A Stacklbrg Stratgy for Routng Flow ovr Tm Umang Bhaskar Lsa Flschr Ellot Anshlvch Abstract Routng gams ar usd to to undrstand th mpact of ndvdual usrs dcsons on ntwork ffcncy. Most pror work on routng gams uss a smplfd modl of ntwork flow whr all flow xsts smultanously, and usrs car about thr thr maxmum dlay or thr total dlay. Both of ths masurs ar surrogats for masurng how long t taks to gt all of a usr s traffc through th ntwork. W attmpt a mor drct study of how comptton affcts ntwork ffcncy by xamnng routng gams n a flow ovr tm modl. W gv an ffcntly computabl Stacklbrg stratgy for ths modl and show that th compttv qulbrum undr ths stratgy s no wors than a small constant tms th optmal, for two natural masurs of optmalty. 1 Introducton In routng gams, playrs rout a fxd amount of flow n a ntwork. A playr suffrs a cost, whch dpnds on ts routng and th routng chosn by th othr playrs. A flow n a routng gam s an qulbrum flow f no playr can choos a dffrnt routng and rduc ts cost. Routng gams modl a varty of problms, ncludng routng on roads [3, 33], computr ntworks [11, 25, 29], and schdulng tasks on machns [23]. For many masurs of th qualty of a routng, th qulbrum n routng gams s known to b nffcnt compard to a routng whch optmzs th masur. Ths nffcncy s quantfd by th prc of anarchy [26]: th worst rato of th objctv valuatd for th qulbrum flow, to th optmal flow. Thr s consdrabl ntrst n obtanng bounds on th prc of anarchy n routng gams. Playrs n a routng gam hav a bottlnck objctv f a playr s cost s th maxmum dlay on th dgs t uss [5]. Th bottlnck objctv modls applcatons whr a playr s cost dpnds largly on th prformanc of th worst rsourc t uss. Ths objctv gnors th ffct of dlay on dgs bsds th bottlnck {umang,lkf}@cs.dartmouth.du. Partally supportd by NSF grants CCF-0728869 and CCF-1016778. anshl@cs.rp.du. Partally supportd by NSF grants CCF- 0914782 and NtSE-1017932. All mssng proofs ar gvn n th full vrson [8]. dg, whch can lad to th countrntutv stuaton whr playrs fal to dstngush btwn two stratgs whch hav th sam bottlnck, but hav consdrably dffrnt dlays. Ths bhavor may rsult n an unboundd prc of anarchy,.g., [2, 5, 11]. In many of ths bad nstancs, th prc of anarchy would b 1 f playr costs dpndd on dgs bsds th bottlnck dg. Modls whr a playr taks nto account th dlays on all dgs hav an mprovd prc of anarchy [11]. Evn modls whr a playr s cost s an aggrgaton of th cost on ach dg assum that th flow s statc: vry dg has flow on t nstantanously and smultanously and, onc stablshd, a flow contnus ndfntly. Howvr, th flow n ntworks s oftn transnt. Flow ntrs a ntwork, uss t for som tm, and thn xts, and th flow on an dg changs wth tm. Ths tm-varyng natur of flows s capturd by flows ovr tm, ntroducd n [15]. In ths modl, flow travrss th path n fnt tm, and xts th ntwork at th snk. Thus, th flow on ach dg of th ntwork vars wth tm. Evry dg has a capacty whch lmts th flow rat on th dg. W consdr routng gams for flows ovr tm. Evry playr controls nfntsmal flow, and n contrast to prvous modls whr usrs car about thr thr maxmum dlay or thr total dlay, n our modl th cost of a playr s th tm at whch t arrvs at th snk. A playr s stratgy s a path from th sourc to th dstnaton. On vry dg, th flow follows frst-n, frst-out (FIFO). Whl th ntwork s capactatd, th modl allows th nflow on an dg to b largr than th capacty of th dg. Th xcss flow forms a quu at th tal of th dg, and must wat for th prcdng flow to xt bfor t can xt th dg. Although th capacts and th dg-dlays ar fxd, th total dlay on an dg vars wth th sz of th quu on th dg. Th quu sz sn by flow arrvng at th dg vars wth tm; hnc th total dlay along any path vars wth tm. Ths gam, whch w call a tmporal routng gam, Thus, unlk th statc flow modl consdrd n [12], th ablty to rduc capacts s not suffcnt to nforc th optmal flow.
Our Contrbuton. W study th qulbrum flow n tmporal routng gams. W show that small constant bounds on th ffcncy loss of qulbrum flow n tmporal routng gams can b nforcd. In partcular, Fgur 1: Flow n xcss of capacty on an dg forms a quu at th tal follows th modl of slfsh routng of flows ovr tm usd n [21]. Th modl posssss a numbr of ntrstng charactrstcs. It follows FIFO, whch s a standard assumpton n traffc routng ltratur. Th modl s basd on dynamc quung, frst usd n [34]. Furthr, th qulbrum flow ovr tm can b charactrzd n trms of spcal statc flows [21], dscrbd n 2. In flows ovr tm, smlar to statc flows, varous objctvs may b usd to compar th prformanc of th qulbrum flow to th optmal flow. A natural objctv s th total dlay: th sum of th costs of th playrs. Th flow whch mnmzs th total dlay s th arlst arrval flow, whch maxmzs th flow that arrvs at th dstnaton by tm θ, for vry tm θ. W call th rato of th total dlay of th worst qulbrum flow to th total dlay of arlst arrval flow th total dlay prc of anarchy. A dffrnt objctv s th tm takn to rout a fxd amount of flow to th snk. Ths s calld th complton tm, and s mnmzd by a quckst flow. Th arlst arrval flow s also a quckst flow. Th rato of th tm takn by th worst qulbrum flow, to th tm takn by th quckst flow to rout a fxd amount of flow s calld th tm prc of anarchy. A thrd objctv s th amount of flow that rachs th dstnaton by tm θ. Th rato of amount of flow whch rachs th dstnaton by tm θ n th worst qulbrum flow to th amount of flow whch rachs th snk n th arlst arrval flow s calld th vacuaton prc of anarchy. In th Stacklbrg modl ntroducd n [32], dffrnt playrs n a gam hav dffrnt prorts. A ladr pcks a stratgy frst, and thn th followrs pck thr stratgs. Importantly, th ladr commts to a stratgy bfor th followrs pck thrs. In thr 1982 book on noncoopratv gam thory [6], Basar and Olsdr dscrb a gnral form of Stacklbrg gams whr playrs may hav dffrnt stratgy spacs. W mbrac ths gnral dfnton. In our sttng, th ntwork managr s th ladr. Gvn som physcal lmt on th capacty of ach dg, th ntwork managr actng as ladr pcks a capacty for ach dg whch dos not xcd ths physcal lmt. Th rmanng playrs, actng as followrs, thn pck a rout from sourc to snk as thr stratgy. W gv a polynomal-tm computabl Stacklbrg stratgy to nforc a bound of /( 1) on th tm prc of anarchy n tmporal routng gams; and W show th sam stratgy also nforcs a bound of 2/( 1) on th total dlay prc of anarchy. Th stratgy w dscrb s basd on th followng ky rsult. In tmporal routng gams whr th dg capacts satsfy crtan proprts wth rspct to th quckst flow, th tm prc of anarchy s boundd by /( 1). Th bound of /( 1) on th tm prc of anarchy statd abov s tght, as thr s a matchng xampl n [21]. Our rsults ar n sharp contrast to two prvous rsults. In [21], th authors show that th vacuaton prc of anarchy s Θ(log n), whr n s th numbr of vrtcs. Furthr, [24] consdrs th maxmum tm takn by any playr to travl through th ntwork. Thy show that for ths objctv, th prc of anarchy s Ω(n). W rstrct our analyss to sngl-sourc, sngl-snk ntworks wth constant nflow. In ths cas, th quckst flow s known to b a tmporally rpatd flow (s 2). Thr ar varabl nflows for whch ths s not tru. Equlbrum for tmporal routng gams wth a sngl sourc and sngl snk xst [21]. Furthr, n sttngs wth constant nflow, thy can b dscrbd n trms of statc flows wth spcal proprts [21]. Ths proprts ar crucally usd n our proofs. Rlatd Work. For slfsh routng of statc flows, th ltratur s vast; s [7, 27, 33] for arly rsults on th qulbrum n slfsh routng. Th trm prc of anarchy was frst usd n [26] to dscrb th ffcncy loss causd by th absnc of a controllng authorty. Snc thn, th prc of anarchy has bn wdly usd as a masur of how systm prformanc dgrads f rsourcs ar usd slfshly. For rsults on th prc of anarchy for statc flows, s [29]. Stacklbrg stratgs hav bn usd n computr scnc ltratur to manag th ffcncy loss at qulbrum [22, 28, 30]. Hr, th ntwork managr s a playr wth flow whch sh routs wth th objctv of rducng ffcncy loss. Coordnaton mchansms, ntroducd n [9], rfr to a choc of systm paramtrs by th dsgnr to nflunc qulbra. Th trm s usd to dscrb stuatons both whr th systm paramtrs
ar chosn bfor th markt powr of th playrs s known, and whr th markt powr of th playrs s known bforhand. In th lattr stuaton, ths concpt fts wthn th noton of a Stacklbrg gam as dfnd by Basar and Olsdr [6]. Our approach n ths papr may also b vwd as a coordnaton mchansm wth th markt powr of playrs known. Coordnaton mchansms and rlatd rsults ar furthr dscussd n [10]. Ford and Fulkrson [15] ntroduc flows ovr tm. Thy consdr th problm of maxmzng th amount of th flow whch can b snt from a sourc s to a dstnaton t by a gvn tm T; th flow whch achvs ths s calld th maxmum dynamc flow. A rlatd problm s th quckst flow problm: fnd th dynamc flow whch routs a fxd amount of flow M from s to t n mnmum tm. Th arlst arrval flow problm gnralzs th maxmum flow and th quckst flow problms. For a sngl sourc and dstnaton, arlst arrval flows xst [16], howvr th flow ovr tm obtand may hav a dscrpton of sz xponntal n th sz of th nput [35]. Ths and othr problms on flows ovr tm ar consdrd n [13, 14, 18, 19, 20]. Tmporal routng gams ar analyzd by Koch and Skutlla n [21] and ar basd on dtrmnstc quung modls usd arlr n traffc smulaton [31, 34]. Th authors n [21] show that f all dgs hav zro dlay, th tm prc of anarchy s 1. In contrast, thy show th vacuaton prc of anarchy s Θ(log n). Macko t al. study th xstnc of Brass s paradox n tmporal routng gams [24]. Thy show that th maxmum dlay suffrd by any playr can b arbtrarly wors than for an optmal flow ovr tm whch mnmzs th maxmum dlay. Anshlvch and Ukkusur [4] analyz a dffrnt dscrt-tm modl of slfsh routng. In thr modl, th dlay of an dg at any tmstp t s a functon of th flow ntrng th dg at t and th hstory of th dg, whch s an ncodng of th flow ntrng th dg n tmstps bfor t. Howvr, n thr modl, dgs ar uncapactatd; and n nstancs wth multpl sourcs and snks, th qulbrum may not xst. In sngl-sourc, sngl-snk nstancs an qulbrum xsts and can b computd ffcntly. Howvr th tm prc of anarchy may b larg. Hofr t al. [17] consdr a dffrnt modl whr thy consdr flow controlld by playrs as tasks and dgs as machns. A playr s task corrsponds to a sgnfcant amount of flow, rathr than nfntsmal as n th modls dscussd prvously, and must b routd on a sngl path. 2 Modl, Notaton and Dfntons Lt G = (V, E) b a drctd acyclc graph wth two spcal vrtcs s and t calld th sourc and snk. Each dg n th graph has a nonngatv capacty c and a nonngatv dg-dlay d. An s-t path n th graph s a squnc of dgs (v 0, w 0 ),..., (v l, w l ) such that v 0 = s, w l = t, w = v +1 and v v j for j. W abus notaton slghtly and dfn d p := p d for any path p. Statc Flows. In a drctd graph G = (V, E) wth capacts c on th dgs and sourc s and snk t, a statc flow f s an assgmnt of nonngatv valus f uv that satsfy f uv = 0 for (u, v) E and capacty constrants (2.1) and flow consrvaton (2.2) for (u, v) E: (2.1) (2.2) f c E f uv = f vw v V \ {s, t} u w Th valu of a statc flow f s f = v f sv. A path flow f p on a path p s a flow on p of valu f p. For an acyclc graph, a statc flow f can b dcomposd nto th sum of path flows on a st of paths P so that f = p P: p f p [1]. W us f = {f p } p P to dnot a flow dcomposton of flow f, whr P s th st of paths wth strctly postv flow. Flows ovr Tm. A flow ovr tm s dnotd (f +, f ) and s dfnd by th functons of tm f uv + and fuv, u, v V. For any tm θ R + and (u, v) E, f uv(θ) + = fuv(θ) = 0. For = (u, v) E and tm θ R +, f + (θ) s th rat of flow nto dg at tm θ, and f (θ) s th rat of flow out of dg at tm θ. A flow ovr tm (f +, f ) s fasbl f t satsfs capacty constrants (2.3) and flow consrvaton (2.4): (2.3) (2.4) f (θ) c E, θ R + fuv (θ) = f vw + (θ) v V \ {s, t}, u w θ R + Th nt flow rat lavng s and ntrng t must b postv. (2.5) (2.6) f us + (θ) fsw (θ) 0 θ R + u w f ut + (θ) ftw (θ) 0 θ R + u w For a vrtx v, dfn f v + (θ) := u f uv (θ) and fv (θ) := w f+ vw(θ). Th rat of flow ntrng an dg f + (θ) may b largr than th capacty of th dg. In ths cas, th
xcss flow forms a quu at th tal of th dg and must wat for th flow bfor t n th quu, bfor t starts travrsng th dg. Dfn th total flow ntrng and xtng dg by tm θ to b F + (θ) = θ 0 f+ (ν)dν and F (θ) = θ 0 f (ν)dν rspctvly. Th dg-dlay d s th tm takn by flow to travrs th dg f thr s no quu on th dg. Thn E, θ R +, (2.7) F (θ) F + (θ d ). To nsur that flow ntrng an dg at any tm also lavs th dg aftr fnt tm, th flow ovr tm must satsy, E, θ R +, (2.8) < : F + (θ) F (θ + d + ). Th quung-dlay q (θ) on dg at tm θ s th mnmum tm flow ntrng th dg at tm θ must wat bfor t starts travrsng th dg: (2.9) q (θ) := mn{ 0 : F + (θ) = F (θ + d + )} Whn flow lavs th quu, th tm takn to travrs th dg s th dg-dlay d. Th total-dlay at of flow ntrng dg at tm θ s d +q (θ). By (2.9), flow ntrng an dg at tm θ must allow all th flow whch ntrd arlr to xt th dg bfor t can xt, hnc flow on an dg follows FIFO. Flow ntrs th graph at th sourc s at a constant rat. Th total amount of flow to b routd to th snk s M. Th complton tm s th tm whn M unts of flow arrv at th snk. Optmal Flows Ovr Tm. Th maxmum flowovr-tm problm wth tm horzon T s to maxmz th amount of th flow snt from s to t by tm T. Ford and Fulkrson [15] show that th maxmum dynamc flow can b obtand n polynomal tm by computng th statc flow ˆf whch maxmzs (T +1) ˆf d ˆf. Thus ˆf s a mnmum cost statc flow wth th cost of dgs bng th dg-dlays. For a flow dcomposton { ˆf p } p P of ˆf, th maxmum dynamc flow snds flow at rat ˆf p along path p from tm 0 to T d p. Such a dynamc flow, obtand by rpatng a statc flow ovr tm, s calld a tmporally rpatd flow. For a maxmum dynamc flow, w call th statc flow rpatd ovr tm th undrlyng statc flow. Th quckst flow problm for flow M s to fnd th flow ovr tm whch mnmzs th tm takn to snd M unts of flow from s to t. Th quckst flow problm can b solvd by a bnary sarch to fnd th mnmum tm T for a maxmum dynamc flow to rout at last M unts of flow. Thus, th quckst flow problm can b solvd by a tmporally rpatd flow. Th arlst arrval flow problm s to fnd a flow ovr tm whch maxmzs th flow that arrvs at th dstnaton by tm θ, for vry tm θ. An arlst arrval flow s also a maxmum flow-ovr-tm and a quckst flow, but th convrs may not b tru. Thus, th arlst arrval flow may not b a tmporally rpatd flow. For a sngl sourc and dstnaton, arlst arrval flows xst [16]. Tmporal Routng Gams. Th tupl Γ = (G, s, t, c, d,, M) forms an nstanc of th tmporal routng gam. Evry playr n ths gam controls nfntsmal flow. A playr s cost s th tm ts flow arrvs at th snk. A playr s stratgy s a path from s to t. W assum an arbtrary ordrng on th playrs whch corrsponds to th ordr n whch thr flow arrvs at th sourc. Lmma 2.1. ([21]) For any dg E, th functon θ + q (θ) s monotoncally ncrasng n θ. By (2.9), th arlst tm that flow ntrng an dg at tm θ can xt th dg s θ + d + q (θ). It follows from Lmma 2.1 that n a tmporal routng gam, flow dos not wat on an dg unlss th dg has a quu. Equlbrum Flow. Informally, a flow ovr tm s an qulbrum flow f vry playr mnmzs ts cost, gvn th stratgs of th othr playrs. To formalz ths, for vry vrtx v th labl functon l v (θ) s th arlst tm that flow startng from s at tm θ can rach v. Thus l s (θ) = θ, and l v (θ) = mn (u,v) E {l u(θ) + d uv + q uv (l u (θ))} From Lmma 2.1 and th dfnton of th labl functons: Lmma 2.2. For ach nod v V, th functon l v s monotoncally ncrasng and contnuous. For vrtx v and tm θ, dfn l v(θ) := lv(θ) θ. Snc l s (θ) = θ, l s(θ) = 1. For a fxd θ and gvn th quus on th dgs, th labls on all th vrtcs can b found n th followng mannr: l s (θ) = θ and l v (θ) = for v s. Thn n 1 tms, for ach = (u, v) E st l v (θ) = mn{l v (θ), l u (θ) + d uv + q uv (l u (θ))}. Th corrctnss of th labls obtand aftr n 1 rpttons follows from th corrctnss of th Bllman-Ford algorthm.
Th shortst-path ntwork at tm θ, G θ, s th subgraph nducd by th st of dgs E θ = {(u, v) E : l v (θ) = l u (θ) + d uv + q uv (l u (θ))}. Flow s snt ovr currnt shortst paths f for vry dg (u, v) E and for all θ R +, f l v (θ) < l u (θ) + d + q uv (l u (θ)) thn f + (l u (θ)) = 0. Dfnton 2.1. Lt (G, s, t, c, d,, M) b a tmporal routng gam. A flow ovr tm (f +, f ) s an qulbrum flow f () (s,v) E f+ sv (θ) = { c0 f θ M/ 0 othrws () flow s snt ovr currnt shortst paths, and () for vry E and θ 0, f q (θ) > 0, thn f (θ + d ) = c. Evry tmporal routng gam has an qulbrum [21]. For a tmporal routng gam Γ, (f + (Γ), f (Γ)) s th qulbrum flow, and EQ(Γ) s th complton tm of th qulbrum flow. If th nstanc s clar from contxt, w smply us (f +, f ) and EQ. Prc of Anarchy. In ths papr w consdr two sparat objctvs. In Scton 4 and Scton 6, our objctv s to mnmz th complton tm. For ths objctv, th optmal flow s a quckst flow. Snc a quckst flow can b rprsntd as a tmporally rpatd statc flow, w us ˆf(Γ) to dnot th undrlyng statc flow for th quckst flow for a tmporal routng gam Γ, and (Γ) to dnot th complton tm of th quckst flow. Th tm prc of anarchy s thn dfnd as max Γ EQ(Γ)/ (Γ). If th nstanc s clar, w us ˆf to rfr to th undrlyng statc flow and for th complton tm. W say th statc flow undrlyng th quckst flow saturats vry dg of th graph f for all E, ˆf = c and v ˆf sv =. W show that f ths condton holds, thn th prc of anarchy s small. Instad of flow ntrng th graph at th sourc s at a constant rat, anothr way to thnk of th modl s that all flow s prsnt at th sam tm at a nod s, and thr s an ntal dg (s, s) of capacty and dlay 0. Thn th arrval tm at t for a playr s also ts dlay. In Scton 5, th objctv s to mnmz th total dlay of a flow ovr tm whch routs a fxd amount of flow M from th sourc to th dstnaton. Th total dlay of a flow ovr tm n a tmporal routng nstanc s th sum of th arrval tms at t of th playrs. For a flow (f +, f ) and nstanc Γ wth complton tm T, th total dlay D((f +, f )) = T 0 f+ t (θ)θdθ. In ths, cas th arlst arrval flow s th optmal flow snc t maxmzs th flow at t at vry tm θ. For a tmporal routng gam Γ, lt (g + (Γ), g (Γ)) b th arlst arrval flow. Th total dlay prc of anarchy s dfnd as max Γ D(f + (Γ), f (Γ))/D(g + (Γ), g (Γ)). In th full vrson [8], w gv an xampl of a tmporal routng gam and ts qulbrum flow. 3 Th Structur of Equlbra Equlbra n tmporal routng gams can b charactrzd n trms of statc flows wth crtan proprts, calld rat flows. W us th proprts of rat flows to obtan our bounds on th prc of anarchy. In ths scton, w ntroduc som of ths proprts, as wll as an algorthm for computng qulbra. Both rat flows and th algorthm w dscuss ar dscrbd n [21]. Rat Flows. For dg = (v, w) E and tm θ R +, dfn x + (θ) := F + (l v (θ)) and x (θ) := F (l w (θ)). Thorm 3.1. ([21]) For a flow ovr tm, flow s snt ovr currnt shortst paths f and only f for all dgs and for all θ, x + (θ) = x (θ). Lt x (θ) := x + (θ). At qulbrum, t follows by ntgratng (2.4) ovr tm and from Thorm 3.1 that for vry θ R +, x (θ) s a statc flow n th uncapactatd ntwork G. For θ such that x(θ) s dffrntabl, (3.10) x (θ) θ = f + (l v (θ))l v(θ) = f (l w(θ))l w (θ). For any tm θ, th flow x(θ) gvn by x (θ) on vry dg s calld th statc flow undrlyng th qulbrum flow. Lt x dx(θ) (θ) := dθ whr th dffrntal xsts. Th followng thorm dscrbs som proprts of x (θ). For θ R +, dfn E 1 (θ) := {(v, w) E : q vw (l v (θ)) > 0} as th st of dgs whch hav postv quus on thm at tm θ. Thorm 3.2. ([21]) For an qulbrum flow n a tmporal routng gam Γ = (G, s, t, c, d, ) lt θ 0 b such that x (θ) and l v (θ) xst for all v V, E. Thn (x (θ)) Gθ s a statc flow of valu n th uncapactatd graph. Furthr, th statc flow (x (θ)) G θ satsfs
l w(θ) l v(θ), { l w(θ) = max l v(θ), x vw (θ) } c vw l w (θ) = x vw(θ) c vw (v, w) E(G θ ) \ E 1 (θ) wth x vw (θ) = 0, (v, w) E(G θ ) \ E 1 (θ) wth x vw > 0, (v, w) E 1 (θ). Th statc flow (x (θ)) Gθ s calld a rat flow. By (3.10), x vw (θ) xsts ff l v (θ) and l w (θ) xst. If at tms θ and θ R + th shortst-path ntworks and th st of dgs wth postv quus ar th sam,.., G θ = G θ and E 1 (θ) = E 1 (θ ), thn th rat flow x (θ) and l v (θ) satsfy th condtons of Thorm 3.2 at tm θ as wll. Computng Equlbra. In [21], th authors dscrb an algorthm to comput qulbrum flow. Th algorthm dvds th tm from θ = 0 to θ = M/ nto a numbr of phass, wth phass dvdd by vnts. Not that θ = M/ s th tm th last flow lavs th sourc; th tm ths flow rachs th snk s th complton tm. An vnt can b of two knds. A quu-vnt occurs at tm θ f for som dg = (u, v), th quu dcrass to zro at tm l u (θ). That s, th quung dlay q (l u (θ)) = 0 and for som δ > 0 and vry 0 < ǫ δ, q (l u (θ ǫ)) > 0. A path-vnt occurs at tm θ f som dg = (u, v) ntrs th shortst-path ntwork at tm θ,.., l v (θ) = l u (θ) + d + q (l u (θ)) and for som δ > 0 and vry 0 < ǫ δ, l v (θ) > l u (θ) + d + q (l u (θ)). Quu vnts and path vnts ar collctvly trmd vnts. Not that n dtrmnng th tm an vnt occurs, w ar usng th sourc as a fram of rfrnc. Whl th vnt actually occurs at a latr tm θ, w say an vnt occurs at tm θ f flow lavng th sourc at tm θ rachs th tal of th dg n th quu-vnt or path-vnt at tm θ. For a gvn nstanc, w ordr th vnts occurrng n an qulbrum flow by th tm of th occurrnc (usng th sourc as th fram of rfrnc) and ndx thm, startng from 0 to r. Th vnt r s a spcal vnt, corrspondng to th last flow lavng th sourc, thus th qulbrum flow nds at vnt r. W dfn θ as th tm vnt occurs, and τ := l t (θ ). Thus, θ r = M/ and τ r = EQ. A phas s th tm ntrval btwn two vnts. Phas as th tm btwn vnts 1 and. Thus, tm θ s n phas f θ 1 < θ < θ. W xclud th vnt tms θ snc th vrtx labls ar l v (θ) and th statc flow x(θ) ar not dffrntabl at ths tms. Wthn a phas, th shortst-path ntwork and th st of dgs wth quus on thm rman constant. Hnc th rat flow x (θ) and th rat of chang of vrtx labls l v (θ) xst and ar fxd for θ wthn a phas. Th frst phas, phas 1, s th tm btwn θ 0 = 0 and θ 1. Thus, for a phas, w dfn th followng notaton: G dnots th shortst-path ntwork n phas. c s th capacty of th shortst path ntwork n phas. s th chang n capacty of th shortst path ntwork whn vnt occurs, thus = c c 1. W dfn := 0. Not that = 0 f vnt 1 s a quu vnt, or 1 s a path vnt but th capacty of shortst path ntwork dos not chang; ths could happn f a mnmum cut s unaffctd by th dg addd to th shortst path ntwork n a path vnt. Snc th rat of chang of vrtx labls l v (θ) s fxd for θ wthn a phas, th vrtx labls l v (θ) ar fxd lnar functons of θ wthn a phas; and thus wthn a phas l v s wll-dfnd for all v. Thus, w can dfn: l v := l v (θ) for any tm θ n phas. Th st E 1 s dfnd as { = (v, w) : q (l v (θ )) > 0}. W us x to dnot th rat flow n phas. For th notaton abov, f th phas s clar from contxt, for smplcty w omt th phas. Thus th rat flow would b dnotd by x. For dg (v, w) n th shortst path ntwork at tm θ, q vw(θ) := qvw(lv(θ)) θ. Snc l w (θ) = l v (θ) + d vw + q vw (l v (θ)), q vw(θ) = l w(θ) l v(θ). Snc th rat of chang of th vrtx labls s constant, w dfn for phas : If dg = (v, w) s n th shortst-path ntwork n phas, dfn q vw := l w l v, othrws q := 0. For an s-t path p, w abus notaton slghtly to dfn qp :=. p q 4 A Stacklbrg Stratgy for th Tm Prc of Anarchy In Scton 6, w prov our man tchncal rsult: Thorm 4.1. For a tmporal routng gam whr th statc flow undrlyng th quckst flow saturats vry dg of th graph, th tm prc of anarchy s /( 1).
In gnral nstancs of th tmporal routng gam whr th rat of qulbrum flow may xcd th optmal flow on som dgs, a bound on th tm prc of anarchy s unknown. Howvr, n any tmporal routng gam, w show how to us Thorm 4.1 to obtan a smpl Stacklbrg stratgy to nforc a bound of /( 1) on th prc of anarchy. Thorm 4.2. For a tmporal routng gam, lt b th tm takn by th quckst flow to rout all flow to th snk. Thr xsts a polynomal-tm computabl Stacklbrg stratgy to nforc an qulbrum flow that routs all flow at th sourc to th snk n tm at most /( 1). Proof. For a tmporal routng gam Γ = (G, s, t, c, d,, M), lt ˆf b th statc flow undrlyng th quckst flow. Ths can b computd n polynomal tm by conductng a bnary sarch to fnd th mnmum tm T such that th maxmum dynamc flow wth tm horzon T gts at last M flow to th snk. Th Stacklbrg stratgy s thn as follows. Th ntwork managr, actng as th ladr, rducs th capacty on ach dg so that th nw capacts c ar th valu of th statc flow on ach dg n Γ: c = ˆf. It s asy to s that th quckst flow rmans unchangd; furthr on ach dg wth th modfd capacts, ˆf saturats vry dg. By Thorm 4.1, th prc of anarchy s now boundd by /( 1). Thus n any nstanc of th tmporal routng gam, by rducng th capacty of dgs, th complton tm of qulbrum flow can b boundd by /( 1) tms th complton tm of th optmal flow. 5 Th Total Dlay Prc of Anarchy W now obtan bounds on th total dlay prc of anarchy of tmporal routng gams. Th total dlay prc of anarchy s th maxmum ovr all nstancs, of th rato of total dlay of th qulbrum flow to th mnmum total dlay. Snc th cost of a playr s th tm t arrvs at th snk, for a flow ovr tm (f +, f ) wth complton tm T th total dlay D((f +, f )) = T 0 θ f+ t (θ)dθ. W frst show that n tmporal routng gams whch satsfy th sam assumpton as n Thorm 4.1, th total dlay prc of anarchy s boundd by a small constant. Thorm 5.1. For a tmporal routng gam whr th statc flow undrlyng th quckst flow saturats vry dg of th graph, th total dlay prc of anarchy s 2/( 1). Th followng lmma gvs a lowr bound on th total dlay of th arlst arrval flow. Th proof, and all mssng proofs, ar gvn n th full vrson of th papr [8]. Lmma 5.1. Th total dlay of th arlst arrval flow (g +, g ) wth complton tm T n an nstanc Γ s at last MT/2. Proof of Thorm 5.1. Lt EQ dnot th complton tm of th qulbrum flow. Thn by Thorm 4.1, EQ T/( 1). Th total dlay of qulbrum flow (f +, f ) s boundd by (5.11) D((f +, f )) = T/ 1 0 T 1 0 MT 1 θ f + t (θ)dθ T/( 1) f + t (θ)dθ Th rsult now follows from (5.11) and Lmma 5.1. Smlar to th proof of Thorm 4.2, Thorm 5.1 can b usd to gv a Stacklbrg stratgy for nforcng a bound of 2/( 1) on th total dlay prc of anarchy n any gnral nstanc. Thorm 5.2. For a tmporal routng gam, thr xsts a polynomal-tm computabl Stacklbrg stratgy to nforc an qulbrum flow wth total dlay at most 2/( 1) tms that of th arlst arrval flow n th unmodfd nstanc. 6 Th Tm Prc of Anarchy In ths scton, w prov Thorm 4.1. W assum that on vry dg, ˆf = c. For a path dcomposton { ˆf p } p P of ˆf along paths p P, by our assumpton, ˆf p P p =. Concptually, w show that for an nstanc Γ of th tmporal routng gam, th rato of EQ to s worst f vry vnt thr occurs at tm 0, or occurs at a fxd tm µ. Thus, w modfy an nstanc Γ to obtan an nstanc Γ whr vry vnt thr occurs at tm 0 or tm µ. W thn obtan a bound on EQ n ths smplr nstanc. Ths concptual vw s smplfd; t may not always b possbl to prsrv th vnts f w nsst on vry vnt occurng at thr tm 0 or tm µ. Howvr, w show a bound on EQ can b obtand n Γ by followng th sam stps analytcally: Stp 1: For any path p, gt a lowr bound on d p n trms of {θ } r =0 and th quus on th path p
(Lmma 6.3). Stp 2: Us th bound n stp 1 to obtan an uppr bound on EQ n trms of th vnt tms {θ } r =0 and th quus on th dgs (Lmma 6.4 and Corollary 6.2). Stp 3: Show that thr s som vnt k r so that f all th vnts bfor and ncludng k happn at tm 0, and all vnts aftr k occur at th sam tm, thn th uppr bound on EQ n ths modfd nstanc also bounds EQ n th orgnal nstanc (Lmma 6.5). Stp 4: Evaluat th uppr bound on EQ for ths modfd, smplr nstanc (Lmma 6.6 and Thorm 4.1). Our frst stp s to show a rlaton btwn th labl on t and th rat of flow nto t. Lmma 6.1. Lt (f +, f ) b th qulbrum flow for a tmporal routng gam Γ wth nflow and corrspondng labls l. Thn l t(θ) = f t + (l t (θ)) for θ R +. W frst show Lmma 6.1 for a path n th graph, and thn us path dcompostons of x (θ) n conjuncton wth (3.10) to gt th rsult. Lmma 6.2. Lt p = (s, v 1, v 2,...,v k ) b a path n G θ. If for vry par of conscutv dgs (u, v), (v, w) n p, fuv(l v (θ)) = f vw(l + v (θ)), thn l v k (θ) = f+ sv 1 (l s (θ)) fv k 1 v k (l vk (θ)). Proof of Lmma 6.1. Lt x and l v b th rat flow and rat of chang of labls at tm θ. Lt {x p} p P b a path dcomposton of x whr P s th st of paths wth postv flow. Instad of th graph G = (V, E), w consdr th qulbrum flow n a graph Ḡ = (V, Ē) wth Ē = E P. Evry dg a Ē corrsponds to a par x (, p) wth E and p P, wth capacty c a = c p x and dlay d a = d. W obtan an qulbrum flow n Ḡ and show that th labls on th vrtcs at any tm φ ar th sam n G and Ḡ. Dfn a modfd flow ovr tm ( f +, f ) n Ḡ as follows: f+ a (φ) = f + ca (φ) c and f a (φ) = f ca (φ) c. Thn th cumulatv flow F a + (φ) := φ f + 0 a (θ) = F + ca c and smlarly F a (φ) := φ f 0 a (φ) = F ca c. Thus, q a (φ) = q (φ) va (2.9). Snc d a = d and l s (φ) = φ, t follows that th labls on th vrtcs n Ḡ for th flow ovr tm ( f +, f ) ar qual to th labls on th corrspondng vrtcs n G for th qulbrum flow (f +, f ), for vry tm φ R +. It s asy to vrfy th condtons for qulbrum flow n Dfnton 2.1 for ( f +, f ) n Ḡ. W show that at tm θ, l t = n H. Snc f + t (lt(θ)) th nod labls ar th sam for (f +, f ) and ( f +, f ), and f v + (φ) = f v + (φ) for vry vrtx v and tm φ R +, ths provs th lmma. Snc ( f +, f ) s an qulbrum flow n Ḡ, th flow y wth y = x ca c s a rat flow n graph Ḡ at tm θ. Consdr a path p P; thr s a corrspondng s-t path q n Ḡ consstng of all th dgs whch corrspond to path p. By our constructon, on vry dg a of path q, y a = x p. On conscutv dgs (u, v), (v, w) n q, y uv = l v f uv(l v (θ)) and y vw = l v f vw(l + v (θ)). Snc y uv = y vw, t follows that f uv (l v(θ)) = f vw + (l v(θ)). Thn by Lmma 6.2, l t f v k t(l t (θ)) = f sv + 1 (l s (θ)). Snc ths s tru for all paths p P, w can sum ovr all ths paths to obtan l t (θ) f t + (l t(θ)) = f s + (θ) =. Thus n graph Ḡ, l t = f + t (lt(θ)). Corollary 6.1. For any vnts, 1, τ τ 1 = c (θ θ 1 ). Proof. By dfnton, τ τ 1 = l t (θ ) l t (θ 1 ) and f t + (l t(θ)) = c n phas. Thus τ τ 1 = θ θ 1 l t (φ)dφ = θ θ 1 c dφ = c0 c (θ θ 1 ). W us Corollary 6.1 to bound d p n trms of {τ } r =0. Lmma 6.3. For any s-t path p, d p τ r ( ) r =1 1 + qp c (τ τ 1 ). Proof. By dfnton of shortst path ntwork, for any tm θ and for any dg = (v, w) n th shortst path ntwork G θ, d = l w (θ) l v (θ) q (l v (θ)). For any vrtx v, l v (θ r ) = r =1 l v (θ θ 1 ) + l v (θ 0 ), and smlarly q (l v (θ r )) = r (θ θ 1 ). Hnc, =1 q for any dg n th shortst path ntwork at tm θ r, d = l w (θ 0 ) l v (θ 0 )+ ( ) r =1 lw l v q (θ θ 1 ). For dgs not n th shortst path ntwork at θ r, d > l w (θ r ) l v (θ r ) = l w (θ 0 ) l v (θ 0 ) + r =1 ( l w l v q ) (θ θ 1 ). Summng ovr all dgs n path p ylds d p τ 0 + r =1 ( l t l s q p ) (θ θ 1 ). Substtutng n from Corollary 6.1 and from Lmma 6.1, and snc l s(θ) = 1, d p τ 0 + r c c =1 0 ( ) c0 1 q p (τ τ 1 ). c Smplfyng ylds th dsrd rsult.
Lmma ˆf 6.4. For a tmporal routng gam wth p P p =, th complton tms of th optmal flow and qulbrum flow ar rlatd as (6.12) EQ = c r 1 EQ r 1 τ +1 =0 p P ˆf p d p. Th proof s basd on th flow arrval rat at t for th qulbrum and optmal flows. For th tmporal routng gam n Fgur 2, ths arrval rats ar plottd n Fgur 3 and Fgur 4. Fgur 2: An nstanc of a tmporal routng gam Fgur 3: Arrval rat at t for qulbrum flow Fgur 4: Arrval rat at t for optmal flow Proof. Consdr th arrval rat at th snk for th optmal flow. For any path p P, th rat of flow arrvng at th snk ncrass by ˆf p at tm d p. Th total flow arrvng at th snk by tm θ s th ara undr ths curv up to tm θ. Thus, M = p ˆf p d p, and smlarly for th qulbrum flow, M = c r EQ r 1 =0 τ +1. Equatng ths ylds = cr EQ + p P r 1 ˆf p d p τ +1, =0 and dvdng both sds by EQ ylds th dsrd qualty. Usng th lowr bound n Lmma 6.3 and dfnng λ r := c r + cr ˆf p p qp r, λ 0 := c1 p ˆf p qp 1 and for 1 r 1, λ := 1 ˆf p p ( c ) qp c+1 qp +1, Corollary ˆf 6.2. For a tmporal routng gam wth p P p =, EQ c r 1 EQ r λ τ. =0 Lmma 6.5 s usd to partton th vnts nto two sts, wth vnts n th frst st occurrng at tm θ = 0 and vnts n th scond st occurrng at tm θ r : Lmma 6.5. For 0 r and λ, y R, f 0 y 0 y 1 y r, thn r =0 λ y y r max k r =k λ. By Lmma 6.5, k r : r =0 λ τ τ r k λ. Thn snc τ r = EQ, substtutng n Corollary 6.2, (6.13) EQ c r 1 λ. k Evaluatng k λ, w obtan k λ = c r f k = 0 and k λ = c r + c k c0 p ˆf p qp f k > 0. If k = 0, thn (6.13) bcoms EQ cr 1 ( c r ) = 1 and hnc n ths cas, = EQ. If k > 0, (6.14) ( ) EQ c r 1 c r + c k ˆf p q p p = 1 c k ( ) 2 p = 1 c k ( ) 2 ˆf p qp ˆf q = 1 c k ( ) 2 c q, snc by assumpton, c = ˆf on vry dg. Lmma 6.6. In any phas k of th qulbrum flow, c q c0 ln c0 c k. Proof Sktch. By th condtons of Thorm 3.2 and th dfnton of q, c q = x l (1 v l ) k w and hnc c q = ( =(v,w) x 1 l v = p P xk p =(v,w) p ( 1 lk v ) l k w l k w ) for a path dcomposton {x k p }p P of x k. W thn show that for any s-t path p, ( =(v,w) p 1 l ) v lnlt = ln c k by Lmma 6.1. l k w Snc p P xk p = c0, th rsult follows. Proof of Thorm 4.1. Lt w = c k /. Thn from (6.14) and Lmma 6.6, EQ 1 w ln 1 w. Lt z = w ln 1 w, thn z s maxmzd whn w = 1/. Hnc, EQ 1 1/ = 1. Obsrvng that EQ s th nvrs of th prc of anarchy, complts th proof.
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