an r = r (Γ 1/k 1)/Γ 1/k Agan, we obtan Inequalty (43): β max{γ1/k, Γ 1/k Γ e /k Γ 1/k Γ 1/k 1 r We o not have to conser the case f f exlctly because the same analyss as for w 2 ales n ths case Lemma 44 Let v () be a rmal noe an Y = a y be the weghte sum of the y-values of the ual neghbors of v () After the man art of the algorthm (e, after the loos at lne 20), Y c { (k + f + 1)Γ 3/k max Γ 1/k, Γ 1/k c max

6 Proof For smlcty, we efne Q := 1 Γ 3/k c max max{γ 1/k, Γ 1/k Before γ s ecrease for the last tme, we have γ 1/Γ (f 1)/k because at least one r n the ual neghborhoo of v () has to be greater than 0 If we assume that the last tme γ s ecrease t s only reuce to γ = 1/Γ (f+1)/k, Lemma 43 stll hols The analyss s exactly the same as for the case w 2 n Lemma 43 By Lemma 43, Y s therefore boune by the area uner the curve c Q/(Γ 1/k 1) 1/x for x between 1/Γ (f+1)/k an Γ : Y = Γ c Q Γ 1/k 1 ( c (k + f + 1)Q ln 1 Γ (f+1)/k Γ 1/k 1 1 x x Γ 1/k ) c (k + f + 1)Q The last nequalty follows from ln(1 + t) t At the en of the algorthm, all rmal constrants are satsfe at least f tmes Further, the rmal an ual obectve functons are the same Lemma 45 After the loos at lne 20, : r = 0 an f f an m =1 c x = c max n =1 y Proof When enterng the e -loo for the last tme, by Lemma 41, Γ ( f+1)/k γ a r r c max c N () γ can only be greater than 0 f there s exactly one r n the ual neghborhoo of v () whch s greater than zero If r s stll greater than 0 when e = 0, x wll be ncrease by 1 whch makes w 1 an therefore r = 0 after the next call to ncrease uals() f counts the number of tmes the th constrant of (PP) s satsfe It s ncrease together wth w n lne 13 of Algorthm 1 Every tme the nteger art of w s ncrease, r s ve by Γ w /k an w s set to w w Therefore, r = 0 mles f f Let v () be a rmal noe whch ncreases x by x + (lne 8) All ual neghbors v () of v () ncrease y + by a r x + / γ Hence, the sum of the y + -ncreases over all ual neghbors of v () s x + a r = x + a r γ a = c x + r c max Because f f, all y + are 0 n the en an thus y s equal to the sum of all ncreases of y + Combnng the above lemmas, we get the followng theorem Theorem 41 For arbtrary k, k 1, Algorthm 1 aroxmates (PP) an (DP) by a factor { Γ 4/k max Γ 1/k, Γ 1/k The tme comlexty of Algorthm 1 s ( ) ( 1 O (k k Γ 1/k 1 Γ 1/k k log Γ For k O(log Γ ), ths smlfes to O(k k ) )) Proof For the aroxmaton rato, we have to look at lne 21 of Algorthm 1 where all x an y values are ve by the largest ossble values to kee/make the rmal/ual soluton feasble By Lemma 45, each rmal constrant s satsfe at least f tmes Therefore, all rmal varables are ve by at least f Due to Lemma 44, for each rmal noe, the sum of the y values of ts ual neghbors s at most c (k +f +1)Q for Q as efne n Lemma 44 Dvng all ual varables by (k + f + 1)Q therefore reners the ual soluton feasble By Lemma 45, the rato between the obectve functons of the rmal an the ual solutons s m =1 c x k + f + 1 n =1 y c max Q f c max k + k+1 Γ 1/k Q k +1 Γ 1/k 1 = c max Γ 1/k Q = Γ 4/k max { Γ 1/k, Γ 1/k Because of the ualty theorem for lnear rogrammng, ths rato s an uer boun on the aroxmaton rato for (PP) an (DP) As for the tme comlexty, note that each teraton of the nner-most loo (e -loo) requres two rouns Hence, the algorthm has tme comlexty O(k (k + f)h) The clam follows from substtutng the actual values for f an h For k O(log Γ ), Γ 1/k 1 s a constant an therefore the tme comlexty smlfes to O(k k ) Corollary 41 For suffcently small ε, Algorthm 1 comutes a (1 + ε)-aroxmaton for (PP) an (DP) n O ( log Γ log Γ /ε 4) rouns In artcular, a constant factor aroxmaton can be acheve n tme O(log Γ log Γ )

7 Remark: Usng methos smlar to the ones escrbe n [2, 14], t s ossble to get r of the eenency on the coeffcents ρ := a max /a mn As a result, the runnng tme an aroxmaton rato woul een on the number of noes m an n nstea of the egrees an Dstrbute Ranomze Rounng We can aly our strbute LP aroxmaton algorthms together wth stanar strbute ranomze rounng technques to obtan strbute aroxmaton algorthms for a number of combnatoral roblems We can rove that gven an α-aroxmate soluton for the LP relaxaton of roblems for whch the matrx elements a {0, 1, we can comute n a constant number of rouns a O(α log )-aroxmaton for the corresonng coverng IP an a O(α )-aroxmaton for the ackng IP 5 Unboune Messages In [12], Lnal an Saks resente a ranomze strbute algorthm to ecomose a grah nto subgrahs of lmte ameter We use ther algorthm to ecomose the lnear rogram nto sub-rograms whch can be solve locally n the LOCAL moel For a general grah G = (V, E) wth n noes, the algorthm of [12] yels a subset S V of V such that each noe u S has a leaer l(u) V an such that the followng roertes hol 3 (I) u S : (u, l(u)) < k (II) u, v S : l(u) l(v) (u, v) E (III) S can be comute n k rouns (IV) u V : Pr[u S] 1 en 1/k (u, v) enotes the stance between two noes u an v on G We aly the algorthm of [12] to obtan connecte comonents of G wth the followng roertes (I) The comonents have small ameter (II) Dfferent comonents are far enough from each other such that we can efne a local lnear rogram for each comonent n a way n whch the LPs of any two comonents o not nterfere (III) Each noe belongs to one of the comonents wth robablty at least, where eens on the ameter we allow the comonents to have Because of the lmte ameter, the LPs of each comonent can then be comute locally We aly the ecomoston rocess n arallel often enough such that wh each noe has been selecte a logarthmc number of tmes 3 We use = 1/n 1/k n the algorthm of Secton 4 of [12], the roertes then rectly follow from Lemma 41 of [12] For the ecomoston of (PP) an (DP), we nee the followng lemma Lemma 51 Let {y 1,, y m be a subset of the ual varables of DP an let x 1,, x n be the rmal varables whch are aacent to the gven subset of the ual varables Further let P P an DP be LPs where the matrx A conssts only of the columns an rows corresonng to the varables n x an y Every feasble soluton for P P makes the corresonng rmal nequaltes n P P feasble an every feasble soluton for DP s feasble for DP (varables not occurrng n P P an DP are set to 0) Further, the values of the obectve functons for the otmal solutons of P P an DP are uer boune by the otmal values for P P an DP We call P P an DP the sub-lps nuce by the subset {y 1,, y m of ual varables We aly the grah ecomoston algorthm of [12] to obtan P P an DP (as n Lemma 51) whch can be solve locally For the ecomoston of the lnear rogram, we efne G such that the noe set V s the set of ual noes of the grah G an the ege set E s E := { (u, v) u, v V G (u, v) 4 By ths, we can guarantee that non-aacent noes n G o not have neghborng rmal noes n G whose varables occur n the same constrant of (PP) Further, a message over an ege of G can be sent n 4 rouns on the network grah G The basc algorthm for a ual noe v to aroxmate P P an DP then works as follows: 1: Run grah ecomoston of [12] on G; 2: f v S then 3: sen IDs of rmal neghbors to l(v) 4: f; 5: f v = l(u) for some u S then 6: comute local PLP/DLP (cf Lemma 51) of varables of u S for whch v = l(u) 7: sen resultng values to noes holng the resectve varables 8: f The rmal noes only forwar messages n stes 1, 3, an 7 an receve the values for ther varables n ste 7 We now have a closer look at the locally comute LPs n lne 6 By Proerty (II) of the grah ecomoston algorthm, ual varables belongng to fferent local LPs cannot occur n the same ual constrant (otherwse, the accorng ual noes ha to be neghbors n G) The analogous fact hols for rmal varables snce ual noes belongng to fferent local LPs have stance at least 6 on G an thus rmal noes belongng to fferent local LPs have stance

8 at least 4 on G Therefore, the local LPs o not nterfere an together they form the sub-lps nuce by S (cf Lemma 51) The comlete LP aroxmaton algorthm now conssts of N neenent arallel executons of the escrbe basc algorthm The varables of the N sub-lps are ae u an n the en, rmal/ual noes ve ther varables by the maxmum/mnmum ossble value to kee/make all constrants they occur n feasble 4 Fnally, N can be chosen to otmze the aroxmaton rato Theorem 51 Let N = αen 1/k ln n for α 451 Executng the basc algorthm N tmes, summng u the varables of the N executon an vng these sums as escrbe, yels an αen 1/k aroxmaton of (PP)/(DP) wh The algorthm requres O(k) rouns Corollary 51 Usng the network ecomoston algorthm of [12], n only O(k) rouns, PP an DP can be aroxmate by a factor O(n 1/k ) wh For k Θ(log n), ths gves a constant factor aroxmaton n O(log n) rouns 6 Lower Boun We erve tme lower bouns for strbute aroxmablty of ackng roblems, even n the LOCAL moel More recsely, we rove lower bouns for the most basc ackng roblems, the fractonal maxmum matchng roblem (FMM) Our general aroach follows [9] n whch smlar results are obtane for mnmum vertex cover whch s a coverng roblem Secfcally, our ackng lower boun grah s structurally smlar (although wth subtle fferences) to the one use n [9] Let E enote the set of eges ncent to noe v FMM s the natural LP relaxaton of MM an efne as max e E y, subect to v E y 1, v V an y 0, e E The outcome of an ege s ecson (y ) n a k-local comutaton s entrely base on the nformaton gathere wthn ts k-neghborhoo The ea for the lower boun s to construct a grah famly G k = (V, E) n whch, after k rouns of communcaton, two aacent eges see exactly the same grah toology Informally seakng, both of them are equally qualfe to on the matchng However, n G k, takng the wrong ecson wll be runous an yels a subotmal global aroxmaton The constructon of G k s a two ste rocess Frst, the general structure of G k s efne usng the concet of a cluster-grah CG k Seconly, we construct an nstance of G k obeyng the roertes mose by CG k 4 The rmal an ual varables x an y are ve by 1 P mn N b l a 1 lx l an max N c Pl a ly l, resectvely 61 The Cluster Grah The noes v V n G k are groue nto sont sets whch are lnke to each other as bartte grahs The structural roertes of G k are escrbe usng a recte cluster grah CG k = (C, A) wth oubly labele arcs l : A N N A noe C C reresents a cluster, e, one of the sont sets of noes n G k An arc a = (C, D) A wth l(a) = (δ c, δ ) enotes that the clusters C an D are lnke as a bartte grah n whch each noe u C has egree δ c an each noe v D has egree δ It follows that C δ c = D δ The cluster grah conssts of two equal subgrahs, so-calle cluster-trees CT k as efne n [9] In CG k, we atonally a an arc l(c, C ) := (1, 1) between two corresonng noes of the two cluster trees Formally, CT k an CG k are efne as follows We call clusters aacent to exactly one other cluster leaf-clusters, an all other clusters nner-clusters Defnton 61 [9] For a gven δ an a ostve nteger k, the cluster tree CT k s recursvely efne as follows: CT 1 := (C 1, A 1 ), C 1 := {C 0, C 1, C 2, C 3 A 1 := {(C 0, C 1 ), (C 0, C 2 ), (C 1, C 3 ) l(c 0, C 1 ) := (δ, δ 2 ), l(c 0, C 2 ) := (δ 2, δ 3 ), l(c 1, C 3 ) := (δ, δ 2 ) Gven CT k 1, CT k s obtane n two stes: For each nner-cluster C, a a new leaf-cluster C wth l(c, C ) := (δk+1, δ k+2 ) For each leaf-cluster C wth (C, C ) A an l(c, C ) = (δ, δ +1 ), a new leaf-clusters C wth l(c, C ) := (δ, δ +1 ) for = 1 k + 1, + 1 Defnton 62 Let T k an T k be two nstances of CT k Further, let C an C be corresonng clusters n T k an T k, resectvely We obtan the cluster grah CG k by ang an arc l(c, C ) := (1, 1) for all clusters C CT k Further, we efne n 0 := C 0 C 0 Ths unquely efnes the sze of all clusters Fgure 1 shows CT 2 an CG 2 The shae subgrahs correson to CT 1 an CG 1, resectvely, the ashe lnes reresent the lnks l(c, C ) := (1, 1) Note that nether CT k nor CG k efne the aacency on the level of noes They merely rescrbe for each noe the number of neghbors n each cluster We efne S 0 := C 0 C 0 an S 1 := C 1 C 1 The layer of a cluster s the stance to C 0 n the cluster tree T k an T k enote the two cluster trees consttutng CG k 62 The Lower Boun Grah G k Havng efne the cluster grah CG k, t s now our goal to obtan

9 δ 3 δ 4 δ 2 δ3 δ δ 2 δ 3 δ 4 δ δ 2 C 2 δ 2 δ 3 C 0 C 0 C 0 δ δ 3 C δ 2 1 δ 4 C 3 δ δ 2 Fgure 1: Cluster-Tree CT 2 an Cluster-Grah CG 2 a realzaton of G k whch has the structure mose by CG k an features the atonal roerty that there are no short cycles As we must rove that the toologes seen by noes n S 0 an S 1 are entcal, the absence of short cycles s of great hel Partcularly, f there are no cycles of length 2k + 1 an less, all noes see a tree locally The grth of a grah G, enote by g(g), s the length of the shortest cycle n G Lemma 61 states that t s nee ossble to construct G k as escrbe above Lemma 61 If k +1 δ/2, G k can be constructe such that the followng contons hol: (I) G k follows the structure of CG k (II) The grth of G k s at least g(g k ) 2k + 1 (III) G k has n 4 2k δ 4k2 noes Next we show that all noes n S 0 an S 1 have the same vew an consequently, all eges n E see the same toology Usng the followng result from [9] facltates ths task Lemma 62 [9] Let G k be an nstance of a cluster tree CT k wth grth g(g k ) 2k + 1 The vews of all noes n clusters C 0 an C 1 are entcal u to stance k Because G k has grth at least 2k + 1 by Lemma 61, the two cluster-trees T k an T k consttutng G k must have grth 2k + 1 as well It follows from Lemma 62 that the esre equalty of vews hols for both T k an T k Base on ths fact, t s now easy to show that equalty of vews hols n G k, too Lemma 63 Let G k be an nstance of a cluster grah CG k wth grth g(g k ) 2k + 1 The vews of all noes n clusters S 0 an S 1 are entcal u to stance k 63 Analyss We now erve the lower bouns on the aroxmaton rato for k-local FMM algorthms Let OPT be the otmal soluton for FMM an let ALG be the soluton comute by any algorthm All noes n S 0 an S 1 have the same vew an therefore, every ege n E sees the same toology V e,k Lemma 64 When ale to G k = (V, E) as constructe n Subsecton 62, any strbute, ossbly ranomze algorthm whch runs for at most k rouns comutes, n exectaton, a soluton of at most ALG S 0 /(2δ 2 ) + ( V S 0 ) Proof The fractonal value assgne to e = (u, v) by an algorthm s enote by y The ecson of whch value y s assgne to ege e eens only on the vew the toologes T u,k an T v,k an the labelngs L(T u,k ) an L(T v,k ), whch e can collect urng the k communcaton rouns Assumng that the labelng of G k s chosen unformly at ranom, the labelng L(T u,k ) for any noe u s also chosen unformly at ranom All eges connectng noes n S 0 an S 1 see the same toology If the labels are chosen unformly at ranom, t follows that the strbuton of the vews an therefore the strbuton of the y s the same for all those eges We call the ranom varables escrbng the strbuton of the y, Y Let u S 1 be a noe of S 1 The noe u has δ 2 neghbors n S 0 Therefore, for eges e between noes n S 0 an S 1, by lnearty of exectaton, E [Y ] 1/δ 2 because otherwse there exst labelngs for whch the calculate soluton s not feasble By Lemma 63, eges e wth both en-onts n S 0 have the same vew as eges between S 0 an S 1 Hence, also for the value y of e, E [Y ] 1/δ 2 must hol There are S 0 /2 such eges an therefore the execte total value contrbute by eges between two noes n S 0 s at most S 0 /(2δ 2 ) All eges whch o not connect two noes n S 0, have one en-ont n V \ S 0 In orer to get a feasble soluton, the total value of all eges aacent to a set of noes V, can be at most V Ths can for examle be seen by lookng at the ual roblem, a kn of mnmum vertex cover where some eges only have one en noe Takng all noes of V (assgnng 1 to the resectve varables) yels a feasble soluton for ths vertex cover roblem The clam now follows by alyng Yao s mnmax rncle We now erve the lower boun Lemma 64 gves an uer boun on the number of noes chosen by any k- local FMM algorthm Choosng all eges wthn S 0 s feasble, hence, OPT S 0 /2 In orer to establsh a relatonsh between n, S 0, δ, an k, we boun n as n S 0 (1 + k+1 δ (k+1) ) usng a geometrc seres The secon lower boun then follows easly from = δ k+2

10 Theorem 61 For all ars (n, k) an (, k), there are grahs G an a constant c 1/4, such that n k communcaton rouns, every strbute algorthm for FMM on G has aroxmaton ratos at least Ω(n c/k2 /k) an Ω ( 1/k /k ), resectvely By settng k = β log n/ log log n an k = β log / log log, resectvely, for a constant β > 0, we obtan the followng corollary Corollary 61 In orer to obtan a olylogarthmc or constant aroxmaton rato, every strbute algorthm for FMM requres at least Ω( log n/ log log n) an Ω(log / log log ) communcaton rouns The same lower bouns hol for the constructon of maxmal matchngs an maxmal neenent sets Remark: The algorthm n Secton 5 acheves a olylogarthmc aroxmaton n O(log / log log ) communcaton rouns Therefore, for olylogarthmc aroxmatons, our lower boun for FMM s tght 7 Conclusons It s nterestng to vew local comutaton n a wer context of comutatonal moels Aroxmaton algorthms an onlne algorthms try to boun the egraaton of a globally otmal soluton cause by lmte comutatonal resources an knowlege about the future, resectvely More recently, the rce of anarchy, has been roose to measure the subotmalty resultng from selfsh nvuals [20] In a smlar srt, our aer shes lght on the rce of localty, e, the egraaton of a globally otmal soluton f each nvual s knowlege s restrcte to ts neghborhoo or local envronment Secfcally, the uer an lower bouns resente n ths aer characterze the achevable trae-off between local nformaton an the qualty of a global soluton of coverng an ackng roblems References [1] N Alon, L Baba, an A Ita A fast an smle ranomze arallel algorthm for the maxmal neenent set roblem J Algorthms, 7(4): , 1986 [2] Y Bartal, J W Byers, an D Raz Global Otmzaton Usng Local Informaton wth Alcatons to Flow Control In Proc of the 38 th Sym on Founatons of Comuter Scence (FOCS), ages , 1997 [3] D Dubhash, A Me, A Pancones, J Rahakrshnan, an A Srnvasan Fast Dstrbute Algorthms for (Weakly) Connecte Domnatng Sets an Lnear-Sze Skeletons In Proc of the ACM-SIAM Symosum on Dscrete Algorthms (SODA), ages , 2003 [4] M Elkn Uncontonal Lower Bouns on the Tme- Aroxmaton Traeoffs for the Dstrbute Mnmum Sannng Tree Problem In Proc of the 36th ACM Symosum on Theory of Comutng (STOC), 2004 [5] F Fch an E Ruert Hunres of mossblty results for strbute comutng Dstrbute Comutng, 16(2-3): , 2003 [6] L Flescher Aroxmatng Fractonal Multcommoty Flow Ineenent of the Number of Commotes SIAM Journal on Dscrete Math, 13(4): , 2000 [7] A Israel an A Ita A Fast an Smle Ranomze Parallel Algorthm for Maxmal Matchng Informaton Processng Letters, 22:77 80, 1986 [8] L Ja, R Raaraman, an R Suel An Effcent Dstrbute Algorthm for Constructng Small Domnatng Sets In Proc of the 20th Symosum on Prncles of Dstrbute Comutng (PODC), ages 33 42, 2001 [9] F Kuhn, T Moscbroa, an R Wattenhofer What Cannot Be Comute Locally! In Proc of the 23 r ACM Sym on Prncles of Dstrbute Comutng (PODC), ages , 2004 [10] F Kuhn an R Wattenhofer Constant-Tme Dstrbute Domnatng Set Aroxmaton In Proc of the 22n ACM Symosum on Prncles of Dstrbute Comutng (PODC), ages 25 32, 2003 [11] N Lnal Localty n Dstrbute Grah Algorthms SIAM Journal on Comutng, 21(1): , 1992 [12] N Lnal an M Saks Low Dameter Grah Decomostons Combnatorca, 13(4): , 1993 [13] M Luby A Smle Parallel Algorthm for the Maxmal Ineenent Set Problem SIAM Journal on Comutng, 15: , 1986 [14] M Luby an N Nsan A Parallel Aroxmaton Algorthm for Postve Lnear Programmng In Proc of the 25th ACM Symosum on Theory of Comutng (STOC), ages , 1993 [15] M Naor an L Stockmeyer What can be comute locally? SIAM Journal on Comutng, 24(6): , 1995 [16] C Paamtrou an M Yannakaks Lnear Programmng wthout the Matrx In Proc of the 25th Sym on Theory of Comutng (STOC), ages , 1993 [17] D Peleg Dstrbute Comutng: A Localty-Senstve Aroach SIAM, 2000 [18] S Plotkn, D Shmoys, an E Taros Fast Aroxmaton Algorthms for Fractonal Packng an Coverng Problems Mathematcs of Oeratons Research, 20: , 1995 [19] S Raagoalan an V Vazran Prmal-Dual RNC Aroxmaton Algorthms for Set Cover an Coverng Integer Programs SIAM Journal on Comutng, 28: , 1998 [20] T Roughgaren an E Taros How Ba s Selfsh Routng? In Proc of the 41 th Sym on Founatons of Comuter Scence (FOCS), ages , 2000 [21] N Young Ranomze Rounng wthout Solvng the Lnear Program In Proc of the 6th Symosum on Dscrete Algorthms (SODA), ages , 1995 [22] N Young Sequental an Parallel Algorthms for Mxe Packng an Coverng In Proc of the 42 n Symosum on Founatons of Comuter Scence (FOCS), ages , 2001