Resolving Braess s Paradox in Random Networks

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1 Reolving Brae Paradox in Random Nework Dimiri Foaki 1, Alexi C. Kapori 2, Thanai Lianea 1, and Paul G. Spiraki 3,4 1 Elecrical and Compuer Engineering, Naional Technical Univeriy of Ahen, Greece. 2 Informaion and Communicaion Syem Dep., Univeriy of he Aegean, Samo, Greece. 3 Deparmen of Compuer Science, Univeriy of Liverpool, UK 4 Compuer Technology Iniue and Pre Diophanu, Para, Greece. foaki@c.nua.gr, kaporia@gmail.com, lianea@mail.nua.gr, P.Spiraki@liverpool.ac.uk, piraki@ci.gr Abrac. Brae paradox ae ha removing a par of a nework may improve he player laency a equilibrium. In hi work, we udy he approximabiliy of he be ubnework problem for he cla of random G n,p inance proven prone o Brae paradox by (Roughgarden and Valian, RSA 2010) and (Chung and Young, WINE 2010). Our main conribuion i a polynomial-ime approximaion-preerving reducion of he be ubnework problem for uch inance o he correponding problem in a implified nework where all neighbor of and are direcly conneced by 0 laency edge. Building on hi, we obain an approximaion cheme ha for any conan ε > 0 and wih high probabiliy, compue a ubnework and an ε-nah flow wih maximum laency a mo (1+ε)L +ε, where L i he equilibrium laency of he be ubnework. Our approximaion cheme run in polynomial ime if he random nework ha average degree O(poly(ln n)) and he raffic rae i O(poly(ln ln n)), and in quaipolynomial ime for average degree up o o(n) and raffic rae of O(poly(ln n)). 1 Inroducion An inance of a (non-aomic) elfih rouing game coni of a nework wih a ource and a ink, and a raffic rae r divided among an infinie number of player. Every edge ha a non-decreaing funcion ha deermine he edge laency caued by i raffic. Each player roue a negligible amoun of raffic hrough an pah. Oberving he raffic caued by oher, every player elec an pah ha minimize he um of edge laencie. Thu, he player reach a Nah equilibrium (a.k.a., a Wardrop equilibrium), where all player ue pah of equal minimum laency. Under ome general aumpion on he laency funcion, a Nah equilibrium flow (or imply a Nah flow) exi and he common player laency in a Nah flow i eenially unique (ee e.g., [14]). Previou Work. I i well known ha a Nah flow may no opimize he nework performance, uually meaured by he oal laency incurred by all player. Thu, in he Thi reearch wa uppored by he projec Algorihmic Game Theory, co-financed by he European Union (European Social Fund - ESF) and Greek naional fund, hrough he Operaional Program Educaion and Lifelong Learning of he Naional Sraegic Reference Framework (NSRF) - Reearch Funding Program: THALES, inveing in knowledge ociey hrough he European Social Fund, by he ERC projec RIMACO, and by he EU FP7/ (DG INFSO G4-ICT for Tranpor) under Gran Agreemen no (Projec ecompa).

2 2 D. Foaki, A.C. Kapori, T. Lianea, and P.G. Spiraki l 1 x x v l 2 x l 1 x x v l 2 x l 3 x l 4 x w l 5 x x l 4 x w l 5 x x Fig. 1. (a) The opimal oal laency i 3/2, achieved by rouing half of he flow on each of he pah (, v, ) and (, w, ). In he (unique) Nah flow, all raffic goe hrough he pah (, v, w, ) and ha a laency of 2. (b) If we remove he edge (v, w), he Nah flow coincide wih he opimal flow. Hence he nework (b) i he be ubnework of nework (a). la decade, here ha been a ignifican inere in quanifying and underanding he performance degradaion due o he player elfih behavior, and in miigaing (or even eliminaing) i uing everal approache, uch a inroducing economic diincenive (oll) for he ue of congeed edge, or exploiing he preence of cenrally coordinaed player (Sackelberg rouing), ee e.g., [14] and he reference herein. A imple way o improve he nework performance a equilibrium i o exploi Brae paradox [3], namely he fac ha removing ome edge may improve he laency of he Nah flow 5 (ee e.g., Fig. 1 for an example). Thu, given an inance of elfih rouing, one naurally eek for he be ubnework, i.e. he ubnework minimizing he common player laency a equilibrium. Compared again Sackelberg rouing and oll, edge removal i impler and more appealing o boh he nework adminiraor and he player (ee e.g., [6] for a dicuion). Unforunaely, Roughgarden [15] proved ha i i NP-hard no only o find he be ubnework, bu alo o compue any meaningful approximaion o i equilibrium laency. Specifically, he proved ha even for linear laencie, i i NP-hard o approximae he equilibrium laency of he be ubnework wihin a facor of 4/3 ε, for any ε > 0, i.e., wihin any facor le han he wor-cae Price of Anarchy for linear laencie. On he poiive ide, applying Alhöfer Sparificaion Lemma [1], Foaki, Kapori, and Spiraki [6] preened an algorihm ha approximae he equilibrium laency of he be ubnework wihin an addiive erm of ε, for any conan ε > 0, in ime ha i ubexponenial if he oal number of pah i polynomial, all pah are of polylogarihmic lengh, and he raffic rae i conan. Inereingly, Brae paradox can be dramaically more evere in nework wih muliple ource and ink. More pecifically, Lin e al. [8] proved ha for nework wih a ingle ource-ink pair and general laency funcion, he removal of a mo k edge canno improve he equilibrium laency by a facor greaer han k + 1. On he oher hand, Lin e al. [8] preened a nework wih wo ource-ink pair where he removal of a ingle edge improve he equilibrium laency by a facor of 2 Ω(n). A for 5 Due o pace conrain, we have rericed he dicuion of relaed work o he mo relevan reul on he exience and he eliminaion of Brae paradox. There ha been a large body of work on quanifying and miigaing he conequence of Brae paradox on elfih raffic, epecially in he area of Tranporaion Science and Compuer Nework. The inereed reader may ee e.g., [15,12] for more reference.

3 Reolving Brae Paradox in Random Nework 3 he impac of he nework opology, Milchaich [11] proved ha Brae paradox doe no occur in erie-parallel nework, which i preciely he cla of nework ha do no conain he nework in Fig. 1.a a a opological minor. Recen work acually indicae ha he appearance of Brae paradox i no an arifac of opimizaion heory, and ha edge removal can offer a angible improvemen on he performance of real-world nework (ee e.g., [7,13,14,16]). In hi direcion, Valian and Roughgarden [17] iniiaed he udy of Brae paradox in naural clae of random nework, and proved ha he paradox occur wih high probabiliy in dene random G n,p nework, wih p = ω(n 1/2 ), if each edge e ha a linear laency l e (x) = a e x + b e, wih a e, b e drawn independenly from ome reaonable diribuion. The ubequen work of Chung and Young [4] exended he reul of [17] o pare random nework, where p = Ω(ln n/n), i.e., ju greaer han he conneciviy hrehold of G n,p, auming ha he nework ha a large number of edge e wih mall addiive laency erm b e. In fac, Chung and Young demonraed ha he crucial propery for Brae paradox o emerge i ha he ubnework coniing of he edge wih mall addiive erm i a good expander (ee alo [5]). Neverhele, he proof of [4,17] i merely exienial; i provide no clue on how one can acually find (or even approximae) he be ubnework and i equilibrium laency. Moivaion and Conribuion. The moivaing queion for hi work i wheher in ome inereing eing, where he paradox occur, we can efficienly compue a e of edge whoe removal ignificanly improve he equilibrium laency. From a more echnical viewpoin, our work i moivaed by he reul of [4,17] abou he prevalence of he paradox in random nework, and by he knowledge ha in random inance ome hard (in general) problem can acually be racable. Deparing from [4,17], we adop a purely algorihmic approach. We focu on he cla of o-called good elfih rouing inance, namely inance wih he properie ued by [4,17] o demonrae he occurrence of Brae paradox in random nework wih high probabiliy. In fac, one can eaily verify ha he random inance of [4,17] are good wih high probabiliy. Raher urpriingly, we prove ha, in many inereing cae, we can efficienly approximae he be ubnework and i equilibrium laency. Wha may be even more urpriing i ha our approximaion algorihm i baed on he expanion propery of good inance, namely he very ame propery ued by [4,17] o eablih he prevalence of he paradox in good inance! To he be of our knowledge, our reul are he fir of heoreical naure which indicae ha Brae paradox can be efficienly eliminaed in a large cla of inereing inance. Technically, we preen eenially an approximaion cheme. Given a good inance and any conan ε > 0, we compue a flow g ha i an ε-nah flow for he ubnework coniing of he edge ued by i, and ha a laency of L(g) (1 + ε)l + ε, where L i he equilibrium laency of he be ubnework (Theorem 1). In fac, g ha hee properie wih high probabiliy. Our approximaion cheme run in polynomial ime for he mo inereing cae ha he nework i relaively pare and he raffic rae r i O(poly(ln ln n)), where n i he number of verice. Specifically, he running ime i polynomial if he good nework ha average degree O(poly(ln n)), i.e., if pn = O(poly(ln n)), for random G n,p nework, and quaipolynomial for average degree up o o(n). A for he raffic rae, we emphaize ha mo work on elfih rouing

4 4 D. Foaki, A.C. Kapori, T. Lianea, and P.G. Spiraki and elfih nework deign problem aume ha r = 1, or a lea ha r doe no increae wih he nework ize (ee e.g., [14] and he reference herein). So, we can approximae, in polynomial-ime, he be ubnework for a large cla of inance ha, wih high probabiliy, include exponenially many pah and pah of lengh Θ(n). For uch inance, a direc applicaion of [6, Theorem 3] give an exponenialime algorihm. The main idea behind our approximaion cheme, and our main echnical conribuion, i a polynomial-ime approximaion-preerving reducion of he be ubnework problem for a good nework G o a correponding be ubnework problem for a 0- laency implified nework G 0, which i a layered nework obained from G if we keep only, and heir immediae neighbor, and connec all neighbor of and by direc edge of 0 laency. We fir how ha he equilibrium laency of he be ubnework doe no increae when we conider he 0-laency implified nework G 0 (Lemma 1). Alhough hi may ound reaonable, we highligh ha decreaing edge laencie o 0 may rigger Brae paradox (e.g., aring from he nework in Fig. 1.a wih l 3(x) = 1, and decreaing i o l 3 (x) = 0 i ju anoher way of riggering he paradox). Nex, we employ Alhöfer Sparificaion Lemma [1] (ee alo [9,10] and [6, Theorem 3]) and approximae he be ubnework problem for he 0-laency implified nework. The final (and crucial) ep of our approximaion preerving reducion i o ar wih he flow-oluion o he be ubnework problem for he 0-laency implified nework, and exend i o a flow-oluion o he be ubnework problem for he original (good) inance. To hi end, we how how o imulae 0-laency edge by low laency pah in he original good nework. Inuiively, hi work becaue due o he expanion properie and he random laencie of he good nework G, he inermediae ubnework of G, connecing he neighbor of o he neighbor of, eenially behave a a complee biparie nework wih 0-laency edge. Thi i alo he key ep in he approach of [4,17], howing ha Brae paradox occur in good nework wih high probabiliy (ee [4, Secion 2] for a deailed dicuion). Hence, one could ay ha o ome exen, he reaon ha Brae paradox exi in good nework i he very ame reaon ha he paradox can be efficienly reolved. Though concepually imple, he full conrucion i echnically involved and require dealing wih he amoun of flow hrough he edge inciden o and and heir laencie. Our conrucion employ a careful grouping-and-maching argumen, which work for good nework wih high probabiliy, ee Lemma 4 and 5. We highligh ha he reducion ielf run in polynomial ime. The ime conuming ep i he applicaion of [6, Theorem 3] o he 0-laency implified nework. Since uch nework have only polynomially many (and very hor) pah, hey ecape he hardne reul of [15]. The approximabiliy of he be ubnework for 0-laency implified nework i an inriguing open problem ariing from our work. Our reul how ha a problem, ha i NP-hard o approximae, can be very cloely approximaed in random (and random-like) nework. Thi reemble e.g., he problem of finding a Hamilonian pah in Erdö-Rényi graph, where again, exience and conrucion boh work ju above he conneciviy hrehold, ee e.g., [2]. However, no all hard problem are eay when one aume random inpu (e.g., conider facoring or he hidden clique problem, for boh of which no uch reul are known in full deph).

5 2 Model and Preliminarie Reolving Brae Paradox in Random Nework 5 Noaion. For an even E in a ample pace, P[E] denoe he probabiliy of E happening. We ay ha an even E occur wih high probabiliy, if P[E] 1 n α, for ome conan α 1, where n uually denoe he number of verice of he nework G o which E refer. We implicily ue he union bound o accoun for he occurrence of more han one low probabiliy even. Inance. A elfih rouing inance i a uple G = (G(V, E), (l e ) e E, r), where G(V, E) i an undireced nework wih a ource and a ink, l e : R 0 R 0 i a non-decreaing laency funcion aociaed wih each edge e, and r > 0 i he raffic rae. We le P (or P G, whenever he nework G i no clear from he conex) denoe he (non-empy) e of imple pah in G. For breviy, we uually omi he laency funcion, and refer o a elfih rouing inance a (G, r). We only conider linear laencie l e (x) = a e x + b e, wih a e, b e 0. We reric our aenion o inance where he coefficien a e and b e are randomly eleced from a pair of diribuion A and B. Following [4,17], we ay ha A and B are reaonable if: A ha bounded range [A min, A max ] and B ha bounded range [0, B max ], where A min > 0 and A max, B max are conan, i.e., hey do no depend on r and V. There i a cloed inerval I A of poiive lengh, uch ha for every non-rivial ubinerval I I A, P a A [a I ] > 0. There i a cloed inerval I B, 0 I B, of poiive lengh, uch ha for every nonrivial ubinerval I I B, P b B [b I ] > 0. Moreover, for any conan η > 0, here exi a conan δ η > 0, uch ha P b B [b η] δ η. Subnework. Given a elfih rouing inance (G(V, E), r), any ubgraph H(V, E ), V V, E E,, V, obained from G by edge and verex removal, i a ubnework of G. H ha he ame ource and ink a G, and he edge of H have he ame laencie a in G. Every inance (H(V, E ), r), where H(V, E ) i a ubnework of G(V, E), i a ubinance of (G(V, E), r). Flow. Given an inance (G, r), a (feaible) flow f i a non-negaive vecor indexed by P uch ha q P f q = r. For a flow f, le f e = q:e q f q be he amoun of flow ha f roue on edge e. Two flow f and g are differen if here i an edge e wih f e g e. An edge e i ued by flow f if f e > 0, and a pah q i ued by f if min e q {f e } > 0. We ofen wrie f q > 0 o denoe ha a pah q i ued by f. Given a flow f, he laency of each edge e i l e (f e ), he laency of each pah q i l q (f) = e q l e(f e ), and he laency of f i L(f) = max q:fq>0 l q (f). We omeime wrie L G (f) when he nework G i no clear from he conex. For an inance (G(V, E), r) and a flow f, we le E f = {e E : f e > 0} be he e of edge ued by f, and G f (V, E f ) be he correponding ubnework of G. Nah Flow. A flow f i a Nah (equilibrium) flow, if i roue all raffic on minimum laency pah. Formally, f i a Nah flow if for every pah q wih f q > 0, and every pah q, l q (f) l q (f). Therefore, in a Nah flow f, all player incur a common laency L(f) = min q l q (f) = max q:fq>0 l q (f) on heir pah. A Nah flow f on a nework G(V, E) i a Nah flow on any ubnework G (V, E ) of G wih E f E.

6 6 D. Foaki, A.C. Kapori, T. Lianea, and P.G. Spiraki Every inance (G, r) admi a lea one Nah flow, and he player laency i he ame for all Nah flow (ee e.g., [14]). Hence, we le L(G, r) be he player laency in ome Nah flow of (G, r), and refer o i a he equilibrium laency of (G, r). For linear laency funcion, a Nah flow can be compued efficienly, in rongly polynomial ime, while for ricly increaing laencie, he Nah flow i eenially unique (ee e.g., [14]). ε-nah flow. The definiion of a Nah flow can be naurally generalized o ha of an almo Nah flow. Formally, for ome ε > 0, a flow f i an ε-nah flow if for every pah q wih f q > 0, and every pah q, l q (f) l q (f) + ε. Be Subnework. Brae paradox how ha here may be a ubinance (H, r) of an inance (G, r) wih L(H, r) < L(G, r) (ee e.g., Fig. 1). The be ubnework H of (G, r) i a ubnework of G wih he minimum equilibrium laency, i.e., H ha L(H, r) L(H, r) for any ubnework H of G. In hi work, we udy he approximabiliy of he Be Subnework Equilibrium Laency problem, or BeSubEL in hor. In BeSubEL, we are given an inance (G, r), and eek for he be ubnework H of (G, r) and i equilibrium laency L(H, r). Good Nework. We reric our aenion o undireced nework G(V, E). We le n V and m E. For any verex v, we le Γ (v) = {u V : {u, v} E} denoe he e of v neighbor in G. Similarly, for any non-empy S V, we le Γ (S) = v S Γ (v) denoe he e of neighbor of he verice in S, and le G[S] denoe he ubnework of G induced by S. For convenience, we le V Γ (), E {{, u} : u V }, V Γ (), E {{v, } : v V }, and V m V \ ({, } V V ). We alo le n = V, n = V, n + = max{n, n }, n = min{n, n }, and n m = V m. We omeime wrie V (G), n(g), V (G), n (G),..., if G i no clear from he conex. I i convenien o hink ha he nework G ha a layered rucure coniing of, he e of neighbor V, an inermediae ubnework connecing he neighbor of o he neighbor of, he e of neighbor V, and. Then, any pah ar a, vii ome u V, proceed eiher direcly or hrough ome verice of V m o ome v V, and finally reache. Thu, we refer o G m G[V V m V ] a he inermediae ubnework of G. Depending on he rucure of G m, we ay ha: G i a random G n,p nework if (i) n and n follow he binomial diribuion wih parameer n and p, and (ii) if any edge {u, v}, wih u V m V and v V m V, exi independenly wih probabiliy p. Namely, he inermediae nework G m i an Erdö-Rényi random graph wih n 2 verice and edge probabiliy p, excep for he fac ha here are no edge in G[V ] and in G[V ]. G i inernally biparie if he inermediae nework G m i a biparie graph wih independen e V and V. G i inernally complee biparie if every neighbor of i direcly conneced by an edge o every neighbor of. G i 0-laency implified if i i inernally complee biparie and every edge e connecing a neighbor of o a neighbor of ha laency funcion l e (x) = 0. The 0-laency implificaion G 0 of a given nework G i a 0-laency implified nework obained from G by replacing G[V m ] wih a e of 0-laency edge direcly connecing every neighbor of o every neighbor of. Moreover, we ay ha a 0-laency implified nework G i balanced, if n n 2n.

7 Reolving Brae Paradox in Random Nework 7 Algorihm 1: Approximaion Scheme for BeSubEL in Good Nework Inpu: Good nework G(V, E), rae r > 0, approximaion guaranee ε > 0 Oupu: Subnework H of G and ε-nah flow g in H wih L(g) (1 + ε)l(h, r) + ε 1 if L(G, r) < ε, reurn G and a Nah flow of (G, r) ; 2 creae he 0-laency implificaion G 0 of G ; 3 if r (B maxn +)/(εa min), hen le H 0 = G 0 and le f be a Nah flow of (G 0, r) ; 4 ele, le H 0 be he ubnework and f he ε/6-nah flow of Thm. 2 applied wih error ε/6 ; 5 le H be he ubnework and le g be he ε-nah flow of Lemma 5 aring from H 0 and f ; 6 reurn he ubnework H and he ε-nah flow g ; We ay ha a nework G(V, E) i (n, p, k)-good, for ome ineger n V, ome probabiliy p (0, 1), wih pn = o(n), and ome conan k 1, if G aifie ha: 1. The maximum degree of G i a mo 3np/2, i.e., for any v V, Γ (v) 3np/2. 2. G i an expander graph, namely, for any e S V, Γ (S) min{np S, n}/2. 3. The edge of G have random reaonable laency funcion diribued according o A B, and for any conan η > 0, P b B [b η/ ln n] = ω(1/np). 4. If k > 1, we can compue in polynomial ime a pariioning of V m ino k e V 1 m,..., V k m, each of cardinaliy V m /k, uch ha all he induced ubnework G[{, } V V i m V ] are (n/k, p, 1)-good, wih a poible violaion of he maximum degree bound by and. If G i a random G n,p nework, wih n ufficienly large and p ck ln n/n, for ome large enough conan c > 1, hen G i an (n, p, k)-good nework wih high probabiliy (ee e.g., [2]), provided ha he laency funcion aify condiion (3) above. A for condiion (4), a random pariioning of V m ino k e of cardinaliy V m /k aifie (4) wih high probabiliy. Similarly, he random inance conidered in [4] are good wih high probabiliy. Alo noe ha he 0-laency implificaion of a good nework i balanced, due o (1) and (2). 3 The Approximaion Scheme and Ouline of he Analyi In hi ecion, we decribe he main ep of he approximaion cheme (ee alo Algorihm 1), and give an ouline of i analyi. We le ε > 0 be he approximaion guaranee, and aume ha L(G, r) ε. Oherwie, any Nah flow of (G, r) uffice. Algorihm 1 i baed on an approximaion-preerving reducion of BeSubEL for a good nework G o BeSubEL for he 0-laency implificaion G 0 of G. The fir ep of our approximaion-preerving reducion i o how ha he equilibrium laency of he be ubnework doe no increae when we conider he 0-laency implificaion G 0 of a nework G inead of G ielf. Since decreaing he edge laencie (e.g., decreaing l 3(x) = 1 o l 3 (x) = 0 in Fig. 1.a) may rigger Brae paradox, we need Lemma 1, in Secion 4, and i careful proof o make ure ha zeroing ou he laency of he inermediae ubnework doe no caue an abrup increae in he equilibrium laency. Nex, we focu on he 0-laency implificaion G 0 of G (ep 2 in Alg. 1). We how ha if he raffic rae i large enough, i.e., if r = Ω(n + /ε), he paradox ha

8 8 D. Foaki, A.C. Kapori, T. Lianea, and P.G. Spiraki a marginal influence on he equilibrium laency. Thu, any Nah flow of (G 0, r) i a (1 + ε)-approximaion of BeSubEL (ee Lemma 2 and ep 4). If r = O(n + /ε), we ue [6, Theorem 3] and obain an ε/6-approximaion of BeSubEL for (G 0, r) (ee Theorem 2 and ep 4). We now have a ubnework H 0 and an ε/6-nah flow f ha comprie a good approximae oluion o BeSubEL for he implified inance (G 0, r). The nex ep of our approximaion-preerving reducion i o exend f o an approximae oluion o BeSubEL for he original inance (G, r). The inuiion i ha due o he expanion and he reaonable laencie of G, any collecion of 0-laency edge of H 0 ued by f o roue flow from V o V can be imulaed by an appropriae collecion of low-laency pah of he inermediae ubnework G m of G. In fac, hi obervaion wa he key ep in he approach of [4,17] howing ha Brae paradox occur in good nework wih high probabiliy. We fir prove hi claim for a mall par of H 0 coniing only of neighbor of and neighbor of wih approximaely he ame laency under f (ee Lemma 4, he proof draw on idea from [4, Lemma 5]). Then, uing a careful laencybaed grouping of he neighbor of and of he neighbor of in H 0, we exend hi claim o he enire H 0 (ee Lemma 5). Thu, we obain a ubnework H of G and an ε-nah flow g in H uch ha L(g) (1 + ε)l(h, r) + ε (ep 5). We ummarize our main reul. The proof follow by combining Lemma 1, Theorem 2, and Lemma 5 in he way indicaed by Algorihm 1 and he dicuion above. Theorem 1. Le G(V, E) be an (n, p, k)-good nework, where k 1 i a large enough conan, le r > 0 be any raffic rae, and le H be he be ubnework of (G, r). Then, for any ε > 0, Algorihm 1 compue in ime n O(r2 A 2 max ln(n+)/ε2 ) + poly( V ), a flow g and a ubnework H of G uch ha wih high probabiliy, wr. he random choice of he laency funcion, g i an ε-nah flow of (H, r) and ha L(g) (1+ε)L(H )+ε. By he definiion of reaonable laencie, A max i a conan. Alo, by Lemma 2, r affec he running ime only if r = O(n + /ε). In fac, previou work on elfih nework deign aume ha r = O(1), ee e.g., [14]. Thu, if r = O(1) (or more generally, if r = O(poly(ln ln n))) and pn = O(poly(ln n)), in which cae n + = O(poly(ln n)), Theorem 1 give a randomized polynomial-ime approximaion cheme for BeSubEL in good nework. Moreover, he running ime i quaipolynomial for raffic rae up o O(poly(ln n)) and average degree up o o(n), i.e., for he enire range of p in [4,17]. The nex ecion are devoed o he proof of Lemma 1 and 5, and of Theorem 2. 4 Nework Simplificaion We fir how ha he equilibrium laency of he be ubnework doe no increae when we conider he 0-laency implificaion G 0 of a nework G inead of G ielf. We highligh ha he following lemma hold no only for good nework, bu alo for any nework wih linear laencie and wih he layered rucure decribed in Secion 2. Lemma 1. Le G be any nework, le r > 0 be any raffic rae, and le H be he be ubnework of (G, r). Then, here i a ubnework H of he 0-laency implificaion of H (and hu, a ubnework of G 0 ) wih L(H, r) L(H, r).

9 Reolving Brae Paradox in Random Nework 9 Proof kech. We aume ha all he edge of H are ued by he equilibrium flow f of (H, r) (oherwie, we can remove all unued edge from H). The proof i conrucive, and a he concepual level, proceed in wo ep. For he fir ep, given he equilibrium flow f of he be ubnework H of G, we conruc a implificaion H 1 of H ha i inernally biparie and ha conan laency edge connecing Γ () o Γ (). H 1 alo admi f a an equilibrium flow, and hu L(H 1, r) = L(H, r). We can alo how how o furher implify H 1 o ha i inermediae biparie ubnework become acyclic. The econd par of he proof i o how ha we can eiher remove ome of he inermediae edge of H 1 or zero heir laencie, and obain a ubnework H of he 0-laency implificaion of H wih L(H, r) L(H, r). To hi end, we decribe a procedure where in each ep, we eiher remove ome inermediae edge of H 1 or zero i laency, wihou increaing he laency of he equilibrium flow. Le u focu on an edge e kl = {u k, v l } connecing a neighbor u k of o a neighbor v l of. By he fir par of he proof, he laency funcion of e kl i a conan b kl > 0. Nex, we aemp o e he laency of e kl o b kl = 0. We have alo o change he equilibrium flow f o a new flow f ha i an equilibrium flow of laency a mo L in he modified nework wih b kl = 0. We hould be careful when changing f o f, ince increaing he flow hrough {, u k } and {v l, } affec he laency of all pah going hrough u k and v l and may deroy he equilibrium propery (or even increae he equilibrium laency). In wha follow, we le r q be he amoun of flow moving from an pah q = (, u i, v j, ) o he pah q kl = (, u k, v l, ) when we change f o f. We noe ha r q may be negaive, in which cae, r q uni of flow acually move from q kl o q. Thu, r q define a rerouing of f o a new flow f, wih f q = f q r q, for any pah q oher han q kl, and f kl = f kl + q r q. We nex how how o compue r q o ha f i an equilibrium flow of co a mo L in he modified nework (where we aemp o e b kl = 0). We le P = P H 1 \ {q kl } denoe he e of all pah in H 1 oher han q kl. We le F be he P P marix, indexed by he pah q P, where F [q 1, q 2 ] = e q 1 q 2 a e e q 1 q kl a e, and le r be he vecor of r q. Then, he q-h componen of F r i equal o l q (f) l q (f ). In he following, we conider wo cae depending on wheher F i ingular or no. If marix F i non-ingular, he linear yem F r = ε 1 ha a unique oluion r ε, for any ε > 0. Moreover, due o lineariy, for any α 0, he unique oluion of he yem F r = α ε 1 i α r ε. Therefore, for an appropriaely mall ε > 0, he linear yem Q ε = {F r = ε 1, f q r q 0 q P, f kl + q r q 0, l qkl (f ) L+b kl ε} admi a unique oluion r. We keep increaing ε unil one of he inequaliie of Q ε become igh. If i fir become r q = f q for ome pah q = (, u i, v j, ) P, we remove he edge {u i, v j } from H 1 and adju he conan laency of e kl o ha l qkl (f ) = L ε. Then, he flow f i an equilibrium flow of co L ε for he reuling nework, which ha one edge le han he original nework H 1. If q r q < 0 and i fir become q r q = f kl, we remove he edge e kl from H 1. Then, f i an equilibrium flow of co L ε for he reuling nework, which again ha one edge le han H 1. If q r q > 0 and i fir become l qkl (f ) = L + b kl ε, we e he conan laency of he edge e kl o b kl = 0. In hi cae, f i an equilibrium flow of co L ε for he reuling nework ha ha one edge of 0 laency more han he iniial nework H 1. Moreover, we can how ha if q kl i dijoin from he pah q P, he fac ha he

10 10 D. Foaki, A.C. Kapori, T. Lianea, and P.G. Spiraki inermediae nework H 1 i acyclic implie ha he marix F i poiive definie, and hu non-ingular. Therefore, if q kl i dijoin from he pah in P,he procedure above lead o a decreae in he equilibrium laency, and evenually o eing b kl = 0. If F i ingular, we can compue r q o ha f i an equilibrium flow of co L in a modified nework ha include one edge le han he original nework H 1. If F i ingular, he homogeneou linear yem F r = 0 admi a nonrivial oluion r 0. Moreover, due o lineariy, for any α R, α r i alo a oluion o F r = 0. Therefore, he linear yem Q 0 = {F r = 0, f q r q 0 q P, f kl + q r q 0} admi a oluion r 0 ha make a lea one of he inequaliie igh. We recall ha he q-h componen of F r i equal o l q (f) l q (f ). Therefore, for he flow f obained from he paricular oluion r of Q 0, he laency of any pah q P i equal o L. If r i uch ha r q = f q for ome pah q = (, u i, v j, ) P, we remove he edge {u i, v j } from H 1 and adju he conan laency of e kl o ha l qkl (f ) = L. Then, he flow f i an equilibrium flow of co L for he reuling nework, which ha one edge le han he original nework H 1. If r i uch ha q r q = f kl, we remove he edge e kl from H 1. Then, f i an equilibrium flow of co L for he reuling nework, which again ha one edge le han H 1. Each ime we apply he procedure above eiher we decreae he number of edge of he inermediae nework by one or we increae he number of 0-laency edge of he inermediae nework by one, wihou increaing he laency of he equilibrium flow. So, by repeaedly applying hee ep, we end up wih a ubnework H of he 0-laency implificaion of H wih L(H, r) L(H, r). 5 Approximaing he Be Subnework of Simplified Nework We proceed o how how o approximae he BeSubEL problem in a balanced 0- laency implified nework G 0 wih reaonable laencie. We may alway regard G 0 a he 0-laency implificaion of a good nework G. We fir ae wo ueful lemma abou he maximum raffic rae r up o which BeSubEL remain inereing, and abou he maximum amoun of flow roued on any edge / pah in he be ubnework. Lemma 2. Le G 0 be any 0-laency implified nework, le r > 0, and le H 0 be he be ubnework of (G 0, r). For any ε > 0, if r > Bmaxn+ A minε, hen L(G 0, r) (1+ε)L(H 0, r). Proof. We aume ha r > Bmaxn+ A, le f be a Nah flow of (G minε 0, r), and conider how f allocae r uni of flow o he edge of E E (G 0 ) and o he edge E E (G 0 ). For impliciy, we le L L(G 0, r) denoe he equilibrium laency of G 0, and le A = e E 1/a e and A = e E 1/a e. Since G 0 i a 0-laency implified nework and f i a Nah flow of (G 0, r), here are L 1, L 2 > 0, wih L 1 + L 2 = L, uch ha all ued edge inciden o (rep. o ) have laency L 1 (rep. L 2 ) in he Nah flow f. Since r > Bmaxn+ A min, L 1, L 2 > B max and all edge in E E are ued by f. Moreover, by an averaging argumen, we have ha here i an edge e E wih a e f e r/a, and ha here i an edge e E wih a e f e r/a. Therefore, L 1 (r/a ) + B max and L 2 (r/a ) + B max, and hu, L r A + r A + 2B max.

11 Reolving Brae Paradox in Random Nework 11 On he oher hand, if we ignore he addiive erm b e of he laency funcion, he opimal average laency of he player i r/a + r/a, which implie ha L(H0, r) r/a + r/a. Therefore, L L(H0, r) + 2B max. Moreover, ince r > Bmaxn+ A, minε A n /A min, and A n /A min, we have ha: L(H 0, r) r A + r A B maxn A min ε A min n + B maxn A min ε Therefore, 2B max εl(h 0, r), and L (1 + ε)l(h 0, r). A min n 2B max /ε Lemma 3. Le G 0 be a balanced 0-laency implified nework wih reaonable laencie, le r > 0, and le f be a Nah flow of he be ubnework of (G 0, r). For any ε > 0, if P b B [b ε/4] δ, for ome conan δ > 0, here exi a conan ρ = 24AmaxBmax δεa 2 min uch ha wih probabiliy a lea 1 e δn /8, f e ρ, for all edge e. Approximaing he Be Subnework of Simplified Nework. We proceed o derive an approximaion cheme for he be ubnework of any implified inance (G 0, r). Theorem 2. Le G 0 be a balanced 0-laency implified nework wih reaonable laencie, le r > 0, and le H0 be he be ubnework of (G 0, r). Then, for any ε > 0, we can compue, in ime n O(A2 max r2 ln(n +)/ε 2 ) +, a flow f and a ubnework H 0 coniing of he edge ued by f, uch ha (i) f i an ε-nah flow of (H 0, r), (ii) L(f) L(H0, r) + ε/2, and (iii) here exi a conan ρ > 0, uch ha f e ρ + ε, for all e. Theorem 2 i a corollary of [6, Theorem 3], ince in our cae he number of differen pah i a mo n 2 + and each pah coni of 3 edge. So, in [6, Theorem 3], we have d 1 = 2, d 2 = 0, α = A max, and he error i ε/r. Moreover, we know ha any Nah flow g of (H 0, r) roue g e ρ uni of flow on any edge e, and ha in he exhauive earch ep, in he proof of [6, Theorem 3], one of he accepable flow f ha g e f e ε, for all edge e (ee alo [6, Lemma 3]). Thu, here i an accepable flow f wih f e ρ+ε, for all edge e. In fac, if among all accepable flow enumeraed in he proof of [6, Theorem 3], we keep he accepable flow f ha minimize he maximum amoun flow roued on any edge, we have ha f e ρ + ε, for all edge e. 6 Exending he Soluion o he Good Nework Given a good inance (G, r), we creae he 0-laency implificaion G 0 of G, and uing Theorem 2, we compue a ubnework H 0 and an ε/6-nah flow f ha comprie an approximae oluion o BeSubEL for (G 0, r). Nex, we how how o exend f o an approximae oluion o BeSubEL for he original inance (G, r). The inuiion i ha he 0-laency edge of H 0 ued by f o roue flow from V o V can be imulaed by low-laency pah of G m. We fir formalize hi inuiion for he ubnework of G induced by he neighbor of wih (almo) he ame laency B and he neighbor of wih (almo) he ame laency B, for ome B, B wih B + B L(f). We may hink of he nework G and H 0 in he lemma below a ome mall par of he original nework G and of he acual ubnework H 0 of G 0. Thu, we obain he following lemma, which erve a a building block in he proof of Lemma 5.

12 12 D. Foaki, A.C. Kapori, T. Lianea, and P.G. Spiraki Lemma 4. We aume ha G(V, E) i an (n, p, 1)-good nework, wih a poible violaion of he maximum degree bound by and, bu wih V, V 3knp/2, for ome conan k > 0. Alo he laencie of he edge in E E are no random, bu here exi conan B, B 0, uch ha for all e E, l e (x) = B, and for all e E, l e (x) = B. We le r > 0 be any raffic rae, le H 0 be any ubnework of he 0-laency implificaion G 0 of G, and le f be any flow of (H 0, r). We aume ha here exi a conan ρ > 0, uch ha for all e E(H 0 ), 0 < f e ρ. Then, for any ɛ 1 > 0, wih high probabiliy, wr. he random choice of he laency funcion of G, we can compue in poly( V ) ime a ubnework G of G, wih E (G ) = E (H 0 ) and E (G ) = E (H 0 ), and a flow g of (G, r) uch ha (i) g e = f e for all e E (G ) E (G ), (ii) g i a 7ɛ 1 -Nah flow in G, and (iii) L G (g) B + B + 7ɛ 1. Proof kech. For convenience and wlog., we aume ha E (G) = E (H 0 ) and ha E (G) = E (H 0 ), o ha we imply wrie V, V, E, and E from now on. For each e E E, we le g e = f e. So, he flow g aifie (i), by conrucion. We compue he exenion of g hrough G m a an almo Nah flow in a modified verion of G, where each edge e E E ha a capaciy g e = f e and a conan laency l e (x) = B, if e E, and l e (x) = B, if e E. All oher edge e of G have an infinie capaciy and a (randomly choen) reaonable laency funcion l e (x). We le g be he flow of rae r ha repec he capaciie of he edge in E E, and ge 0 l e(x)dx. Such a flow g can be compued in rongly minimize Po(g) = e E polynomial ime (ee e.g., [18]). The ubnework G of G i imply G g, namely, he ubnework ha include only he edge ued by g. I could have been ha g i no a Nah flow of (G, r), due o he capaciy conrain on he edge of E E. However, ince g i a minimizer of Po(g), for any u V and v V, and any pair of pah q, q going hrough u and v, if g q > 0, hen l q (g) l q (g). Thu, g can be regarded a a Nah flow for any pair u V and v V conneced by g-ued pah. To conclude he proof, we adju he proof of [4, Lemma 5], and how ha for any pah q ued by g, l q (g) B + B + 7ɛ 1. To prove hi, we le q = (, u,..., v, ) be he pah ued by g ha maximize l q (g). We how he exience of a pah q = (, u,..., v, ) in G of laency l q (g) B + B + 7ɛ 1. Therefore, ince g i a minimizer of Po(g), he laency of he maximum laency g-ued pah q, and hu he laency of any oher g-ued pah, i a mo B + B + 7ɛ 1, i.e., g aifie (iii). Moreover, ince for any pah q, l q (g) B + B, g i an 7ɛ 1 -Nah flow in G. Grouping he Neighbor of and. Le u now conider he enire nework G and he enire ubnework H 0 of G 0. Lemma 4 can be applied only o ube of edge in E (H 0 ) and in E (H 0 ) ha have (almo) he ame laency under f. Since H 0 doe no need o be inernally complee biparie, here may be neighbor of (rep. ) conneced o dijoin ube of V (rep. of V ) in H 0, and hu have quie differen laency. Hence, o apply Lemma 4, we pariion he neighbor of and he neighbor of ino clae V i and V j according o heir laency. For convenience, we le ɛ 2 = ε/6, i.e., f i an ɛ 2 - Nah flow, and L L H0 (f). By Theorem 2, applied wih error ɛ 2 = ε/6, here exi a ρ uch ha for all e E(H 0 ), 0 < f e ρ+ɛ 2. Therefore, L 2A max (ρ+ɛ 2 )+2B max i bounded by a conan. We pariion he inerval [0, L] ino κ = L/ɛ 2 ubinerval, where he i-h ubinerval i I i = (iɛ 2, (i + 1)ɛ 2 ], i = 0,..., κ 1. We pariion he verice of V (rep. of

13 Reolving Brae Paradox in Random Nework 13 V ) ha receive poiive flow by f ino κ clae V i (rep. V i ), i = 0,..., κ 1. Preciely, a verex x V (rep. x V ), conneced o (rep. o ) by he edge e x = {, x} (rep. e x = {x, }), i in he cla V i (rep. in he cla V i ), if l ex (f ex ) I i. If a verex x V (rep. x V ) doe no receive any flow from f, x i removed from G and doe no belong o any cla. Hence, from now on, we aume ha all neighbor of and receive poiive flow from f, and ha V 0,... V κ 1 (rep. V 0,..., V κ 1 ) i a pariioning of V (rep. V ). In exacly he ame way, we pariion he edge of E (rep. of E ) ued by f ino k clae E i (rep. E), i i = 0,..., κ 1. To find ou which par of H 0 will be conneced hrough he inermediae ubnework of G, uing he conrucion of Lemma 4, we furher claify he verice of V i and V j baed on he neighbor of and on he neighbor of, repecively, o which hey are conneced by f-ued edge in he ubnework H 0. In paricular, a verex u V i belong o he clae V (i,j) wih f {u,v} > 0. Similarly, a verex v V j, for all j, 0 j κ 1, uch ha here i a verex v V j belong o he clae V (i,j), for all i, 0 i κ 1, uch ha here i a verex u V i wih f {u,v} > 0. A verex u V i (rep. o V (i,j) ), and ha he (rep. v V j ) may belong o many differen clae V (i,j) cla V (i,j) i non-empy iff he cla V (i,j) of pair (i, j) for which V (i,j) i.e., doe no depend on V and r. We le E (i,j) verice in V (i,j) and E (i,j) i non-empy. We le k κ 2 be he number and V (i,j) are non-empy. We noe ha k i a conan, be he e of edge connecing o he be he e of edge connecing o he verice in V (i,j). Building he Inermediae Subnework of G. The la ep i o replace he 0-laency implified par connecing he verice of each pair of clae V (i,j) and V (i,j) in H 0 wih a ubnework of G m. We pariion, a in condiion (4) in he definiion of good nework, he e V m of inermediae verice of G ino k ube, each of cardinaliy V m /k, and aociae a differen uch ube V m (i,j) wih any pair of non-empy clae V (i,j) and V (i,j). For each pair (i, j) for which he clae V (i,j) and V (i,j) are nonempy, we conider he induced ubnework G (i,j) G[{, } V (i,j) V m (i,j) V (i,j) ], which i an (n/k, p, 1)-good nework, ince G i an (n, p, k)-good nework. Therefore, V (i,j) ] in he 0 in he role of he flow f, and ρ = we can apply Lemma 4 o G (i,j), wih H (i,j) 0 H 0 [{, } V (i,j) role of H 0, he rericion f (i,j) of f o H (i,j) ρ + ɛ 2. Moreover, we le B (i,j) = max e E (i,j) l e (f e ) and B (i,j) = max e E (i,j) l e (f e ) correpond o B and B, and inroduce conan laencie l e(x) = B (i,j) for all e E (i,j) and l e(x) = B (i,j) for all e E (i,j), a required by Lemma 4. Thu, we obain, wih high probabiliy, a ubnework H (i,j) of G (i,j) and a flow g (i,j) ha roue a much flow a f (i,j) on all edge of E (i,j) E (i,j), and aifie he concluion of Lemma 4, if we keep in H (i,j) he conan laencie l e(x) for all e E (i,j) E (i,j). The final oucome i he union of he ubnework H (i,j), denoed H (H ha he laency funcion of he original inance G), and he union of he flow g (i,j), denoed g, where he union i aken over all k pair (i, j) for which he clae V (i,j) and V (i,j) are non-empy. By conrucion, all edge of H are ued by g. Uing he properie of he conrucion above, we can how ha if ɛ 1 = ε/42 and ɛ 2 = ε/6, he flow g i an ε-nah flow of (H, r), and aifie L H (g) L H0 (f) + ε/2. Thu, we obain:

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