Stacklbrg stratgis for slfish routing in gnral multicommodity ntworks Gorg Karakostas Stavros G. Kolliopoulos Abstract A natural gnralization of th slfish routing stting ariss whn som of th usrs oby a cntral coordinating authority, whil th rst act slfishly. Such bhavior can b modld by dividing th usrs into an α fraction that ar routd according to th cntral coordinator s routing stratgy (Stacklbrg stratgy), and th rmaining 1 α that dtrmin thir stratgy slfishly, givn th routing of th coordinatd usrs. On sks to quantify th rsulting pric of anarchy, i.., th ratio of th cost of th worst traffic quilibrium to th systm optimum, as a function of α. It is wll-known that for α = 0 and linar latncy functions th pric of anarchy is at most 4/3 [20]. Ifα tnds to 1, th pric of anarchy should also tnd to 1 for any rasonabl algorithm usd by th coordinator. W analyz two such algorithms for Stacklbrg routing, LLF and SCALE. For gnral topology ntworks, multicommodity usrs, and linar latncy functions, w show a pric of anarchy bound for SCALE which dcrass from 4/3 to1asα incrass from 0 to 1, and dpnds only on α. Up to this work, such a tradoff was known only for th cas of two nods connctd with paralll links [18], whil for gnral ntworks it was not clar whthr such a rsult could b achivd, vn in th singl-commodity cas. W show a wakr bound for LLF and also som xtnsions to gnral latncy functions. Th xistnc of a cntral coordinator is a rathr strong rquirmnt for a ntwork. W show that w can do away with such a coordinator, as long as w ar allowd to impos taxs (tolls) on th dgs in ordr to str th slfish usrs towards an improvd systm cost. As long as thr is at last a fraction α of usrs that pay thir taxs, w show th xistnc of taxs that lad to th simulation of SCALE by th tax-payrs. Th xtnsion of th rsults mntiond abov quantifis th improvmnt on th systm cost as th numbr of tax-vadrs dcrass. Dpartmnt of Computing and Softwar, McMastr Univrsity, 1280 Main st. W., Hamilton, Ontario L8S 4K1, Canada. Rsarch supportd by an NSERC Discovry Grant and MITACS; (karakos@mcmastr.ca). Dpartmnt of Informatics and Tlcommunications, National and Kapodistrian Univrsity of Athns, anpistimiopolis Ilissia, Athns 157 84, Grc; (www.di.uoa.gr/ sgk).
1 Introduction W study th prformanc of a ntwork shard by noncooprativ nonatomic usrs. Evry slfish usr nds to rout an infinitsimal amount of flow from a spcifid origin to a spcifid dstination nod. Lt f b a flow vctor dfind on th ntwork paths, which dscribs a givn traffic pattrn according to th standard multicommodity flow convntions. Evry path has a latncy function l (f) which xprsss th disutility (dlay) xprincd by all usrs on th path du to th aggrgatd flow of all usrs using som dg of. Each slfish usr wants to choos a minimumlatncy path from hr origin to hr dstination nod. Th usr intraction is modlld by studying th systm in th stady stat capturd by th classic traffic quilibrium concpt [24]. Th traffic quilibrium is charactrizd by th Wardrop principl: for vry origin-dstination pair (s i,t i )th disutility on vry usd s i t i path is qual and lss than or qual to th disutility on any unusd s i t i path. Hnc, in quilibrium no usr has an incntiv to unilatrally switch paths. Thr is a larg litratur on traffic quilibria in transportation scinc, s [19]. Slfish bhavior inducs infficincy from th systm prspctiv. Motivatd by dcntralizd data ntworks, Koutsoupias and apadimitriou [13] wr th first to propos as a masur of this infficincy th worst-possibl ratio btwn th systm cost of an quilibrium and of an optimal routing dsignd by a cntral coordinator. This ratio is calld th th pric of anarchy. In th contxt of slfish routing, w dfin th systm (social) cost as th total latncy of th usrs. Th pric of anarchy for slfish routing was studid in th sminal papr by Roughgardn and Tardos [20]. Thy showd that for linar latncy functions th pric of anarchy is at most 4/3, and this is tight. For arbitrary continuous latncy functions th pric of anarchy is unboundd [20]. Svral othr rsults hav pinpointd th pric of anarchy ρ(l) for various familis L of latncy functions [17, 5, 19]. S th rcnt survy by Roughgardn [15] for a comprhnsiv ovrviw. aramtrizing th pric of anarchy solly by th latncy typ is lgitimat: undr mild assumptions th pric of anarchy is indpndnt of th ntwork topology [17]. Ths rsults rfr to on xtrm of slfish routing, namly to th cas whr all usrs ar slfish. Th othr xtrm is th systm optimum whr all usrs ar coordinatd and follow th prdtrmind optimal routing. Th natural qustion that ariss thn is: what happns whn only a fraction of th usrs ar slfish, whil th rst follow a prdtrmind policy? Ar thr such policis that can always improv th pric of anarchy, givn that a non-zro fraction of usrs can b coordinatd? If so, how dos th improvmnt dpnd on this fraction? For xampl, if th improvmnt is insignificant vn whn almost all usrs ar coordinatd, thn such a policy is obviously of littl valu. Issus lik ths ar important for ral ntworks [12], sinc in gnral, it is quit possibl that thy do not fall in on of th two xtrms, but that thir usrs ar a mixtur of slfish and coordinatd ons. As w shall s, surprisingly littl is known for such ntworks. In fact, th only cas that has bn thoroughly studid, vn for th cas of linar latncy functions, is th cas of a ntwork with two nods connctd by a numbr of paralll links [18]. To our knowldg, th currnt work is th first that dals with th issus abov for gnral topology ntworks, and with multipl origin-dstination pairs for th usrs (multicommodity cas). Stacklbrg routing. Our main rsults dal with Stacklbrg routing, a notion first proposd by Korilis, Lazar and Orda [12]. An α fraction of th usrs ar coordinatd and th rst ar slfish. Th coordinatd usrs ar controlld by a coordinator who assigns thm to routs computd by an algorithm of choic. This algorithm is th Stacklbrg policy. Lt f b th corrsponding flow vctor output by th algorithm. Th rmaining 1 α fraction of th usrs choos paths slfishly by taking into account th spcifid routs of th coordinatd usrs: if th slfish usrs rach a traffic 1
pattrn x, thy xprinc latncy l (x + f) onapath. Th concpt is inspird by Stacklbrg gams (s,.g., [2]) whr playrs ar asymmtric and ar dividd into ladrs and followrs. Th followrs ract rationally (in our trms slfishly) to th stratgis imposd on thm by th ladrs. An important diffrnc btwn Stacklbrg gams and Stacklbrg routing is that in th formr stting, ach ladr is slfishly intrstd in hr own individual utility. In Stacklbrg routing, coordinatd usrs aim to improv th social cost. A givn Stacklbrg policy σ inducs an associatd quilibrium in which th Wardrop principl holds for vry slfish usr. This is a Stacklbrg quilibrium. Givn a Stacklbrg policy σ, th worst-possibl ratio btwn th cost of a Stacklbrg quilibrium, and th minimum total latncy is an xprssion that should dpnd on α, and w call it th pric of anarchy curv of σ. For convninc, w trat th pric of anarchy curv intrchangably as a scalar (if on thinks of α as fixd) and as a function (if on thinks of α as variabl). Lt L b th family of latncy functions at hand. Clarly (i) th curv of any policy σ passs through th points (0, ρ(l)) and (1, 1). Concivably, for any rasonabl Stacklbrg policy (ii) th curv also has to b a continuous nonincrasing function of α. W call a curv fulfilling Conditions (i) and (ii) normal. rvious rsults on Stacklbrg routing. As mntiond abov, rathr littl is known for Stacklbrg routing. Roughgardn [18] dfind two natural Stacklbrg policis SCALE and Largst Latncy First (LLF). SCALE simply sts th flow on vry path qual to α tims th optimal flow. LLF in th contxt of paralll links ordrs th links in trms of thir latncy in th optimal solution and saturats thm on-by-on, from largst to smallst, until thr ar no cntrally controlld usrs rmaining. Roughgardn [18] not only obtaind normal curvs for LLF on paralll links but h also provd th optimality of LLF for such ntworks with linar latncy functions. Mor spcifically h obtaind a 4/(3 + α) pric of anarchy for linar latncy functions and a 1/α bound for gnral latncy functions. Both bounds ar tight [18]. For non-linar latncy functions, thr ar xampls of four-nod ntworks whr it is impossibl to achiv th 1/α bound (roposition B.3.1 in [16]). On multicommodity ntworks th prformanc can b arbitrarily bad for latncy functions with ngativ cofficints [19], but w do not considr such functions hr. Obtaining a 4/3 ɛ guarant for linar latncis or a bound wakr than 1/α for gnral latncy functions ar mntiond in [18] and [15], rspctivly as opn problms, vn for th singl-commodity cas. Our rsults. This papr stablishs for th first tim th xistnc of a normal curv for Stacklbrg routing with linar latncy functions on gnral multicommodity ntworks. This should b contrastd with th arlir rsults that applid only to th singl-commodity paralll links ntwork [18]. Mor spcifically, w analyz SCALE and a vrsion of LLF which w call strong LLF (cf. Sc. 2) for linar latncy functions. For SCALE, our pric of anarchy curv is 4 3 X 3 whr X = (1 1 α)(3 1 α+1) 2. S Fig. 2 for a plot. Hnc w show that a vry simpl policy to implmnt (SCALE) achivs a vry significant improvmnt of th pric of anarchy for th linar latncy 1 α+1 cas. In viw of th simplicity of th policy (SCALE) that achivs such an improvmnt, it is rathr surprising that virtually no progrss has bn mad sinc [18] appard. On possibl xplanation is th fact that our analysis xamins th ntwork as a whol, and avoids th dg-by-dg bounding that has bn th stapl of classic rsults on th pric of anarchy,.g., [20, 17, 5]. Th tchnical machinry that maks our approach possibl is th analysis of slfish routing by rakis [14]. W us th spcial structur of SCALE in ordr to rlax on of th hard constraints that th bound of [14] nds to satisfy. Morovr, w dmonstrat that our uppr bound analysis for SCALE is 2
narly tight for vry α, by giving a st of linar latncy functions on th Brass graph for which SCALE prforms vry clos to our uppr bound. Mor dtails appar in Sction 3. Our analyss of SCALE and strong LLF can b xtndd to gnral latncy functions using th concpt of Jacobian similarity [14], a notion adaptd from Hssian similarity in intrior point mthods [22, 14]. Th lattr approach, which is outlind in Sction 5, has th potntial of yilding bounds that ar spcific to individual familis of latncy functions. For paralll links Roughgardn [18] givs an xampl whr LLF outprforms SCALE. On th othr hand on th instanc of th Brass paradox (cf. Sction 4), SCALE outprforms LLF. Finally on our hard xampl for SCALE (cf. Sction 3) which shars th sam undrlying graph with th Brass paradox, LLF outprforms SCALE. Hnc th two policis ar incomparabl, in th sns that no policy dominats th othr on all possibl inputs. Th thr typs of instancs w dscribd suggst that in ordr to achiv th bst possibl curv, both th ntwork topology and th latncy functions mattr. Although this appars to b in stark contrast with th indpndnc of th pric of anarchy for slfish routing from th ntwork topology [17], it should not com as a surpris: Stacklbrg routing has an algorithmic componnt which is lacking from traditional slfish routing. Slfish routing with tax vasion. Th xistnc of a cntral coordinator is a rathr strong rquirmnt for a ntwork. A wll-studid altrnativ for mitigating th ffcts of slfishnss gos back to th origins of traffic quilibria (s [3]): impos montary taxs (tolls) pr-unit-of-flow on th dgs. Slfish usrs ar conscious both of th travl latncy and th montary cost on a path. It is by now known that such taxs xist vn whn usrs ar htrognous, i.., thy ar dividd into classs whr ach class has a diffrnt snsitivity lvl towards paying taxs [25, 10, 8]. S also th work in [4]. In th sam way Stacklbrg routing stablishs partial control ovr th usrs by cntrally coordinating only an α fraction of thm, w xamin whthr similar ffcts can b achivd whn only an α fraction of th usrs pay taxs. Equivalntly, on can think of th rmaining 1 α fraction of th usrs as tax-vadrs having a zro snsitivity to taxs. In Sction 6 w show that thr is a st of dg taxs so that th pric of anarchy obtaind is qual to th pric of anarchy of th SCALE policy. As th fraction of law-abiding citizns incrass from 0 to 1, th systm cost is improvd accordingly. This work was first publishd as a McMastr Univrsity tchnical rport [11]. Indpndntly of our work, Corra and Stir-Moss [6] and Swamy [23] hav obtaind rsults for gnral latncy functions. Othr rcnt work on Stacklbrg routing includs [9, 21]. 2 rliminaris A dirctd ntwork G =(V,E), with paralll dgs allowd, is givn on which a st of usrs ach want to rout an infinitsimal amount of flow (traffic) from a spcifid origin to a dstination nod in G. Usrs ar dividd into k classs (commoditis). Th dmand of class i =1,...,k, is d i > 0 and th corrsponding origin dstination pair is (s i,t i ). A fasibl vctor x is a valid flow vctor (dfind on th path or dg spac as appropriat) that satisfis th standard multicommodity flow convntions and routs dmands d i for vry commodity i. W us fasibl flow vctors throughout th papr to charactriz traffic pattrns. W us K to dnot th (convx) st of all th fasibl vctors. As in [18] w assum sparabl costs: ach dg is assignd a nonngativ, nondcrasing latncy function l (f ) that givs th dlay xprincd by any usr on du to congstion causd 3
by th total flow f that passs through. W also assum th standard additiv modl: for a path, l (f) = l (f ). Stacklbrg policis can b classifid as wak or strong [19]. A wak Stacklbrg policy controls dmand αd i from ach commodity for a paramtr α (0, 1). A strong Stacklbrg policy givs mor powr to th coordinator: h can control as much dmand from ach commodity as h ss fit undr th condition that th total dmand controlld quals α k i=1 d i. In th singl-commodity cas, strong and wak policis coincid. Lt f b th flow vctor of th slfish usrs and f th stratgic flow of th coordinatd usrs. Th additiv cost modl maks it asy to viw our flows somtims as path flows and somtims as dg flows. Th systm cost of fasibl flow x is dfind as x l (x). Lt C q := (f + f )l (f + f) dnot th cost at quilibrium. W dnot by a flow that optimizs th systm cost and th optimum itslf as C opt, i.., C opt := l ( ). Th SCALE policy is a wak on dfind by stting f := αf opt for vry. Not that this quivalnt to stting f := α for all paths. Th strong LLF policy imposs a total ordr on th paths usd by all commoditis basd on nondcrasing l ( ) valus and braking tis arbitrarily. It thn saturats paths on by on from th largst latncy to th smallst until th total dmand of th controlld usrs quals α k i=1 d i. By saturating a path,wmanthat f :=. Not that th last path in th ordring that is assignd positiv flow may carry lss than. is assignd as much flow as ndd to rach a total of α k i=1 d i. 3 Linar latncy functions In this sction, w xamin th cas of linar (or affin) latncy functions. That is, for all, l (f )=a f + b, with a,b 0. A first attmpt. Existing uppr bounds on th pric of anarchy dpnd to a larg xtnt on th bhavior of th latncy function on individual dgs. This is what w call th dg-by-dg approach. Th dfinitions of th anarchy valu α(l) by Roughgardn [17] and th β(l) paramtr by Corra t al. [5], whr L is a class of latncy functions, ar particularly rvaling in this contxt. In ordr to gain intuition into th problm w initially try an analysis of Stacklbrg routing using similar argumnts. W assum that th coordinator uss th SCALE policy. Lt β = β(l). Th dfinition of β implis that for any dg l (f + f ) β(f + f )l (f + f )+f opt l (f opt ). (1) W can gt a bttr uppr bound whn dg is undrutilizd by th slfish usrs. Dfin an dg to b light if f c f for a suitabl c>0. An dg which is not light is calld havy. Dfin δ [0, 1] such that light f opt l(f opt )=(1 δ)c opt and havy f opt l(f opt )=δc opt. Lmma 1 Lt c, δ b dfind as abov. Thn for a gnral ntwork with linar latncy [ functions ] and a fraction α of coordinatd usrs, SCALE achivs a pric of anarchy Cq C opt 4 3 1 α2 4 (1 δ). roof: Sinc th l ( ) functions ar nondcrasing, w hav that for th light dgs f opt l (f + f ) f opt l (α(1 + c)f opt ) f opt l (f opt ) (2) light light light 4
undr th assumption that α(1 + c) 1. For havy dgs, (1) yilds that f opt l (f + f ) β (f + f )l (f + f )+ f opt l (f opt ). (3) havy havy havy For linar latncy functions it is wll-known that β 1/4 [5], hnc latr w will us th valu β =1/4. Th analysis is affctd by th amount of cost that pays on th light and havy dgs, rspctivly. From now on, w mak us of th assumption that th dg latncy functions ar linar. By summing (2), (3) ovr all th dgs w obtain that f opt l (f + f ) βc q β (f + f )l (f + f )+C opt βc q βα 2 light light l ( )+C opt βc q +[1 βα 2 (1 δ)]c opt (4) whr th scond inquality is du to th fact that th l s ar linar and α 1. Lt ˆf := f b th flow that rmains if w rmov flow f from th optimal flow. Not that ˆf is a flow that satisfis dmands ˆd i d i, for all commoditis i =1,...,k. In th spcial cas, whr f = α, ˆd i =(1 α)d i, for i =1,...,k. Thn from th variational inquality l (f + f )(x f ) 0, x = flow that satisfis dmands ˆd i,i=1,...,k (5) that f satisfis as a traffic quilibrium [7], w gt th following for x := ˆf: C q = (f + f )l (f + f ) ( ˆf + f )l (f + f )= l (f + f ). (6) By using (6) in (4), and rplacing β by 1/4 w gt th lmma. If δ<1, Lmma 1 yilds a normal curv. Hnc w would lik to hav δ as small as possibl. Th paramtr c must satisfy α(c +1) 1, and, by dfinition, th biggr c is th smallr δ potntially is. Thrfor w should pick c := 1 α α. Not, though, that, vn with this choic of c, itmaystillbthcasthatδ =1. Inthiscas, th bound w hav calculatd is not bttr than th classic 4/3 that holds whn no Stacklbrg policy is usd. Th dg-by-dg approach ld us to bliv that, at last for SCALE, th asy cas is whn pays a substantial fraction of its cost on dgs that ar undrutilizd by th slfish usrs. Aftr complting our uppr and lowr bound drivations w will hav dmonstratd instad that a small δ is th difficult cas. An improvd uppr bound for SCALE. In ordr to prov our main rsult for SCALE and linar latncy functions w will hav to dpart from th approach usd abov and xamin th ntwork as a whol. If for vry dg, th latncy function is of th form l (f )=a f, it is wll known that th pric of anarchy is 1. Th infficincy of slfish routing for gnral affin functions could b attributd, in a sns, to th xistnc of load-indpndnt latncy trms b > 0insom l () functions. Our analysis xploits th fact that th SCALE and LLF policis dcras th total influnc of ths trms on th social cost at quilibrium. 5
Thorm 1 For gnral multicommodity ntworks with linar latncy functions and a fraction α of usrs coordinatd by th SCALE policy, th pric of anarchy is boundd as follows C q C opt 4 3 X 3 (1 1 α)(3 1 α +1) whr X = 2. 1 α +1 roof: A lmma of rakis [14] provids us with an important tool for our analysis. It was originally drivd to dal with asymmtric and non-sparabl cost functions. Considr th latncy function as a vctor-valud function L : R m + Rm +, with L(f) =Gf + b and m = E. In our cas, G is a diagonal matrix containing th a s and b T =[b ] E. In this notation C q =[L(f + f)] T (f + f). From th proof of Thorm 3 in [14] w can abstract away th following fact, which isolats th contribution of th flow-dpndnt part of th latncy to th total cost: Lmma 2 [14] Givn th notation abov, lt f K b a vctor that satisfis [L(f)] T ( f) 0. [ ] a1 G For any scalars a 1,a 2 0 that satisfy T GT 2 G 2 a 2 G T 0 w hav that f T G T a 1 f T Gf + a 2 ( ) T G. In our cas, G is symmtric, and G 0sincG is a diagonal matrix with ntris G[, ] =a 0, for all E. In this cas, Lmma 2 can b rducd to a mor mallabl form, which is implicit in [14]: Lmma 3 [14] If for all dgs, l (f )=a f + b with a,b 0, thn for any a 1,a 2 0 that satisfy a 1 a 2 1/4 C q a 1 a (f + f ) 2 + a 2 a (f opt ) 2 + b. roof: For vry a 1,a 2 0, w show that th smidfinit constraint of Lmma 2 is quivalnt to th following holding for vry 2m-dimnsional vctor X =[X 1 X 2 ] T,whrX 1,X 2 ar m- dimnsional vctors: [X 1 X 2 ] [ a1 G G 2 G 2 a 2 G ] [ X1 X 2 If a 1 a 2 1 4, thn th following holds for any two numbrs x 1,x 2 : ] 0 a 1 X1 T GX 1 + a 2 X2 T GX 2 X1 T GX 2 0. (7) a (a 1 (x 1 )2 + a 2 (x 2 )2 x 1 x 2 ) a ( a 1 x 1 a 2 x 2 )2 0. (8) By considring th coordinats x 1,x 2 of X 1,X 2 sparatly, applying (8) to thm, and finally adding ovr all dgs, w gt (7). W show that for f = f + f th scond hypothsis of Lmma 2 is also satisfid. By (5) Inquality (9) yilds that [L(f + f)] T (( f) f ) 0 [L(f + f)] T ( (f + f)) 0. (9) C q =[L(f + f)] T (f + f) [L(f + f)] T =(f + f) T G T + b T. By using Lmma 2 to upprbound th right-hand sid th proof is complt. 6
Not that in ordr to apply Lmma 3 w ar fr to pick a 1,a 2 subjct to th constraints of th Lmma. This is xactly th point whr th SCALE policy hlps us to gt a bttr bound for th pric of anarchy: whil [14] also gts to pick a 1,a 2 subjct to ths constraints and th xtra constraint a 2 1, w will not hav to oby th lattr constraint. Th dtails of th proof follow. W rwrit th right-hand sid of Lmma 3 in trms of paths: C q a 1 (f + f ) a (f + f )+a 2 = a 1 C q a 1 (f + f ) b + a 2 Lt := a 1 (f + f ) b + Thn (10) can b writtn as (1 a 1 )C q a 2 a a b. + + b b. (10) a +. (11) Sinc w hav assumd that a 1 0andb 0, for all E, w gt that a 1 f b + b. (12) By th dfinition of SCALE, w hav that on vry, f = α. Thrfor (1 a 1 α) b. (13) From Equations (11), (13), if w rquir that a 1 1, w hav that C q C opt max{a 2, 1 αa 1 } 1 a 1. To obtain th bst possibl pric of anarchy w solv th program: By stting min max{a 2, 1 αa 1 } s.t. 1 a 1 a 1 a 2 1 4 a 1 1 a 1,a 2 0 a 1 := 1 1 α,a 2 := 1+ 1 α 2 all constraints ar satisfid (not that a 1 1 2 ), th two xprssions in th max of th objctiv function bcom qual, and Thorm 1 follows. 7
l(x) =x u l(x) =1 s l(x) =0 t l(x) =1 v l(x) =x Figur 1: Th Brass paradox instanc. A narly tight xampl for SCALE. Considr th graph of th Brass paradox (Fig. 1). This is a dirctd graph with four vrtics s, t, u, v and fiv dgs (s, u), (u, t), (u, v), (s, v), (v, t). Thr is a singl commodity to b routd from s to t of total dmand 1. W st th latncy of dg (u, v) to b idntically qual to zro. For th othr dgs w dfin a latncy function l(x) which α+2 1 α is paramtrizd by α. For (s, u), (v, t) th latncy is 2 α 2 x, and th rmaining two dgs 1 α hav latncy x + 2 1 α 2 α 2. On can vrify that in th optimal solution th uppr and lowr 1 α paths carry flow 1/2 ach, thrfor C opt = 1 2 + α+6 1 α 2(2 α 2. In th Stacklbrg quilibrium th 1 α) coordinatd usrs push α/2 units of flow along ach of th paths s u t and s v t. Th slfish usrs push 1 α units of flow along th path s u v t. Th rsulting pric of anarchy is α 2 1 α+4 1 α 1+2, and this lowr bound is at most an additiv factor of 0.0323 away from our 1 α uppr bound (this maximum gap happns for α =0.81...). S Fig. 2. Rcall th quantity δ w dfind arlir. In th xampl just producd, on can vrify that th fraction of C opt that is paid on th havy dgs (s, u), (v, t) is a+2 1 α 2+4, which for vry α [0, 1] is 1 α lss than 1/2. Morovr, in th cas whr δ is at last som constant fraction, on can modify th proof of Thorm 1 to obtain an improvd pric of anarchy. In particular, th right-hand sid of (11) can b writtn as a sum of two parts, on for th havy dgs and on for th light ons. Th us of th additiv modl maks asy th transition from a a sum ovr paths to a sum ovr dgs and vic vrsa. (1 a 1 )C q a 2 a (f opt ) 2 + h + a 2 a (f opt ) 2 + l, (14) whr and havy h := a 1 l := a 1 havy (f + f )b + light havy (f + f )b + light light f opt b b. For th contribution of th light dgs to th right-hand sid of (14) on procds as bfor by using that f 0, whil for th contribution of th havy dgs on uss that f >c f. Th 8
improvd uppr bound is obtaind by minimizing max{a 2,1 a 1 (αc+α)}δ+max{a 2,1 αa 1 }(1 δ) 1 a 1 subjct to th constraints a 1 a 2 1 4, a 1 1, a 1,a 2 0. Th quantity c>0 is th on from th dfinition of th light dgs. By th prcding argumnts and th plot in Fig. 2 w conclud that SCALE hits its worst-cas prformanc whn δ is rathr small. This occurs whn th optimal solution pays most of its cost on th light dgs, i.., dgs that ar undrutilizd by th slfish usrs. Natural as this insight is, it appars to contradict th bound of Lmma 1 which was obtaind by th dg-by-dg approach. Th apparnt contradiction is rsolvd whn on notics that th bound of Lmma 1 is a wak on: vn for δ =0, it is largr than th bound of Thorm 1 for all α 0.919 and bcoms only marginally bttr for largr valus of α. An uppr bound for strong LLF. Lt a path b good if it is usd by th coordinatd usrs as dictatd by strong LLF. Thrfor, path is good iff f > 0. aths that ar not good ar calld bad. Thr is a λ [0, 1] such that bad [ (a f opt + b )] = (1 λ)c opt and good [ (a f opt + b )] = λc opt. Thorm 2 Lt λ b dfind as abov. Thn for gnral multicommodity ntworks with linar latncy functions and a fraction α of usrs coordinatd by th strong LLF policy, th pric of anarchy is boundd as follows: { 4 C q 3, if λ [0, 1 3 ) 2(1 λ) C 2 opt 2 λ, if λ [ 1 4λ 3λ 2 3, 1]. roof: By dcomposing th right hand sid of (12) into two parts, on for th good and on for th bad paths, w gt a 1 f b + good good b a 1 f b + bad bad b. (15) Undr th LLF policy, all good paths but on ar saturatd, maning = f. W can rplac th offnding path Π (i.., th on on which 0 < f Π < Π ) by two copis of th sam path in th flow dcomposition of, both with th sam latncy. On copy gts flow f Π out of a total of opt Π and is includd in th st of good paths, and th othr copy gts th rst fπ f Π and is includd in th bad paths. With this nw path st, all good paths ar saturatd, i.., = f. All of th abov can b sn as just a chang of th st of indics usd in th notation for th paths of flow. W us this nw st of indics (dcomposition) of from now on. Thn a 1 f b + b =(1 a 1 ) b, good good good and (15) bcoms (1 a 1 ) good b a 1 bad f b + bad b. (16) Rcall that on a bad path, f =0. If in addition w rquir that a 2 1, quations (11), (16) yild (1 a 1 )C q a 2 a f opt +(1 a 1 ) b + b good bad 9
bad which, in turn, implis that [ (a f opt + b )] + max{a 2, 1 a 1 } [ (a f opt + b )] good C q C opt 1 λ +max{a 2, 1 a 1 }λ 1 a 1. (17) W will pick a 1,a 2, subjct to all th constraints on thm w hav assumd so far, so that w gt th smallst possibl uppr bound on th pric of anarchy from (17). First w assum that a 2 1 a 1 and thrfor (17) implis that Cq C opt 1 (1 a 2)λ 1 a 1. Hnc w would lik to minimiz 1 (1 a 2)λ 1 a 1 subjct to th constraints a 2 1 a 1, a 1 a 2 1 4, 0 a 1,a 2 1. For th cas λ [ 1 3, 1] th minimum is achivd by picking a 1 := 4λ 3λ 2 λ 4(1 λ), a 2 := 1 4a 1, whil for λ [0, 1 3 ) th minimum is achivd by picking a 1 := 1 4, a 2 := 1. If w assum that a 2 < 1 a 1, thn w do not gt a bttr uppr bound. Thrfor our analysis of LLF yilds th uppr bounds givn in th statmnt of Thorm 2. Sinc LLF picks th most xpnsiv paths of to saturat, and f satisfis a fraction α of th ovrall dmand, w hav that λ α (not that in th dfinition of λ abov, ach flow path in th dcomposition pays th latncy of th path du to th whol flow through th dgs of th path). Th uppr bound for th pric of anarchy computd abov is a dcrasing function of λ, hnc w can rplac λ with α in it, and still gt valid uppr bounds that dpnd only on α. If λ>αour analysis yilds a pric of anarchy bound that is vn bttr. 4 lots In this sction, w provid various plots of th curvs for th linar latncy functions mntiond arlir. All th plots wr obtaind using Gnuplot. Our hard xampl of Sction 3 was a modification of th Brass paradox instanc. Th xact Brass paradox instanc (s Fig. 1) is dfind on th sam undrlying four-nod ntwork but with th following latncy functions. On dg (u, v) th latncy is idntically qual to zro; on dgs (s, u), (v, t) l(x) =x andonthrmainingdgs l(x) =1. Thsourciss, th dstination is t, and th dmand to b routd is 1. On can asily vrify that on this instanc th pric of anarchy curv of SCALE is 4/3 (1/3)( α 2 ). For LLF, both paths usd by th optimum solution hav qual latncy. Rgardlss of ti braking, th curv of LLF is 4/3 (1/3)( 2 )forα 1/2 and4/3 (4α/3 2 /3 1/3) for α>1/2. Figur 2 shows th uppr and lowr bounds w obtaind in Sction 3. Figur 3 shows our LLF uppr bound whn λ = 1 in Thorm 2 and th corrsponding lowr bound obtaind from th Brass paradox. Finally, Figur 4 compars th xisting lowr bounds for th two policis on gnral ntworks, by drawing thir prformanc on th Brass paradox dfind in th prvious paragraph. For comparison rasons, in th sam plot w also giv th prformanc of SCALE in our narly tight xampl from Sction 3. Not that th lattr is a significantly strongr lowr bound for SCALE than th lowr bound obtaind by th prformanc of th policy on th Brass paradox instanc dfind abov. It is worth rmarking that thr is a valu of α for which our uppr bound for SCALE from Sction 3 is vry clos to th lowr bound for both policis. For α =1/2 our uppr bound is within an additiv 0.027 factor from 7/6 which is th prformanc of LLF on th Brass paradox instanc. 10
1.35 uppr bound lowr bound 1.3 ric of Anarchy for SCALE 1.25 1.2 1.15 1.1 1.05 1 0 0.2 0.4 0.6 0.8 1 Figur 2: Our uppr and lowr bounds for SCALE, as obtaind in Sction 3 plottd as functions of th fraction α of th usrs that ar coordinatd. α 1.35 uppr bound lowr bound 1.3 ric of Anarchy for strong LLF 1.25 1.2 1.15 1.1 1.05 1 0 0.2 0.4 0.6 0.8 1 Figur 3: Our uppr bound for strong LLF, as obtaind in Sction 3, undr th furthr assumption that λ = α. Th lowr bound is th xact prformanc of LLF on th instanc of th Brass paradox. α 11
1.35 1.3 SCALE our instanc LLF Brass SCALE Brass 1.25 All lowr bounds 1.2 1.15 1.1 1.05 1 0 0.2 0.4 0.6 0.8 1 Figur 4: Th prformanc of th SCALE policy on our hard instanc from Sction 3 vs. th prformanc of th LLF and SCALE policis on th instanc of th Brass paradox. Obsrv that SCALE outprforms LLF on th lattr instanc. 5 Gnral latncy functions Th analysis of th linar cas can b xtndd to gnral latncy functions that satisfy crtain proprtis. Rcall th vctor-valud function notation L() for th latncy function. According to rakis [14], L(x) satisfis th Jacobian similarity proprty if it has a positiv smidfinit Jacobian matrix ( L(x) 0, for vry x K) andforallw R m, for all x, x K, thr xists A 1 satisfying 1 A wt L(x)w w T L( x)w Aw T L(x)w. Th concpt of Jacobian similarity originats from th Hssian similarity notion in intrior point mthods (s.g., [22]). Th valu A is known to b indpndnt of th matrix dimnsion, for positiv dfinit L(x). If th Jacobian matrix is strongly positiv dfinit, i.., it has ignvalus boundd away from zro, thn A is uppr-boundd by α max x K λ max (S(x)) min x K λ min (S(x)), whr S(x) = L(x)+ L(x)T 2.IfL(x) =Gx + b with G 0, thn A =1[14]. In our cas, L(x) is a diagonal matrix with th diagonal ntry corrsponding to dg bing qual to dl(x) dx. Such a matrix is positiv smidfinit if ths drivativs ar nonngativ for all x K. This is th natural and common assumption that th latncy functions ar incrasing. Thrfor our rsults blow will assum only that th latncy functions ar incrasing. Gnralizing th arlir rmarks on th affin cas w can abstract th following from rakis [14]: 12
Lmma 4 [14] If (i) for all dgs, l (f ) is a continuously diffrntiabl function with dl(f) df 0, and l (0) 0 for all f K and (ii) th matrix L(x) satisfis th Jacobian similarity proprty for som A 1, thn C q a 1 A (f + f )[l (f + f ) l (0)] + C opt +(a 2 1)A f opt [l (f opt ) l (0)] for any a 1,a 2 0 that satisfy a 1 a 2 1/4. W can dfin Z := a 1 A (f + f )l (0) + f opt l (0), and th lmma yilds that (1 a 1 A)C q [(a 2 1)A +1] f opt [l (f opt ) l (0)] + Z. (18) For th SCALE policy Z (1 αa 1 A) f opt l (0), and thrfor w can obtain that (1 Aa 1 )C q [(a 2 1)A +1]C opt A(αa 1 + a 2 1) l (0). undr th conditions a 1 a 2 1/4, a 1 1/A, a 1,a 2 0. W distinguish two cass: 1. αa 1 + a 2 1:Inthiscas,whavthat (1 Aa 1 )C q [(a 2 1)A +1]C opt. Hncwarlookingfora 1,a 2 that solv th following minimization problm: min Aa 2 +1 A s.t. 1 Aa 1 αa 1 + a 2 1 a 1 a 2 1 4 a 1 1 A a 1,a 2 0 If w st a 2 =1/4a 1 thn w must hav that a 1 / ( 1 1 α Th objctiv function is incrasing for [ a 1, 1+ 1 α ). A A 2 +4(1 A), A + A 2 +4(1 A) 4(A 1) 4(A 1) and dcrasing for th othr valus of a 1 in [0, 1 A ). If A A 2 +4(1 A) 4(A 1) a 1 := A A 2 +4(1 A) 4(A 1) othrwiswsta 1 := 1 1 α. If w st a 2 =1 αa 1, thn w must hav If 1 1 α a 1 ( 1 1 α, 1+ 1 α ). ] 1 1 α, thn w st 1 A thn w st a 1 := 1 1 α, ls th problm is infasibl. So this cas dos not add somthing nw to th prvious bound. 13
2. αa 1 + a 2 1: In this cas, sinc l (0) l (f opt ), for all E, w hav that (1 Aa 1 )C q [1 Aαa 1 ]C opt. Hncwarlookingfora 1,a 2 that solv th following minimization problm: min 1 Aαa 1 1 Aa 1 s.t. αa 1 + a 2 1 a 1 a 2 1 4 a 1 1 A a 1,a 2 0 This cas producs th sam bounds as th prvious on. Hnc w gt th following thorm for gnral (incrasing) latncy functions and th SCALE stratgy: Thorm 3 Lt A 1 b th Jacobian similarity proprty paramtr for th latncy functions matrix L(x), and α th coordinatd fraction of flow that follows th SCALE stratgy. Thn th pric of anarchy is uppr-boundd as follows: whr C q A +4(1 A)a 1 C opt 4a 1 (1 Aa 1 ) 1. If A A 2 +4(1 A) 4(A 1) 1 1 α thn a 1 = A A 2 +4(1 A) 4(A 1). 2. If A A 2 +4(1 A) 4(A 1) > 1 1 α thn a 1 = 1 1 α. Not that th bound of Thorm 3 dpnds only on th family of latncy functions (through A) and α. Also obsrv that for A slightly gratr than 1, part 1 of Thorm 3 holds. In that cas, th bound dpnds only on A. For th cas A = 1, it is asy to s that th analysis coincids with th analysis of th linar latncy functions cas. 6 Th ffct of tax vasion on ntworks So far w hav assumd that th ntwork is subjct to a cntral coordinating authority that can dcid th routing of a fraction α of th ovrall traffic, whil allowing th rst to act slfishly. In this sction, w xplor whthr th sam ffcts can b achivd whn no such cntral authority xists, i.., thr is no notion of ladr and followr in th Stacklbrg sns. Instad w wish to us taxs (tolls) on th dgs of th ntwork assuming that all usrs ar slfish but an α fraction of thm ar still law-abiding tax-paying citizns. Th rmaining 1 α fraction dos not bliv in paying taxs. In this sction, w show that such taxs do xist. W assum throughout th sction that th latncy functions l () ar continuous, incrasing and tak only nonngativ valus. Th flow for vry origin-dstination pair (commodity) i =1,...,k 14
of rat d i in th ntwork is split into two parts: f i corrsponds to th st of tax-payrs with rat αd i and f i corrsponds to th st of tax-vadrs with rat (1 α)d i. Th tax payrs can b htrognous: thy attach an importanc factor a(i) > 0 to thir disutility du to taxation. Lt f := i f i, f := f i i, for all. W ar looking for th xistnc of nonngativ dg taxs b, for all E, such that for vry commodity i (i) th tax-paying usrs f i prciv dg costs l (f + f )+a(i) b, for all E, (ii) th tax-vadrs f i prciv dg costs l (f + f ), and (iii) th b s ar such that th tax-payrs ar forcd to implmnt th SCALE policy. Th lattr mans that at th traffic quilibrium w must hav k i=1 f i = α, E. (19) A ky obsrvation is that if, in addition, w assum that all latncy functions l ( ) arstrictly incrasing, thn conditions (19) ar quivalnt to k i=1 f i α, E. (20) W prov this similarly to Claim 1 in [10]. Assum for th sak of contradiction that som inqualitis in (20) ar strict. Dfin th flow f/α, by routing for ach commodity i, flow qual to f i /α. Thn for vry dg, f /α f opt, and l ( f /α) l (f opt ). By nonngativity, ( f /α)l ( f /α) f opt l (f opt ), E. If for som dg, f /α < f opt, it follows that 0 l ( f /α) <l (f opt ). Thrfor for this particular dg ( f /α)l ( f /α) <f opt l (f opt ). It follows that E ( f /α)l ( f /α) < E f opt l (f opt ), which contradicts th optimality of. W us th framwork of [1, 10] to incorporat constraints (20) into a complmntarity problm that dscribs th traffic quilibrium in our cas. Th complmntarity problm (C) is dfind by constraints (21)-(31): f i ( l (f + αf opt ) + b ū i )=0 i, i (21) a(i) f i ( l (f + α ) u i )=0 i, i (22) l (f + αf opt ) a(i) u i ( ū i ( + b ū i i, i (23) l (f + α ) u i i, i (24) i f i αd i )=0 i (25) i f i (1 α)d i )=0 i (26) i f i αd i i (27) 15
b ( i f i i f i (1 α)d i i (28) αf opt )=0 E (29) i f i α E (30) f i, f i,b,u i, ū i 0,, i (31) To provid intuition w dscrib brifly th maning of th constraints of (C). For dtails th radr is rfrrd to [1, 10]. (C) dfins two traffic quilibria that must b rachd simultanously. Constraints (21), (23), (25), (27), (29), (30) xprss th quilibrium problm for th tax-payrs and constraints (22), (24), (26), (28) th quilibrium problm for th tax-vadrs. Lt us tak a closr look at th quilibrium problm for th tax-payrs. Constraints (21), (23) xprss th Wardrop principl. Th variabl ū i is th common disutility xprincd by all tax-vadrs who blong to commodity i. Constraint (27) nforcs that tax-payrs in usr class i satisfy rat at last αd i. If thy satisfy a rat strictly gratr than αd i, this coms for fr sinc by Constraint (25) ū i must b zro. In [1] it is shown that ths constraints form an xact formulation of th traffic quilibrium problm for th tax-vadrs. Constraints (30) nforc (20). Th Lagrang multiplirs b for (30), apparing in (29), will b th dsird taxs. Similar considrations apply to th quilibrium problm for th tax-vadrs xcpt of cours that for thm thr ar no capacity constraints lik (30). To prov th xistnc of th tax vctor b with th proprtis (i)-(iii) dscribd abov, it is nough to show that (C) has a solution. By using th fact that αf opt is a known constant for vry dg whn is known, it follows from [1] that th complmntarity problm (C )blowwithvariablsf i,u i has a solution (f,u ): f i ( l (f + α ) u i )=0 i, i l (f + α ) u i i, i (C ) u i ( f i (1 α)d i )=0 i i f i (1 α)d i i i f i,u i 0,, i In turn, by using th argumnts from th proof of Thorm 2 in [10] w can show that th following complmntarity problm (C ) with variabls f i,b, ū i is quivalnt to pair of primal and dual linar programs and also has a solution ( f,b, ū ): f i ( l (f + αf opt ) + b ū i )=0 i, i a(i) l (f + αf opt ) + b ū i i, i (C ) a(i) ū i ( i f αd i )=0 i i 16
b ( i f i i f i αd i i αf opt )=0 E i f i α f i,b, ū i 0 E,, i Now it is clar that (f, f,b,u, ū ) is a solution of (C), and w can us taxs b on ach dg to induc th tax-payrs to follow th SCALE policy. Thn all our rsults about th ffcts of SCALE hold also for this stting. If th latncy functions ar non-strictly monoton, w hav obtaind that thr xists on tax-inducd quilibrium in which th tax-paying usrs implmnt th SCALE policy [1, 10]. If th latncy functions ar strictly monoton, in vry tax-inducd quilibrium th tax payrs implmnt SCALE [1, 10]. W summariz our findings in th nxt thorm. Thorm 4 Lt x b th total flow through dg for som traffic assignmnt. If for all E, th functions l () ar strictly monoton, and l () 0, thr is a b R E + such that if an α fraction of th usrs (calld th tax-payrs) xprincs dg disutility l (x )+a(i) b, E, whil th rst xprinc disutility l (x ), for all E, thn th tax payrs induc in quilibrium th flow vctor α. Hr is th flow that minimizs th systm cost x l (x). 7 Discussion rakis [14] drivs th pric of anarchy for non-sparabl asymmtric latncy functions. Thrfor our rsults from Sction 3 ar bound to xtnd to that stting as wll. Thr ar svral issus that ar lft opn. Can on gt a strictly dcrasing curv for LLF? Morovr th diffrnc btwn th uppr and lowr bounds for LLF is currntly considrabl. For SCALE it would b intrsting to clos th rathr small gap that xists btwn our uppr and lowr bounds. It would b intrsting also if on could dtrmin th instancs on which SCALE outprforms LLF and vic vrsa. Finally, and prhaps mor importantly, is thr an optimal Stacklbrg stratgy for gnral multicommodity ntworks? Rfrncs [1] H. Z. Aashtiani and T. L. Magnanti. Equilibria on a congstd transportation ntwork. SIAM Journal of Algbraic and Discrt Mthods, 2:213 226, 1981. [2] T. Başar and G. J. Olsdr. Dynamic Noncooprativ Gam Thory. SIAM, 1999. [3] M.Bckmann,C.B.McGuir,andC.B.Winstn.Studis in th Economics of Transportation. Yal Univrsity rss, 1956. [4] R. Col, Y. Dodis, and T. Roughgardn. ricing ntwork dgs for htrognous slfish usrs. In rocdings of th 35th Annual ACM Symposium on Thory of Computing, pags 521 530, 2003. 17
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