In Procdings of th 32nd Confrnc on Artificial Intllignc (AAAI 2018), Nw Orlans, Lousiana, USA. Fbruary 2018 Traffic Optimization For a Mixtur of Slf-intrstd and Compliant Agnts Guni Sharon 1, Michal Albrt 2, Tarun Rambha 3, Stphn Boyls 4, Ptr Ston 1 1 Dpartmnt of Computr Scinc, Univrsity of Txas at Austin, Austin, TX 78712, USA 2 Dpartmnt of Computr Scinc, Duk Univrsity, Durham, NC 27708, USA 3 Civil and Environmntal Enginring, Cornll Univrsity, Ithaca, NY 14853, USA 4 Civil, Architctural and Environmntal Enginring, Th Univrsity of Txas at Austin, Austin, TX 78712, USA gunisharon@gmail.com, malbrt@cs.duk.du, tr244@cornll.du, sboyls@mail.utxas.du, pston@cs.utxas.du Abstract This papr focuss on two commonly usd path assignmnt policis for agnts travrsing a congstd ntwork: slfintrstd routing, and systm-optimum routing. In th slfintrstd routing policy ach agnt slcts a path that optimizs its own utility, whil in th systm-optimum routing, agnts ar assignd paths with th goal of maximizing systm prformanc. This papr considrs a scnario whr a cntralizd ntwork managr wishs to optimiz utilitis ovr all agnts, i.., implmnt a systm-optimum routing policy. In many ral-lif scnarios, howvr, th systm managr is unabl to influnc th rout assignmnt of all agnts du to limitd influnc on rout choic dcisions. Motivatd by such scnarios, a computationally tractabl mthod is prsntd that computs th minimal amount of agnts that th systm managr nds to influnc (compliant agnts) in ordr to achiv systm optimal prformanc. Morovr, this mthodology can also dtrmin whthr a givn st of compliant agnts is sufficint to achiv systm optimum and comput th optimal rout assignmnt for th compliant agnts to do so. Exprimntal rsults ar prsntd showing that in svral larg-scal, ralistic traffic ntworks optimal flow can b achivd with as low as 13% of th agnt bing compliant and up to 54%. Introduction In multiagnt systms, thr ar gnrally two paradigms of intraction. Cntralizd control paradigms assum that a singl dcision making ntity is abl to dictat th actions of all th agnts, thus lading thm to a coordinatd social optimum. Dcntralizd control paradigms, on th othr hand, assum that ach agnt slcts its own actions, and whil it is in principl possibl for thm to act altruistically, thy ar gnrally assumd to b slf-intrstd. In this papr, w considr a routing scnario in which a subst of agnts ar controlld cntrally (compliant agnts), whil th rmaining ar slf-intrstd agnts. W modl th systm as a Stacklbrg routing gam (Yang, Zhang, and Mng 2007) in which th dcision makr for th cntrally controlld agnts is th ladr, and th slf-intrstd agnts ar th followrs. In this papr, w provid a computationally tractabl mthodology for 1) dtrmining th maximum Ths authors contributd qually. Copyright c 2018, Association for th Advancmnt of Artificial Intllignc (www.aaai.org). All rights rsrvd. numbr of slf-intrstd agnts that a systm can tolrat at optimal flow, 2) dtrmining whthr a givn subst of cntrally controlld agnts ar sufficint to achiv systm optimum (), and 3) computing th actions th ladr should prscrib to a sufficint st of compliant agnts in ordr to achiv. A known fact in routing gams is that agnts sking to minimiz thir privat latncy nd not minimiz th total systm s latncy (Pigou 1920; Roughgardn and Tardos 2002). That is, slf-intrstd agnts may rach a usr quilibrium (UE) that is not optimal from a systm prspctiv. Howvr, if all agnts ar assignd paths with minimum systm marginal cost thn th systm will achiv optimal prformanc (Pigou 1920; Bckmann, McGuir, and Winstn 1956; Ditrich 1969). Thrfor, from a systm managr prspctiv, it is dsirabl that all agnts travrsing a ntwork would strictly utiliz minimal marginal cost paths, vn if such paths ar not of minimum latncy for an individual agnt. Howvr, in many important scnarios, it will not b possibl to nforc path assignmnt on all agnts, but it may b possibl to affct th bhavior of a subst (th compliant agnts). As a motivating xampl, considr an opt-in tolling systm whr drivrs ar givn positiv incntivs to nroll but, in xchang, thy will b subjct to tolls that affct thir rout choic (Sharon t al. 2017a; 2017b). Anothr rlvant xampl is virtual privat ntwork (VPN) path allocation. Whil ach packt within th VPN might b slfintrstd, a pro-social ntwork managr might allocat virtual paths that ar diffrnt from thos prfrrd by th slf-intrstd packts (Fingrhut, Suri, and Turnr 1997; Duffild t al. 1999). W show that, in th gnral cas, computing th optimal assignmnt of compliant agnts is NP-hard. Thrfor, w focus on th spcific scnario whr th portion of compliant agnts is sufficintly larg to achiv. W prsnt a novl linar program (LP ) rprsntation for computing th maximal portion of slf-intrstd agnts that allow th systm to achiv and to dtrmin whthr a givn st of compliant agnts is sufficint to achiv. Furthrmor, w provid a mthod to tractably comput th flow assignmnt for th compliant agnts such that prformanc is guarantd. Exprimntal rsults, prformd using a standard traffic
simulator, ar providd and dmonstrat that th numbr of compliant agnts ncssary to achiv systm optimum can b a rlativly small prcntag of total flow (btwn 13% and 53%). Motivation Rcnt advancs in GPS basd tolling tchnology (Numrich, Ruja, and Voß 2012) opn th possibility of implmnting micro-tolling systms in which spcific tolls ar chargd for th us of links within a road ntwork. Such tolls can b chargd on many or all ntwork links, and changd frquntly in rspons to ral-tim obsrvations of traffic conditions. Toll valus and traffic conditions can thn b communicatd to vhicls which might chang routs in rspons, ithr autonomously, or by updating dirctions givn to th human drivr. Stting tolls appropriatly can influnc slf-intrstd drivrs to prfr paths with minimum systm marginal cost and thus, lad to improvd systm prformanc (Sharon t al. 2017a; 2017b). Unfortunatly, political factors dtr public officials from allowing such a micro-tolling schm to b ralizd. Road pricing is known to caus a grat dal of public unrst and is thus opposd by govrnmntal institutions (Schallr 2010). To tackl this issu and avoid public unrst, w suggsts an opt-in micro-tolling systm whr, givn som initial montary sign-up incntiv, drivrs choos to opt-in to th systm and b chargd for ach journy thy tak basd on thir chosn rout. Th vhicls blonging to such drivrs would nd to b quippd with a GPS dvic as wll as a computrizd navigation systm. Givn th toll valus and drivr s valu of tim, th navigation systm would suggst a minimal cost rout whr th cost is a function of th travl tim and tolls. Whil addrssing th issu of political accptanc, an opt-in systm would rsult in traffic that is composd of a mixtur of slf-intrstd and compliant agnts (compliant in th sns that th systm managr can influnc thir rout choic). Such a scnario raiss som practical qustions which ar th focus of this papr, namly, what portion of slf-intrstd agnts can th systm tolrat whil still raching optimum prformanc? Th answr to this qustion can hlp practitionrs to dtrmin both th lvl and th targting of incntivs in an opt-in systm. Problm dfinition and trminology Th trminology in this papr follows that of Roughgardn and Tardos (2002). W rviw th rlvant concpts and notation in this sction. Th flow modl Th flow modl in this work is composd of a dirctd graph G(V, E), and a dmand function R(s, t) R + mapping a pair of vrtics s, t V 2 to a non-ngativ ral numbr rprsnting th rquird amount of flow btwn sourc, s, and targt, t. 1 An instanc of th flow modl is a {G, R} 1 Th dmand btwn any sourc and targt, R(s, t), can b viwd as an infinitly divisibl st of agnts (also known as a nonatomic flow (Rosnthal 1973)). pair. P s,t dnots th st of acyclic paths from s to t. Dfin P as th collction of all P s,t (i.., s,t V 2P s,t ). Th variabl f p rprsnts th flow volum assignd to path p. Similarly, f is th flow volum assignd to link. By dfinition, th flow on ach link (f ) quals th summation of flows on all paths of which is a part. Dfin th systm flow vctor as f = vct{f p }. f is said to b fasibl if for all s, t V 2, p P s,t f p = R(s, t). Each link E has a latncy function l (f ) which, givn a flow volum (f ), rturns th latncy (travl tim) on. Following Roughgardn and Tardos (2002) w mak th following assumption: Assumption 1. Th latncy function l (f ) is non-ngativ, diffrntiabl, and non-dcrasing for ach link E. Th latncy of a simpl path p for a givn flow f, is dfind as l p (f) = p l (f ). A fasibl flow f is dfind as a usr quilibrium (UE) if for vry s, t V 2 and p a, p b P s,t with f pa > 0 it holds that l pa (f) l pb (f) (s Lmma 2.2 in (Roughgardn and Tardos 2002)). In othr words, at UE, no amount of flow can b rroutd to a path with lowr latncy whn th rst of th flow is fixd. Dfin th systm cost associatd with link as c (f ) = l (f )f, th cost of a path p as c p (f) = p c (f ) and th cost of a flow f as c(f) = E c (f ). Dfin c (x) = d dx c (x) and c p(f) = p c (f ). A fasibl flow f is dfind as a systm optimum () flow if for vry s, t V 2 and p a, p b P s,t with f pa > 0 it holds that c p a (f) c p b (f) (s Lmma 2.5 in (Roughgardn and Tardos 2002)). In othr words, at, th bnfit from rducing th flow along any path is always lss than or qual to th cost of adding th sam amount of flow to a paralll, altrnativ path. W follow Roughgardn and Tardos (2002), and mak th following assumption: Assumption 2. Th cost function c (f ) is convx for ach link E. Assumptions 1 and 2 imply that th st of flows corrspond to th st of solutions of a convx program whr th objctiv is to minimiz c(f) = E c (f ) (s Roughgardn and Tardos (2002) Corollary 2.7). Problm Dfinition Th focus of this papr is a scnario whr th dmand is partitiond into slf-intrstd and compliant agnts. W dfin two typs of controllrs that assign paths to all of th agnts. Ths controllrs ar viwd as playrs in a Stacklbrg gam (Yang, Zhang, and Mng 2007). -controllr - Stacklbrg ladr, th -controllr aspirs to assign paths to th compliant subst of agnts that, taking into account th slf-intrstd agnts raction, optimizs th systms prformanc (i.. minimizs total latncy). W rfr to flow assignd by th -controllr as compliant flow. UE-controllr - Stacklbrg followr, considring th compliant agnts path assignmnt as fixd, th UEcontrollr assigns paths to th slf-intrstd agnts, th
UE flow, such that a stat of usr quilibrium (as dfind abov) is achivd. 2 Th problms addrssd in this papr ar: 1. Givn an instanc of th flow modl {G, R}, what is th maximum amount of slf-intrstd agnts that can b assignd to th UE controllr and still prmit th optimal flow. 2. Givn a st of compliant agnts and an instanc of th flow modl {G, R}, can th controllr assign paths to thm in such a way that th systm achivs. 3. If is achivabl, how should th -controllr assign th compliant flow. Equivalntly, what is th optimal Stacklbrg quilibrium. To th bst of our knowldg, this work is th first to answr ths qustions in a gnral stting. Rlatd Work Prvious work xamind mixd quilibrium scnarios whr traffic is composd of: UE and Cournot-Nash (CN) controllrs. A CN-controllr assigns flows to a givn subst of th dmand with th aim of minimizing th total travl tim only for that subst. For instanc, a logistic company with many trucks can b viwd as a CN-controllr. It was shown that th quilibrium for a mixd UE, CN scnario is uniqu and can b computd using a convx program (Hauri and Marcott 1985; Yang and Zhang 2008). On th othr hand, no tractabl algorithm is known for computing th optimal Stacklbrg quilibrium for scnarios that also includ a -controllr. Korilis t. al. (1997) xamind mixd quilibrium scnarios that do includ a -controllr. In thir work, a tchniqu for computing a solution for th abov qustions #1 and #3 was suggstd for spcific typs of flow modls. Thir tchniqu was provn to work for ntworks with a common sourc and a common targt with any numbr of paralll links. Morovr, th latncy functions wr assumd to b of a vry spcific form (linar function with a capacity bound). As a rsult, thir solution is not applicabl whn gnral ntworks with arbitrary latncy functions ar considrd. Othr work (Roughgardn 2004; Immorlica t al. 2009) studid a variant of th schduling problm whr infinitsimal jobs must b assignd to a st of shard machins ach of which is affiliatd with a non-ngativ, diffrntiabl, and non-dcrasing latncy function that, givn th machin load, spcify th amount of tim ndd to complt a job. Whn considring a scnario whr part of th jobs ar assignd to machins by a UE-controllr whil th rst ar assignd by a -controllr, thy show it is NP-hard to comput th optimal Stacklbrg quilibrium (Roughgardn 2004). Thir problm can b viwd as a spcial cas of our problm, spcifically a ntwork with a singl sourc and targt with multipl paralll links btwn thm. Givn that in 2 Th UE nforcd by th UE-controllr applis only for th slf-intrstd subst of agnts. That is, no slf-intrstd agnt can bnfit from unilatrally dviating from its assignd path. this, mor rstrictiv stting, computing th optimal Stacklbrg quilibrium is intractabl, th sam yt gnral qustion in our stting will also b computationally intractabl. Computing th Maximal UE Flow Givn that finding th optimal Stacklbrg quilibrium is NP-hard for an arbitrary siz of compliant flow, this work focuss on scnarios whr th siz of th compliant flow is sufficint to achiv. As w will show, finding th optimal Stacklbrg quilibrium can b don in polynomial tim for such cass. In this sction, w will prsnt a computationally tractabl mthod to comput th maximal UE flow givn an instanc of a flow modl {G, R}, and w will provid a mthod to chck, for a givn lvl of compliant flow, whthr is achivabl. W dfin rue as th maximal amount of dmand comprisd of slf-intrstd agnts that th systm can tolrat and still achiv. Additionally, w dfin rs,t as th amount of dmand from sourc s to targt t that is assignd to th UE-controllr. That is, computing rue is quivalnt to maximizing s,t r s,t. W can cast th problm of maximizing s,t r s,t as an optimization problm, spcifically a linar program (LP ). Assigning valus to all variabls of typ rs,t must follow som constraints. Spcifically, th UE flow from ach origin to ach dstination must b both a subflow of th flow, and must follow a last latncy path. Dfinition 1 (Subflow of flow f). For a dirctd graph G(V, E) and dmand function R, a flow f is a subflow of flow f if for all links E, 0 f f and for ach pair of nods s, t V 2, thr xists 0 r s,t R(s, t) such that f and out(s) in(t) f in(s) f = t out(t) f = s r s,t r s,t. A path p, lading from vrtx s to vrtx t, is said to b zro rducd cost if thr is no othr path, p, lading from s to t with lowr latncy or lowr marginal cost. Dfinition 2 (Zro rducd cost path). For a flow modl {G, R}, a zro rducd cost path with rgard to flow assignmnt f is a path p P s,t such that p P s,t : l p (f) l p (f) and c p(f) c p (f). A link,, is dfind as a zro rducd cost link, with rspct to sourc s, if it is part of any zro rducd cost path originating from s and trminating at t for som origin-dstination pair (s, t) V 2. W dnot th st of zro rducd cost links with rspct to sourc s as ERC s W rquir that th UE flow (flow routd by th UEcontrollr) is routd solly via zro rducd cost links/paths. This is bcaus th UE controllr can only assign flow to minimal latncy paths (othrwis slf-intrstd agnts would dviat). th UE flow is also rquird to follow minimal marginal cost paths ls it cannot b a subflow of th flow.
Not that it is sufficint to only considr whthr or not a link is part of a rducd cost path from th origin s to som dstination t (not a spcific t) bcaus ithr link is along a rducd cost path from (s, t), or thr is no path only along links in ERC s that includs. W can fficintly comput th st of zro rducd cost links for any origin dstination pair (s, t) by applying uniform cost sarch from s to t and marking all links that ar part of optimal paths, onc with rgard to minimal total latncy (arg min p Ps,t (l p (f )), and scond with rgard to minimal marginal cost (arg min p Ps,t (c p(f )). Lt th constant f dnot th flow vctor at a solution. 3 Th flow is not uniqu whn latncy functions ar non-dcrasing, and th maximal amount of UE flow prmittd may, in gnral, dpnd on th spcific flow. Thrfor, w must fficintly sarch ovr th spac of flows. This is possibl du to th following lmmas. Lmma 1. For any two flows that achiv, f and ˆf, l (f ) = l ( ˆf ). Proof. Givn Assumption 2, a flow is th solution to a convx program (Roughgardn and Tardos 2002). Th solutions to a convx program form a convx st. Suppos that thr ar two flows that both achiv, but for which f ˆf. Thn c (f ) = l (f )f must b a linar function btwn f and ˆf (to s this, not that any convx combination of f and ˆf is also an solution, but if c (f ) is not linar, thn th total systm travl tim would b strictly lss, a contradiction). Sinc l (f ) is a nondcrasing function, th only way for c (f ) to b linar is for l (f ) to b constant btwn f and ˆf. Lmma 2. Th st of zro rducd cost paths is idntical for all solutions. Proof. By Lmma 1, all flows hav th sam latncy on ach link, so th solutions can diffr by at most flows along a st of links with constant latncy ovr th rang of which th two flows diffr on thos links. Sinc w assum that th latncy functions ar diffrntiabl, th drivativs of th latncy function ar zro ovr th rang at which thy ar constant. Thrfor, c (f ) = l (f ) + f l (f ) is constant ovr th rang as wll. This implis that any path that is rducd cost in on flow is also rducd cost in th othr flow, sinc th latncy functions and c (f ) ar constant for vry link. f Dfin th constant = sup{f : l (f) = l (f )}, i.. f is th largst flow valu such that th latncy on link is qual to th latncy at a solution. Not that if l is strictly incrasing at f, thn f = f. Howvr, if l is constant at f, thn f > f. Givn that th zro rducd cost paths ar th sam for all flows (Lmma 2), and any flow has th sam latncy on all links (Lmma 1), it will b sufficint to only sarch ovr flows that ar lss than f on ach link E. 3 A flow can b fficintly computd as a solution to a convx program (Roughgardn and Tardos 2002; Dial 2006). For ach vrtx, s, and link,, dfin variabl x s dnoting th amount of UE flow originating from sourc s that is assignd to link. Lt in(v) dnot th st of links for which v is th tail vrtx and out(v) th st of links for which v is th had vrtx. Dfinition 3. For a givn flow modl {G, R}, th UE linar program is: max r rs,t s,t (1),xs s,t V 2 subjct to rs,t R(s, t) s, t V 2 (2) rs,t s V (3) x s = out(s) t V x s in(t) x s s out(t) x s = r s,v s, t V 2 (4) f E, s V (5) x s 0, r s,t 0 s, t V, E (6) x s = 0 s V, E \ E s RC (7) Th flow f UE = v xv dfind by a fasibl solution to th UE linar program (givn constraints (2)-(7)) is a UE subflow. Th flow dfind by an optimal solution to th UE linar program is an optimal UE subflow. Not that th numbr of variabls is { s V, t V, E : r s,t, x s } = O( V 2 + V E ), and th numbr of constraints is also O( V 2 + V E ). Thrfor, sinc th numbr of variabls and constraints ar polynomial in th flow modl, th optimal solution to th UE linar program can b computd in polynomial tim (Karmarkar 1984). Thorm 1. A UE subflow, f UE, dfind by a fasibl solution to th UE linar program is a subflow of a flow. Proof. First, not that by quations (2) (4), th UE subflow, f UE, satisfis flow consrvation constraints. Equation (2) stats that th flow along all zro rducd cost paths from origin s to dstination t must b lss thn total dmand for (s, t). Thn quations (3) and (4) stat that th flow out of nod v must ithr b du to th dmand gnratd by nod v or th flow into it, minus th flow that rachs v as a dstination. Thrfor, f UE is a subflow of a fasibl flow. What must b shown is that thr must xist a flow, f, such that f UE f for all. If is such that l is strictly incrasing at an solution, and thrfor will b strictly incrasing at all solutions by Lmma 1, thn f = f and constraint (5) guarants this claim. Lt E b th st of links such that th latncy function is constant at a flow. Thrfor, it only nds to b shown that thr xists a solution, f, such that for E, f UE f. Suppos that thr xistd a st of links E such that for all flows f, f UE > f. Lt ˆf b an flow. Thn thr must xist an origin dstination pair
(s, t) such that thr ar two sts of paths P >, P < P s,t for which for all p P >, fp UE > ˆf p, and for all p P <, fp UE < ˆf p and all paths only diffr by links in E. This is bcaus th total flow btwn any origindstination is largr in th flow by quation (2). Morovr, p P > (fp UE ˆf p ) p P < ( ˆf p fp UE ) sinc th flow along non-constant latncy links constrains th total flow. Mov p P > (fp UE ˆf p ) units of flow from paths in st P > to paths in st P < in th flow ˆf. Dnot th nw flow by f. Th total travl tim for f cannot incras bcaus th flow has only incrasd on constant latncy links, and th nw flow dos not xcd f on any link. Th total travl tim also cannot hav dcrasd bcaus ˆf was an flow, so f is also an flow. Continu this procdur until thr dos not xist a link E for which f UE xcds th transformd flow. Thn w hav constructd an flow, f, in which, for all links E, f, a contradiction. f UE Lmma 3. For a ntwork {G, R}, lt f b a subflow of a fasibl flow f. Thn th flow f such that f = f f is also a subflow of f. Proof. First, 0 f f, by th dfinition of a subflow. Now st r s,t = R(s, t) rs,t. Thn for all s, t V 2, out(s) f in(s) f = t (R(s, t) r s,t) = t r s,t, and similarly for in(t) f out(t) f Thorm 2. Th optimal valu of th UE linar program for a ntwork instanc {G, R} is th maximum amount of UE agnts that th ntwork can support and achiv. Proof. First, by Thorm 1, thr xists an flow such that th optimal UE subflow, f UE, is a subflow of th flow, and by Lmma 3, thr xists a subflow of compliant agnts that can achiv th solution. Morovr, by th dfinition of th UE linar program and Lmma 2, th UE flow is only along zro rducd cost paths. By th dfinition of zro rducd cost paths, all UE agnts ar willing to tak th assignd paths. Thrfor, th solution is achivabl with th UE flow, and thr is som volum of UE flow that is qual to th objctiv of th UE linar program. Now, suppos that thr was anothr UE flow assignmnt, f, for which compliant flow could b assignd in such a way that th total systm travl tim was achivd and th total UE flow volum was largr than th valu rturnd by th UE linar program. Not that this flow assignmnt (f ) must b a subflow of som flow, f. Morovr, by th dfinition of UE flow and th fact that all paths in a solution ar minimum marginal cost paths, all paths assignd with a UE flow gratr than zro must b a zro rducd cost path. Thrfor, th flow f satisfis th quations (2)-(6), and sinc th UE linar program rturns th optimal UE flow assignmnt undr ths constraints, this is a contradiction. Whil w v dmonstratd that w can comput th maximal UE flow that prmits an solution givn th appropriat assignmnt of th compliant flow, it is likly that a mor common problm would b to dtrmin, for a givn st of compliant agnts, whthr or not it is possibl to achiv with that st. Our mthodology also provids an answr to this qustion, as th following Corollary dmonstrats. Corollary 1. For a givn ntwork instanc {G, R} and givn a st of compliant dmand, rs,t, C from ach origin dstination pair s, t V 2, thr xists a compliant flow f C such that th ntwork achivs if and only if thr xists an x s for all s V and E such that rs,t UE = R(s, t) rs,t C and x s ar a solution to th UE linar program. Proof. By Thorm 1, any solution to th UE linar program dfins a subflow of an flow. Thrfor, if rs,t UE and x s is a solution, thr xists an assignmnt of th compliant flow that achivs. Morovr, if thr xists an assignmnt of th complaint flow, f C, such that a UE subflow with dmands rs,t UE achivs systm optimum, thn th UE flow is only along zro rducd cost paths by dfinition of UE flow and, and th UE subflow is fasibl. Thrfor, th dcomposd UE flow satisfis th constraints of th linar program. Flow Assignmnt for Compliant Agnts Givn that w can now dtrmin both th maximal amount of UE flow that a systm can tolrat and achiv systm optimum and, for a givn st of compliant agnts, whthr or not a systm can achiv optimum, w ar only lft with assigning th compliant flow to paths. This sction tackls th qustion of how to assign paths to a, sufficintly larg, st of compliant agnts such that is achivd. Th mthodology from th prvious sction immdiatly suggsts a solution. Givn a ntwork instanc {G, R}, suppos that w hav compliant dmand qual to rs,t C for all s, t V 2. Thn w must find a flow, f, such that rs,t C and rs,t UE = R(s, t) rs,t C prmit subflows of th solution. Such a flow must xist by Thorm 1 and Corollary 1. Th first stp is to comput th UE subflow, f UE, givn UE dmand. From th prvious sction: this xists and is computationally tractabl. Any fasibl subflow, f C, with dmand rs,t C such that th total flow along link satisfis f C + f UE f has latncy qual to th solution, and th flow f C + f UE, by Lmma 1, is an solution. W can comput f C with th following linar program: subjct to f C out(v) in(v) 0 f C max f C 1 out(v) f C = s (r C v,t) v V f C in(v) = t f C (rs,v) C v V f f UE E W know that a solution to th abov linar program xists and it can b computd tractably. Th final stp is to dcompos th compliant flow, f C, into a pr path assignmnt for ach origin-dstination pair
Figur 1: Thr rprsntativ ntwork topologis: I - Sioux Falls, SD, II - Eastrn Massachustts (Ellipsoids rprsnt diffrnt zons), III - Anahim, CA. (s, t) in ordr to assign individual agnts to a path. This can b don in tim O( V E ) using standard flow dcomposition algorithms (s Sction 3.5 of Ahuja, Magnanti, t. al. (1993) for a discussion). Exprimntal Rsults W ar intrstd in th viability of opt-in micro-tolling schms to mor fficintly utiliz road ntworks. As such, w havn undrtakn an mpirical study to invstigat th minimal amount of compliant flow rquird for (r UE ) in six ralistic traffic scnarios ovr actual road ntworks. Scnarios Each traffic scnario is dfind by th following attributs: 1. Th road ntwork, G(V, E), spcifying th st of vrtics and links whr ach link is affiliatd with a lngth, capacity and spd limit. Ntworks ar, following standard practic, partitiond into traffic analysis zons (TAZs) and ach zon contains a nod blonging to V calld th cntroid. All traffic originating and trminating within th zon is assumd to ntr and lav th ntwork at th cntroid. 2. A trip tabl which spcifis th traffic dmand btwn pairs of cntroids. Th dmand function R btwn nods othr than cntroids is st to zro. Th following bnchmark scnarios wr chosn both for thir divrsity of topology and traffic volum and thir widsprad us within th traffic litratur: Sioux Falls, Eastrn Massachustts, Anahim, Chicago Sktch, Philadlphia, and Chicago-rgional. All traffic scnarios ar availabl at: https://github.com/bstablr/ TransportationNtworks. Figur 1 dpicts thr rprsntativ ntwork topologis (th thr smallst ntworks). Th Traffic Modl A macroscopic modl was usd in ordr to valuat traffic formation. Macroscopic modls calculat th UE in a givn scnario using algorithm B (Dial 2006). For all scnarios, th modl assumd that travl tims follow th Burau of Public Roads (BPR) function (Moss and Mtoi 2017) with th commonly usd paramtrs β = 4, α = 0.15. Th solution is computd by rplacing th latncy functions with c (x) and using algorithm B to obtain th quilibrium solution (Dial 1999). Sinc solving for th UE and solutions rquirs solving a convx program (Dial 2006), w only solv thm to a crtain prcision. To masur convrgnc, givn an assignmnt of agnts to paths, w dfin th avrag xcss cost (AEC) as th avrag diffrnc btwn th travl tims on paths takn by th agnts and thir shortst altrnativ path. Th algorithm trminats whn th AEC is lss than 1E-12 minuts (xcpt for Chicago-rgional for which 1E-10 was usd du to th siz of th ntwork). Thrfor, a minimum marginal cost path is only a minimum up to a thrshold. A link is dfind to b zro rducd cost with rspct to s if it carris flow originating at s in th solution (i.., th link blongs to a minimum marginal cost path) and if th diffrnc btwn th last latncy path that includs and th last latncy unrstrictd path, both lading from s to th had vrtx of, is lss than a thrshold T. Th thrshold T is dfind as follows: for ach origin s and link w calculat th last marginal cost path (c ) lading from s to th had vrtx of at th solution. W do this onc whil rstricting th path to includ and onc without such rstriction. Th diffrnc btwn ths two valus is stord and T is st to b th maximum of ths diffrnc across all th links and origins in th ntwork.
Scnario Vrtics Links Zons Total Flow UE TTT TTT % Improv Thrshold % compliant Sioux Falls 24 76 24 360,600 7,480,225 7,194,256 3.82 6.19E-11 13.04 Eastrn MA 74 258 74 65,576 28,181 27,323 3.04 3.04E-13 19.73 Anahim 416 914 38 104,694 1,419,913 1,395,015 1.75 8.05E-11 19.76 Chicago S 933 2,950 387 1,260,907 18,377,329 17,953,267 2.31 9.14E-10 27.29 Philadlphia 13,389 40,003 1525 18,503,872 335,647,106 324,268,465 3.39 4.20E-09 49.59 Chicago R 12,982 39,018 1790 1,360,427 33,656,964 31,942,956 5.09 4.14E-07 53.34 Tabl 1: Rquird fraction of compliant agnts givn as % compliant for diffrnt scnarios along with ntwork spcifications for ach scnario: numbr of vrtics, links and zons followd by th Total Travl Tim (TTT) at UE (0% compliant agnts) and (100% compliant agnts). Th prcntag of improvmnt of th TTT ovr th UE TTT is givn as % improv. Rsults Tabl 1 prsnts th prcntag of flow that must b compliant in ordr to guarant an solution for six diffrnt traffic scnarios. Each scnario is affiliatd with th numbr of vrtics, links, and zons comprising th affiliatd road ntwork as wll as th numbr of trips that mak up th affiliatd dmand. Th columns UE TTT and TTT rprsnt th total travl tim (in minuts) ovr all agnts for th cas whr 100% of th agnts ar controlld by th UE controllr (UE solution) and whn 100% of th agnts ar controlld by th controllr ( solution) rspctivly. Th prcntag of improvmnt in total travl tim btwn UE TTT and TTT is also shown undr % improv. Th prcntag of rquird compliant flow (formally rue / R whr R = s,t R(s, t)) as computd by th UE linar program (Dfinition 3) is prsntd for ach scnario undr % compliant. 4 Th rsults suggst that as th siz of th ntwork (i.., th numbr of nods and vrtics) incrass, a gratr fraction of compliant travlrs ar ndd to nsur th ntwork achivs systm optimum. This appars to b du to an incrasing numbr of usd paths at th solution as th ntwork siz incrass. As th numbr of paths grow, th st of zro rducd cost paths grows mor slowly, and, thrfor, a highr prcntag of compliant agnts is rquird. Summary This papr discussd a scnario whr a st of agnts travrs a congstd ntwork, whil a cntralizd ntwork managr is sking to optimiz th flow (minimizs total latncy) by influncing th rout assignmnt of a st of compliant agnts. A mthodology was prsntd for computing th minimal volum of traffic flow that nds to b compliant in ordr to rach a stat of optimal traffic flow. Morovr, th mthodology xtnds to infrring which agnts should b compliant and how xactly th compliant agnts should b assignd to paths. Exprimntal rsults dmonstrat that th rquird prcntag of agnts that ar compliant is small for som scnarios but can b gratr than 50% in othrs. Going forward, it would b worthwhil to xplor th possibility of approximation algorithms for assigning compliant flow whn th UE flow volum is too larg to achiv a stat 4 Statistical analysis for Tabl 1 is not prsntd, as th macroscopic modl is dtrministic. of systm optimum. Givn that th optimal solution to this problm is known to b NP-hard, an fficint approximation algorithm would b a usful tool as opt-in ntwork routing systms ar implmntd. Furthr, in ordr to limit th ncssary opt-in incntivs, thr is work ndd to dvlop systms that targt particularly influntial usrs to opt-in to ths systms. Acknowldgmnts A portion of this work has takn plac in th Larning Agnts Rsarch Group (LARG) at UT Austin. LARG rsarch is supportd in part by NSF (IIS-1637736, IIS- 1651089, IIS-1724157), Intl, Raython, and Lockhd Martin. Ptr Ston srvs on th Board of Dirctors of Cogitai, Inc. Th trms of this arrangmnt hav bn rviwd and approvd by th Univrsity of Txas at Austin in accordanc with its policy on objctivity in rsarch. Th authors would lik to thank th Txas Dpartmnt of Transportation for supporting this rsarch undr projct 0-6838, Bringing Smart Transport to Txans: Ensuring th Bnfits of a Connctd and Autonomous Transport Systm in Txas. Th authors would also lik to acknowldg th support of th Data-Supportd Transportation Oprations & Planning Cntr and th National Scinc Foundation undr Grant No. 1254921. Finally, w wish to acknowldg th hlp of Michal Lvin and Josiah Hana to this projct. Rfrncs Ahuja, R. K.; Magnanti, T. L.; and Orlin, J. B. 1993. Ntwork Flows: Thory, Algorithms, and Applications. Prntic Hall, 1 dition. Bckmann, M.; McGuir, C. B.; and Winstn, C. B. 1956. Studis in th Economics of Transportation. Yal Univrsity Prss. Dial, R. B. 1999. Ntwork-optimizd road pricing: Part i: a parabl and a modl. Informs 47. Dial, R. B. 2006. A path-basd usr-quilibrium traffic assignmnt algorithm that obviats path storag and numration. Transportation Rsarch Part B: Mthodological 40(10):917 936. Ditrich, B. 1969. Übr in Paradoxon aus dr Vrkhrsplanung. Untrnhmnsforschung 12:258 268. Duffild, N. G.; Goyal, P.; Grnbrg, A.; Mishra, P.; Ramakrishnan, K. K.; and van dr Mriv, J. E. 1999. A flxibl modl for rsourc managmnt in virtual privat
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