A propos de la difficulte du routage egal par plus courts chemins

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1 A propo de la difficule du rouage egal par plu cour chemin Tahiri Iam, Séphane Pérenne, Frédéric Giroire To cie hi verion: Tahiri Iam, Séphane Pérenne, Frédéric Giroire. A propo de la difficule du rouage egal par plu cour chemin. [Reearch Repor] RR-8175, <hal > HAL Id: hal hp://hal.inria.fr/hal Submied on 10 Dec 2012 HAL i a muli-diciplinary open acce archive for he depoi and dieminaion of cienific reearch documen, wheher hey are publihed or no. The documen may come from eaching and reearch iniuion in France or abroad, or from public or privae reearch cener. L archive ouvere pluridiciplinaire HAL, e deinée au dépô e à la diffuion de documen cienifique de niveau recherche, publié ou non, émanan de éabliemen d eneignemen e de recherche françai ou éranger, de laboraoire public ou privé.

2 On he Hardne of Equal Shore Pah Rouing Frédéric Giroire, Séphane Pérenne, Iam Tahiri RESEARCH REPORT N 8175 November 2012 Projec-Team MASCOTTE ISSN ISRN INRIA/RR FR+ENG

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4 On he Hardne of Equal Shore Pah Rouing Frédéric Giroire, Séphane Pérenne, Iam Tahiri Projec-Team MASCOTTE Reearch Repor n 8175 November page Abrac: In elecommunicaion nework packe are carried from a ource o a deinaion on a pah ha i deermined by he underlying rouing proocol. Mo rouing proocol belong o he cla of hore pah rouing proocol. In uch proocol, he nework operaor aign a lengh o each link. A packe going from o follow a hore pah according o hee lengh. For beer proecion and efficiency, one wihe o ue muliple (hore) pah beween wo node. Therefore he rouing proocol mu deermine how he raffic from o i diribued among he hore pah. In he proocol called OSPF-ECMP (for Open Shore Pah Fir -Equal Co Muliple Pah) he raffic incoming a every node i uniformly balanced on all ougoing link ha are on hore pah. In ha conex, he operaor ak i o deermine he be link lengh, oward a goal uch a maximizing he nework hroughpu for given link capaciie. In hi work, we how ha he problem of maximizing even a ingle commodiy flow for he OSPF-ECMP proocol canno be approximaed wihin any conan facor raio. Beide hi main heorem, we derive ome poiive reul which include polynomial-ime approximaion and an exponenial-ime exac algorihm. We alo prove ha depie heir weakne, our approximaion and exac algorihm are, in a ene, he be poible. Key-word: OSPF-ECMP, max flow, NP-Hard, approximaion. Thi work ha been founded by ANR-DIMAGREEN, ANR-AGAPE, ANR-GRATEL and RAISOM firname.laname@inria.fr RESEARCH CENTRE SOPHIA ANTIPOLIS MÉDITERRANÉE 2004 roue de Luciole - BP Sophia Anipoli Cedex

5 A propo de la difficulé du rouage égal par plu cour chemin Réumé : Le réeaux de communicaion on devenu rè complexe de no jour. Cependan, depui leur ou débu avec ARPANET, le chercheur e le ingénieur on conçu pluieur proocole aux diver équipemen de ce réeaux afin de réduire cee complexié e d aeindre la meilleure performance poible du réeau. OSPF ("Open Shore Pah Fir") [1] e l un de plu ancien proocole qui on enu bon avec cee évoluion. Il apparien à la ou-famille de proocole de rouage. Le bu d un proocol de rouage e de rendre chaque roueur capable de décider, à chaque paque recu, le prochain roueur voiin à qui ce paque doi ere ranmi. Quand le proocole de rouage acivé ur le réeau e OSPF, ou le paque uiven de plu cour chemin pondéré, ou le poid (à voir comme de longueur) on fixé par l adminiraeur réeau. Quand ce poid on défini de manière à avoir pluieur plu cour chemin pour une paire de noeud, le rouage dépendra de la règle qui e implémenée avec OSPF. Il y a pluieur règle permean de balancer le rafic enre le différen plu cour chemin, e l une de plu connue e ECMP ("Equal Co Muliple Pah"): un noeud qui a pluieur lien oran ur le plu cour chemin enver une deinaion d diviera le rafic qui lui arrive e qui e deiné à d de manière égale enre ou le chemin. Si le rafic néceie de ne pa êre paragé enre différen chemin, le poid doiven ere affecé de manière à ce que pour chaque paire de noeud il y ai un unique plu cour chemin. Afin de comprendre la difficulé du rouage OSPF, nou avon regardé un problème aez imple, mai qui ree fondamenal, où le bu e de maximier le débi quand une eule e unique paire de noeud communique de donnée ur le réeau. A nore meilleure connaiance, ce problème a éé prouvé NP-comple [14], uilian une réducion au problème de la couverure d un enemble, e aucune approximaion, avec garanie non rivial, n a éé propoée juqu à préen. Nou monron, en uilian une réducion différene que ce problème ne peu êre approximée à un faceur conan e nou donnon une approximaion qui dépend de la plu longue diance du graphe. Mo-clé : OSPF-ECMP, max flo, NP-Hard, approximaion.

6 On he Hardne of Equal Shore Pah Rouing (a) capaciie (b) Max-Opf-Flow (c) correponding lengh 1 Inroducion Figure 1: Example of maximum opf-flow of value 14. In hi paper we udy he complexiy of rouing according o he Open Shore Pah Fir proocol (OSPF) [1]. The goal of a rouing proocol i o make every rouer capable of deciding, whenever i receive a packe, he nex hop rouer. The deciion mu be aken locally and quickly, while allowing an efficien uage of he nework reource. When OSPF i he rouing proocol ued in he nework, all packe follow hore pah, according o he link lengh ha are e by he nework adminiraor. When here are everal hore pah beween wo node u and v, he rouing depend on he rule ha i ued for load balancing he raffic among he hore pah. There are everal rule. One of he mo ued i ECMP (Equal Co Muliple Pah). According o hi rule, a rouer which ha everal ougoing link on hore pah oward a deinaion v, balance he incoming raffic direced o v evenly among all of hem. In order o underand he algorihmic difficuly of OSPF rouing, we look ino he mo eenial problem in which he goal i o maximize he hroughpu when only a ingle pair of node i communicaing over he nework. We call hi problem Max-Opf-Flow: INSTANCE: a direced capaciaed graph D = (V, A, c) where V i he e of node, A i he e of arc, and c i he capaciy funcion ha aign o each arc (u, v) a capaciy c(u, v), and wo verice, V, repecively he ource and he ink of he flow. QUESTION: find he maximum flow going from o along wih i correponding arc-lengh aignmen under he proocol Open Shore pah Fir" uing he raegy Equal Co Muliple Pah". We give an example of maximum opf-flow in Figure 1 and a formal decripion of he problem in Secion 2. Le u poin ou ha here may be a large gap beween he Max-Opf-Flow and he andard maximum flow: Figure 2 how a flow graph wih n+2 verice in which he value of he Max-Opf-Flow i 2 while he value of he andard maximum flow i n. To he be of our knowledge, hi problem ha been only proved o be NP-Hard [14]. In hi paper, we how ha he Max-Opf-Flow canno be approximaed wihin any conan facor raio. Neverhele, we derive everal poiive reul. Fir, he problem can be ρ-approximaed. Thi implie ha, when he capaciie of all arc are equal, he problem can even be olved exacly in polynomial ime. We hen preen an exac exponenial algorihm of complexiy Õ(2 A ) and how ha any exac algorihm i likely o co 2 Θ( A ). Finally, we provide an approximaion for nework wih mall longe diance of facor 1 cmax 2H( c min ) 1 L, where L i he lengh of a longe imple pah beween he ource and he deinaion of he flow. Õ(g(n)) = O(g(n) log k g(n)), for k N. Logarihmic facor are ignored. Here H(K) = K i=1 1 = O(log(K)) denoe he harmonic number of order K. i RR n 8175

7 4 Giroire & Perenne & Tahiri Figure 2: A flow graph wih a big gap beween Max-Opf-Flow and he andard maximum flow. Arc on he pah u 1 u 2 u n have all capaciie equal o n while arc enering have all capaciie equal o 1. Thi udy how ha he difficuly of Max-Opf-Flow highly depend on he range of he capaciy funcion and of he deph" of he nework. Relaed Work Some previou work [10, 13, 4] howed ha finding an OSPF rouing ha opimize ome crieria (load, capaciy,... ) of he nework performance i a difficul ak; and hi hold for boh unpliable and pliable (wih ECMP rule) rouing. When he raffic i unpliable, he lengh have o be e in a way ha for every pair of node here i a unique hore pah. In [10], he auhor udied he problem of opimizing OSPF-ECMP lengh o a o minimize he oal co of he nework when he co funcion on arc i convex and increaing wih he congeion. They howed ha hi problem i NP-Hard. In l(a) paricular, hey defined he maximum uilizaion of he nework a max a A c(a) where l(a) i he load of arc a, and hey proved ha minimizing he maximum uilizaion i NP-Hard. They alo provided worcae reul abou he performance of OSPF-ECMP rouing v. an opimal muli-commodiy flow rouing in erm of congeion. In [4], he auhor udied he unpliable OSPF ha minimize he maximum uilizaion of he nework. They howed ha hi problem i hard o approximae wihin a facor of O( V 1 ε ). A hee problem are difficul, he mehod o olve hem in pracice uually follow a wo-phae approach baed on linear and ineger programming [6, 12, 8]: In he fir phae a rouing ha i no necearily feaible wih OSPF i found and in he econd phae lengh ha achieve hi rouing hrough OSPF are compued, when poible. Thi econd phae aemp o olve wha i called he invere hore pah problem (ISP). Thi problem can be formulaed a a linear program and herefore can be olved in polynomial ime. Even if he weigh are conrained o ake non-negaive ineger value, ISP remain polynomial hank o a rounding cheme preened in [2]. However finding a oluion o ISP ha minimize he maximum lengh over all arc i NP-Hard [3]. In hi paper, we conider he problem of approximaing he maximum opf-flow for a nework wih a ingle commodiy. Thi problem ha been proved o be NP-Hard in [14], uing a reducion o e cover. To he be of our knowledge, no approximaion wih non rivial guaraneed raio ha been uggeed before for olving hi problem. To conclude, noe ha, more generally, he reul preened here are valid wih minor modificaion for oher hore pah rouing proocol a IS-IS [9] or PNNI [13]. For more informaion, a urvey on he opimizaion of OSPF rouing can be found in [5]. Inria

8 8 8 On he Hardne of Equal Shore Pah Rouing 5 2 Problem Definiion In hi ecion, we how ha in he conex of our udy, which aume a ingle commodiy, he invere hore pah problem i guaraneed, under ome condiion on he rouing, o have a oluion and can be eaily olved. Therefore, even hough Max-Opf-Flow remain difficul, we can focu on finding an adequae rouing funcion while forgeing abou arc-lengh aignmen. In he cae of a ingle commodiy flow, olving he invere hore pah problem i rivial for any flow f going from o. For hi we define A = {a A f(a) > 0}, and e a A, l(a) = 0 and a A, l(a) = 1. Then he hore pah from he ource o he ink are all he pah uing only arc in A. One may wih for a beer lengh funcion, uch ha a A, l(a) > 0, bu i exi iff A i acyclic. Fir, if A conain a cycle hen l(a) mu be null on all he arc of he cycle. Second, if A i acyclic, one eaily ge a lengh funcion ince here exi hen a poenial funcion p : V R uch ha p(u) < p(v) if here i a pah from u o v (hi poenial funcion i given by he dual problem, ee [7]). Then, one e a = (u, v) A, l(u, v) = p(v) p(u). Noice ha all pah from o have lengh p() - p(). Combining hoe wo fac, one can hence alway define lengh uch ha (u, v) V 2, all he pah from u o v in A have he ame lengh and wih l(a) > 0 on all arc a no conained in a cycle. Noe ha, even if i may eem range, here exi digraph for which any oluion of Max-Opf-Flow conain a flow loop (ee Figure 3). + f 16 f f 16 g g/ f/2 8 g/ f/2 f/4 f/ g/2 g/6 g/ (a) capaciie (b) loop-free flow (c) flow wih loop Figure 3: Example of an inance where he opimal flow can only be achieved uing a loop: when avoiding he loop, one can eaily check ha he limiaion come from he conrain f/4 4 which mean ha he flow en from o i a mo 16; while uing a loop he ame conrain allow g o be equal o 24 which lead o a flow going from oward ha i equal o f = g g/6 = 20. Oher configuraion wih le arc are no depiced bu lead o even lower opf-flow value. Therefore, for any flow funcion, here alway exi lengh for which all he pah ued by he flow are hore pah, and hi lengh can be non-negaive if he flow i acyclic. Neverhele, i doe no mean ha any flow funcion can be achieved uing OSPF-ECMP. Indeed f mu alo fulfill he Equally Balanced Condiion". Thi mean ha we mu have: a any node a V and (u, v) A, eiher f(u, v) = 0 or f(u, v) = ou(u) where ou(u) depend only on u. Thi allow u o reformulae he Max-Opf-Flow in a impler way, wihou uing meric explicily. Bu before we need o inroduce a few definiion. Definiion 1 (flow graph) Given a direced capaciaed graph D = (V, A, c) and wo paricular node RR n 8175

9 6 Giroire & Perenne & Tahiri and (o be repecively een a he ource and he ink), heir correponding flow graph i defined a he 5-uple D = (V, A, c,, ). Definiion 2 (flow funcion/value) Given a flow graph D = (V, A, c,, ), a funcion f : A R + i a flow funcion if a A, f(a) c(a) and v V \ {, }, (u,v) A f(u, v) = (v,u) A f(v, u). The flow value i val(f) = (,v) A f(, v) (v,) A f(v, ). Definiion 3 (regular funcion) A funcion over a e S i aid o be regular if i ake only wo value, 0 and anoher poible one. Definiion 4 (balanced funcion) Given graph G = (V, A) and V V, a funcion f i aid o be balanced on V iff for any u V he funcion f u, defined by (u, v) A, f u (v) = f(u, v), i regular. Definiion 5 (opf-flow funcion) For a flow graph D = (V, A, c,, ), a funcion f : A R + i an opf-flow if i i a flow funcion and if i i balanced on V. Now, we are ready o give an equivalen formulaion of he problem Max-Opf-Flow: Inance : a flow graph D = (V, A, c,, ). Queion : find he maximum flow value of an opf-flow funcion in A. The meric problem ha only apparenly vanihed, ince chooing on which arc he flow i null and on which arc i i ricly poiive acually deermine he underlying hidden lengh funcion. Some noaion For f a funcion from a e S o R, we denoe f(s) = S f(). Given a digraph wih verex e V and arc e A we alo abuively denoe (U, V ) he e of couple {(u, v) A u U, v V }. We alo noe he e of ineger {1,..., N} by [N]. 3 Inapproximabiliy reul We ue a reducion o he MAX-3-SAT problem in order o prove fir ha Max-Opf-Flow i hard o approximae, in a polynomial ime, wihin an approximaion raio of Then we ue a elf-amplifying mehod o prove ha hi problem i no in APX. We recall ha MAX-3-SAT i he problem defined by: INSTANCE : a e F = {B j } j [m] of claue, buil on a finie e {x i } i [n] of binary variable, uch ha each claue conain exacly hree lieral. QUESTION : find a ruh aignmen of he variable {x i } i [n] ha maximize he number of aified claue in F. Unle P = NP, Problem MAX-3-SAT canno be approximaed wihin a facor ε for any conan ε > 0, ee [11]. In paricular, even when he opimum value i m, i i ill NP-Hard o find an aignmen ha aifie more han ε claue of F, ε > 0. Propoiion 1 I i NP-hard o approximae Max-Opf-Flow wihin a facor 1 l 3 8l + ε, ε > 0, even if a A, c(a) {1, l}. A example, i i NP-hard o approximae Max-Opf-Flow wihin a facor + ε, ε > 0, even if a A, c(a) {1, 4}, Inria

10 On he Hardne of Equal Shore Pah Rouing 7 Proof. For eek of impliciy, our reducion ue a muligraph. To ge a graph from i one can imply iner a node on every arc. Thi will make he graph imple and will only double i iniial ize. Given an inance I of MAX-3-SAT, defined by he e of binary variable {x i } i [n] and he e of claue {B j } j [m], we build he following flow graph D I = (V, A, c,, ): - For each i [n], we creae a verex a i repreening variable x i. - For each j [m], we creae a verex b j repreening claue B j. - For each i [n], we add k i parallel arc wih capaciy 1 going from o a i, where k i i he number of ime he variable x i appear in he claue. - We add an arc wih capaciy 1 from every b j o, wih j [m]. In addiion o hoe arc, we add arc ha depend on he rucure of he claue : - (i, j) [n] [m], if B j conain he poiive lieral x i we add an arc wih capaciy 1 from a i o b j ; - (i, j) [n] [m], if B j conain a negaive lieral x i we add l parallel arc wih capaciy 1 l from a i o b j An example of he conrucion i diplayed in Figure 4. Noe ha he ize of D I i linear in he number of claue. Figure 4: Flow graph D I conruced from a e F conaining he claue B 1 = (x 1 x 2 x 4 ), B 2 = (x 2 x 3 x 4 ) and B 3 = (x 2 x 3 x 4 ). [The example aume l = 4. Dahed arc have capaciy 1 4. The oher have capaciy 1]. The idea of he reducion i o emulae binary variable a follow: For (i, j) [n] [m] we define he virual link vl(a i, b j ) a he e of arc connecing a i o b j, we ay ha he virual link vl(a i, b j ) i poiive (rep. negaive) virual link if x i (rep. x i ) i a lieral of B j. Then, one mu chooe a a i eiher o end ignificaive flow on he poiive virual link or he negaive one, and hi i excluive. Thi formalize wih he following propery: Propery 1 Any opf-flow funcion mu aify RR n 8175

11 8 Giroire & Perenne & Tahiri a) If for ome (i, j) [n] [m] he flow on a poiive virual link vl(a i, b j ) i ricly greaer han 1 l hen any flow on a negaive virual link vl(a i, b k ) b k, k [m] i a mo 0. b) If for ome (i, j) [n] [m] he flow on a negaive virual link vl(a i, b j ) i ricly poiive hen any flow on a poiive virual link vl(a i, b k ) b k, k [m] i a mo 1 l. To prove (a) remark ha poiive virual link are made of a ingle arc wih capaciy 1, o if (a) premie happen we mu have f(a i, b j ) > 1 l and ince f a i i regular no flow can leave a i on he arc wih capaciy 1 l. Cae (b) i imilar, ince (b) premie mean ha he arc wih capaciy 1 l are ued, and ince f ai i regular, o f(a i, b j ) 1 l (ince here i a unique arc for he poiive virual link). We fir prove ha he Max-Opf-Flow on D I i m if I ha a ruh aignmen ha aifie all claue. To do o, we aociae o hi ruh aignmen a flow funcion f on D I a follow: for each aified claue B j, we chooe one of i aified lieral, and if i i aociaed o he variable x i we end one uni of flow on he pah a i b j uing he virual virual link correponding o he lieral. The flow f ha hen a value m and i balanced ince: - i [n] wih x i rue all arc leaving a i have flow 0 or 1. - i [n] wih x i fale all arc leaving a i have flow 0 or 1 l. Therefore f i an opf-flow. Moreover, hi flow i maximum ince he cu ({}, V \ {}) ha a capaciy m. Conequenly, he value of he Max-Opf-Flow on D I i m. We now aume ha we are able o find (uing a polynomial algorihm) an opf-flow funcion f. To ha opf-flow flow we aociae a binary aignmen for each i [n] a follow: a) If a verex a i end ricly more han 1 l uni of flow along a poiive virual link we e x i o rue. b) If a verex a i end ricly more han 1 l uni of flow along negaive virual link we e x i o fale. According o Propery 1 hi aignmen i conien. Noe ha he above rule may le ome variable unaigned, in which cae we imply dicard hem (or le hem unaigned). Conider now one of he node b j, j [m] for which he flow f(b i, ) i greaer han 3 l. Since here are a mo hree virual link enering he node b j one virual link (a i, b j ), i [n] carrie a flow ricly greaer han 1 l. Then, eiher vl(a i, b j ) i a poiive virual link and x i wa e o rue (by rule (a)) and he claue B j i aified. Or vl(a i, b j ) i a negaive virual link and x i i e o fale (by rule (b)) and he claue B j i aified. So uch a node b j correpond a claue B j aified by our aignmen. So (unle P = NP) we have a mo ( ε)m uch node. Bu hen, he flow i a mo ( ε) m 1 + ( 1 ( ε)) m 3 l = m ( l 3 1 l 8 ε) m = ( ) ( 1 l 3 8l m + ε l 3 ) l m. So we conclude ha l 3 and ε, (unle P = NP) no polynomial algorihm compue a flow wih value greaer han (1 l 3 8l + ε )m. Noe ha he conruced graph ue capaciie { 1 l, 1} o we muliply all he capaciie by l o ge an equivalen graph wih capaciie in {1, l}. Theorem 1 I i NP-Hard o approximae Max-Opf-Flow wihin any conan facor. Proof. To prove he reul, we ue a elf-amplifying error echnic. Indeed, he flow graph D I aociaed o a 3-SAT inance I connec o i like a virual arc (, ) wih capaciy m, bu for which i i NP- Hard o ue a capaciy larger han 31m 32. Conidering a flow graph D, we will replace each of i edge by hoe virual arc. Two facor of error will hen ge muliplied: one becaue no polynomial algorihm can ue he full capaciy of he virual arc, he oher becaue here i anoher error on he original nework ielf. We noe OPT(D) he value of Max-Opf-Flow on D. Le u inroduce wo definiion: Inria

12 On he Hardne of Equal Shore Pah Rouing 9 For a flow graph D = (V, A, c,, ), we define x D = (V, A, c x,, ) where c x i defined by a A, c x (a) = x c(a). For any poiive number x, here i a rivial bijecion which relae every flow funcion f for D o a flow funcion g for x D, defined by e A, g(a) = x f(a). Given wo flow graph D 1 and D 2, we define D 1 D 2 a he graph in which every arc a = (u, v) A (A i in D 1 ) wih capaciy c(a) i ubiued wih he following equence (ee Figure 5): a copy of c(a) D 2 wih ource (u, v) and ink (u, v), wo arc (u, (u, v)) and ((u, v), v) wih capaciie c(a)m, where M i an upper bound of OPT(D 2 ). Figure 5: A virual arc of D 1 D 2 (Righ) ubiuing he ingle arc (u, v) of D 1 (Lef). Suppoe ha one had proven ha he Max-Opf-Flow i hard o approximae wihin ρ 1 and ρ 2 ; namely we can aociae o any inance I of an NP-Complee problem a flow digraph D 1 (rep. D 2 ) uch ha a ρ 1 (rep. ρ 2 ) approximaion of he Max-Opf-Flow allow o olve I. We build he flow graph D 1 D 2. Noe fir ha he ize of D 1 D 2 i he produc of D 1 and D 2 ize (hence if D 1 and D 2 have polynomial ize hen o doe D 1 D 2 ). The maximum opf-flow of D 1 D 2 ha value OPT(D 1 )OPT(D 2 ). Suppoe now ha we can find in polynomial ime an opf-flow of value ricly greaer han ρ 1 ρ 2 OPT(D 1 )OPT(D 2 ) for every inance I. For any inance I, wo differen cae can happen: Eiher, here exi a lea one virual arc wih a flow value ricly larger han ρ 2 OPT(D 2 ). In hi cae, we ue reducion 2 and olve he NP-problem for inance I. Or, for all virual arc, he flow i le han ρ 2 OPT(D 2 ). In hi cae, we can define a valid opfflow for he graph OPT(D 2 ) D 1 by imply aking a flow value of arc (u, v) he flow value of arc (u, (u, v)) in D 1 D 2. We can build a valid opf-flow for D 1 by dividing he value of he flow on each arc by OPT(D 2 ). Thi opf-flow i of value greaer han ρ 1 OPT(D 1 ). We hen ue reducion 1 and we olve he NP-problem for inance I. Hence, by conradicion, i i no poible o find in polynomial ime an opf-flow of value greaer han ρ 1 ρ 2 OPT(D 1 )OPT(D 2 ) for every inance I. To complee he proof we ar wih ρ 1 = ρ 2 = ρ = And aring from hardne o approximae wihin ρ we prove hardne o approximae wihin ρ 2 i follow ha Max-Opf-Flow canno be approximaed wihin any conan unle P = NP. 4 Poiive reul Even hough i i hard o find he opimal oluion o Max-Opf-Flow, we can ill ge an approximaion algorihm. Theorem 2 Given a flow graph D = (V, A, c,, ). Le c min and c max be repecively he malle and he large value of he capaciy funcion c. Then Max-Opf-Flow can be approximaed on D in polynomial ime wihin a facor c min /c max. RR n 8175

13 10 Giroire & Perenne & Tahiri Proof. Le OPT be he maximum value of an opf-flow on D, and le D i o be he direced graph ha have he ame elemen a D bu where he capaciy funcion i he conan funcion ha aign i o every arc. We compue a maximum inegral flow funcion F 1 on D 1. On he one hand, he value of an opf-flow on D canno exceed he maximum value of a flow on D cmax, which i c max v(f 1 ), o OPT c max v(f 1 ). On he oher hand, c min F 1 i a feaible opf-flow for D wih value c min v(f 1 ) cmin c max OPT. When c min = c max we ge he nex corollary : Corollary 1 Any inance of Max-Opf-Flow where all capaciie of he flow graph are equal can be olved in polynomial-ime. We now give an exac algorihm wih exponenional olving ime. Theorem 3 Given a flow graph D = (V, A, c,, ), we have an exac algorihm for olving Max-Opf- Flow on D which run in Õ(2 A ). Proof. Conider an opf-flow funcion f x wih a flow value x and remark ha he knowledge of he configuraion A = {a A f x (a) > 0} enirely deermine f x in he following ene: For a given configuraion, here exi a mo one opf-flow of value x. Indeed if d(v) denoe he ou-degree in A of a verex v he following equaion are aified: v {, }, (v, w) A 1 f x (v, w) = d(v) (u,v) A f x(u, v) (, w) A x f x (, w) = d() Noe ha he above yem i well-defined iff all node adjacen o ome edge in A belong o a pah in A from o. Since he configuraion aociaed o he Max-Opf-Flow aifie hi condiion, we can dicard any degeneraed e A violaing he above condiion. If f 1 i he funcion ha olve he above yem for x = 1, hen for x R he oluion i given by he funcion f x = x f 1. Therefore, he maximum flow value of an opf-flow uing he configuraion A, noed val(a ), expree a he maximum among all x R ha repec he following capaciy conrain: (v, w) A, xf 1 (v, w) c(v, w) Hence, we have val(a c(v,w) ) = min (v,w) A f. 1(v,w) Finally, o compue he max-opf-flow, we can imply compue val(a ), in polynomial ime, for each non degeneraed configuraion and ake he maximum. Remark 1 In he above proof, when A form an acyclic graph, he yem i diagonal and can be olved very efficienly by puhing he flow value from he ource node. Bu, a menionned in he inroducion, we canno limi our earch o uch acyclic configuraion ince here exi flow graph for which he maximum opf-flow conain cycle. We now give a more efficien approximaion algorihm when he range of capaciie i large. The algorihm i baed on a ampling argumen given in Lemma 1 and on he monooniciy of a (muliource) opf-flow, meaning ha if a opf-flow exi, opf-flow of maller value alway exi in he ene of Lemma 2. Definiion 6 Given wo funcion f, g from S R, we ay ha g i f-dominaed if x S, f(x) g(x). Õ(g(n)) = O(g(n) log k g(n)) Inria

14 On he Hardne of Equal Shore Pah Rouing 11 Lemma 1 Le S be a finie e, and a funcion p : S [MAX], hen here exi an inegral regular p(s) p-dominaed funcion q wih q(s) H(MAX). Proof. Aume ha he unique non zero value ha we pick for q i i. Then, ince q p, he be choice for q (in order o maximize q(s)) i o imply e q(x) = 0 iff p(x) < i and q(x) = i iff p(x) i. So, le u define he wo e: (i) = p 1 (i) and G(i) = j [MAX],j i (j). Then, from definiion, for any i [MAX], we can build a regular funcion q, p-dominaed, uch ha q(s) = i G(i). I follow ha he be value ha we can ge for q(s) i: OPT = max i G(i). i [MAX] Noe alo ha, wih hoe noaion (uing a andard double couing argumen), we have: p(s) = i [MAX] i (i) = i [MAX] G(i). We have i [MAX], G(i) OPT i. So p(s) = i [MAX] G(i) OPT i [MAX] 1 i = H(MAX)OPT. I follow ha OPT. Noe ha by conrucion he opimal funcion q p(s) H(MAX) i inegral and OPT i inegral, o we indeed have OPT p(s) H(MAX). Definiion 7 (Muliource opf-flow) Conider a digraph G = (V, A) and a capaciy funcion on he arc c, a muliource opf-flow wih n ource { i } i [n] and a ink i a funcion f : A R + uch ha a A, f(a) c(a) and v V \ { i } i [n] {}, uv A f(u, v) = vu A f(v, u). balanced on V. For i [n], val i = ( i,v) A f( i, v) (v, i) A f(v, i) i called he oupu value of ource i The flow value of f i defined a val(f) = i [n] val i. Lemma 2 (monooniciy) Conider a flow graph wih k ource { i } i [k] and a ink. If i exi a muliource opf-flow in which he ource i ha an oupu value val i, hen, for any {val i} i [k] uch ha i [k], val i val i, here exi a muliource opf-flow in which each ource ha an oupu value val i. Proof. We ake he configuraion A aociaed o he iniial opf-flow, i.e. A = {a A; f(a) > 0}. We hen wan o deermine if an opf-flow wih oupu value val i, i [k] on he ource i feaible wih configuraion A. Such a flow i enirely deermined by he value val i, i [k] (ee he proof of Theorem 3), and he flow on an arc a i of he form i [k] λ a,ival i where λ a,i are all poiive or null. So he load of an arc i increaing wih he ource oupu and he reul follow (he e of feaible opfflow, wih configuraion A, i defined by a A, λ a,i val i c(a)). Definiion 8 For l N, we ay ha a flow graph D = (V, A, c,, ) i l-layered if i verex e can be pariionned ino V = i [l] val i uch ha V 1 = {}, V l = {} and he arc e A aifie A = i [l 1] A i where i [l 1], A i (val i, V i+1 ). Here H(K) denoe he harmonic number of order K. RR n 8175

15 12 Giroire & Perenne & Tahiri Propoiion 2 Given an ineger l 3, le D = (V, A, c,, ) be an l-layered flow graph. Given an inegral flow funcion F ha ha value in [c], we can compue in polynomial-ime an opf-flow of value val(f) greaer han H(c) l val(f ). Proof. The idea i o ample he original flow funcion F level per level. Le ρ = H(c) 1 be our ampling facor. We build by inducion on a flow funcion g ha ha he following properie: - g i balanced on he la level (i.e. on he verex e B = i [l,l] V i ). - g i F -dominaed, ha i a A, g (a) F (a). - v(g ) ρ val(f ). - g i inegral ill level (i.e a i [] A i, g (a) i an ineger). For = 0, we ake g 0 = F. Since B 0 i reduced o he ink, he inducive hypohei hold. We now build g from g 1 for 1. We ar eing g = g 1. By inducion hypohei, g i balanced on all he level in [l + 1, l] (i.e on B 1 ). We wan o make g balanced on level V l. So, we look a he iuaion of he ougoing flow from level V l o level V l +1. We conider he flow funcion g 1 on he e A l. By inducion hypohei, g 1 i inegral wih value in [c]. So we can ample g 1 on he e A l uing Lemma 1 and ge an inegral, regular, g 1 dominaed funcion q uch ha q(a l ) ρg 1 (A l ). We hen e a A l, g (a) = q (a). Hence, he flow from level l o level l + 1 i a lea ρv(g 1 ) = ρ val(f ). The funcion g i no anymore a flow ince i doe no repec he conervaion anymore on level V l and V l +1. Node in V l receive oo much flow, and node in V l +1 end oo much flow. - For node in V l +1, we ue monoony (Lemma 2) which allow u o decreae he flow en oward he ink while keeping he opf conrain aified. - For node in V l we can eaily decreae he flow ince he flow funcion i no required o fulfill he regulariy condiion. Neverhele, we have o keep he flow inegral. Bu ince q i an inegral funcion, he exce flow received a any verex v V l i an ineger. So we can decreae he flow uni per uni and keep g inegral on all level ill l. So all he inducive condiion are fulfilled for, he reul follow by inducion. We define L D (u, v) a he lengh of he longe imple pah beween u and v in a flow graph D. Theorem 4 Le D = (V, A, c,, ) be a flow graph, and aume ha c : A [c], hen Max-Opf-Flow can be approximaed in polynomial ime wihin a facor H(c) 1 L D(,) Proof. We fir compue a maximum inegral flow F wihou flow loop (uch alway exi). The arc ued by he flow hen form an acyclic digraph, and we only conider hi digraph. We now aign o each verex a level a follow: a ep i 0, we mark any verex for which all he in-neighbor were marked a previou ep and aign o uch a verex he Level i. The procedure enure ha he arc all go from a level o a greaer level. Noe ha he ource i he only verex a Level 0 and ha he ink i he only verex a Level L D (, ), ince here i a pah going from he ink o all verice and ha all pah go o he ink. So we have almo layered our digraph, o complee he proce we add k 1 inermediary node on any arc going from level i o level i + k in order o ge only arc beween conecuive level. See Figure 6 for an example. Then we apply Propoiion 2 o he layered digraph wih l = L D (, ) + 1. We obain in polynomial ime an opf-flow f on he layered digraph of value greaer han H(c) l val(f ). We can eaily build an Inria

16 On he Hardne of Equal Shore Pah Rouing 13 opf-flow of ame value on D (he flow on all arc of a pah wih inermediary node i equal, we aigne hi flow value o he correponding arc of he original digraph). Thi end he proof ince he value of a maximum flow F i an upper bound on he value of a maximum opf-flow. a b F 4 a F 2 F 4 F 5 F 3 F 1 F 3 F 2 b F 5 F 1 F 2 F 1 F 1 Figure 6: Example of he conrucion of Theorem 5 Lef: D. Righ: D. A verex u i placed a he level L D (, v), wih L D (, v) he lengh of a longe pah beween and v. Each arc (u, v) A i hen ubdivided ino a pah of lengh L D (u, v). Theorem 5 Le D = (V, A, c,, ) be a flow graph, and aume ha c : A [c min, c max ], hen he maximum value of an opf-flow be approximaed in polynomial ime wihin a facor 1 cmax 2H( c min ) 1 L D(,) Proof. Le f 0 be he maximum value of a (andard) flow on D. We conider he flow graph D = (V, A, c,, ) wih capaciie a A, c (a) = 1 c min c(a). We have a A, c (a) [ cmax c min ]. Since a A, c (a) c(a) 2, he maximum value of a flow on D i a lea 1 c max 2 c min f 0. We hen apply Theorem 5 o D wich ue inegral capaciie in [ cmax c min ] and ge an opf-flow wih value a lea H( cmax c min ) 1 L D(,) 1 c max 2 c min f 0. We hen muliply hi flow by c min and ge he reul. 5 Concluion & Open problem In hi udy, our main incenive wa o inveigae he difficuly of rouing he daa opimally in nework ha have oped for OSPF wih ECMP raegy, a a dynamic rouing proocol. To do o, we concenraed on he fondamenal problem of maximizing he flow from a ingle ource o a ingle deinaion on a nework where hi proocol i aumed o be running. The main reul how ha here exi no conan facor approximaion if he capaciie are no bounded. When all capaciie are ineger in [c], There i a c-facor approximaion, however, we have been unable o find an algorihm wih raio beer han c So we conjecure ha for any µ Max-Opf-Flow canno be approximaed wihin a facor beer han c 1 µ. Neverhele, if he longe pah from o i bounded by L, hen we derive an approximaion algorihm wih he raio 1 cmax 2H( c min ) 1 L. Addiionally, we ugge an exac algorihm ha ha a complexiy of Õ(2 A ). I i an open problem o find an algorihm wih a beer exponenial bae. However, here i a limiaion on he efficiency of uch algorihm, ince i would be able o olve MAX-3-SAT (ee he reducion of Theorem 1) and he be known algorihm for olving an inance wih m claue ha complexiy Õ(1.3m ). For he proof of Theorem 5, we ue he noion of muli-ource opf-flow. In fac, mo of he reul of hi paper could be exended o hi cae, in which here ill i a ingle commodiy, bu in which he flow can be en by everal ource and, poenially, alo en o everal deinaion. Thi cae could RR n 8175

17 14 Giroire & Perenne & Tahiri be inereing o udy furher a i model Conen Delivery Nework (CDN), eimaed o accoun nowaday for 30 o 50% of Inerne global raffic. In hi conex, daa i replicaed in everal daa cener and uer in differen geographical place ge he daa from hee differen ource. Reference [1] Opf verion 2, rfc2328. hp:// [2] BEN-AMEUR, W., AND GOURDIN, E. Inerne rouing and relaed opology iue. SIAM J. Dicre. Mah. 17, 1 (Jan. 2004), [3] BLEY, A. Inapproximabiliy reul for he invere hore pah problem wih ineger lengh and unique hore pah. In Proceeding of he Second Inernaional Nework Opimizaion Conference (2005). [4] BLEY, A. On he approximabiliy of he minimum congeion unpliable hore pah rouing problem. In Proceeding of he 11h inernaional conference on Ineger Programming and Combinaorial Opimizaion (Berlin, Heidelberg, 2005), Springer-Verlag, pp [5] BLEY, A., FORTZ, B., GOURDIN, E., HOLMBERG, K., KLOPFENSTEIN, O., PIORO, M., TOMASZEWSKI, A., AND UMIT, H. Opimizaion of OSPF rouing in IP nework. Springer, 2009, pp in Graph and Algorihm in Communicaion Nework: Sudie in Broadband, Opical, Wirele and Ad Hoc Nework. [6] BLEY, A., AND KOCH, T. Ineger programming approache o acce and backbone IP-nework planning. Tech. Rep. ZR 02-41, Zue Iniue Berlin, [7] COOK, W. J., CUNNINGHAM, W. H., PULLEYBLANK, W. R., AND SCHRIJVER, A. Combinaorial opimizaion. Wiley, [8] DZIDA, MATEUSZ MYCEK, M.,, PIÓRO, M., TOMASZEWSKI, A., AND ZAGOZDZON, M. Valid inequaliie for a hore-pah rouing opimizaion problem. In Inernaional Nework Opimizaion Conference (2007). [9] FORTZ, B., AND THORUP, M. Opimizing opf/i-i weigh in a changing world. IEEE Journal on Seleced Area in Communicaion 20, 4 (2002), [10] FORTZ, B., AND THORUP, M. Increaing inerne capaciy uing local earch. Compuaional Opimizaion and Applicaion 29 (2004), [11] HÅSTAD, J. Some opimal inapproximabiliy reul. Journal of he ACM 48, 4 (2001), [12] HOLMBERG, K., AND CARBONNEAUX, Y. Opimizaion of inerne proocol nework deign and rouing. Nework 43, 1 (Jan. 2004), [13] IZMAILOV, R., SENGUPTA, B., AND IWATA, A. Adminiraive weigh allocaion for pnni rouing algorihm. In IEEE Workhop on High Performance Swiching and Rouing (2001), pp [14] PIÓRO, M., SZENTESI, Á., HARMATOS, J., JÜTTNER, A., GAJOWNICZEK, P., AND KOZ- DOROWSKI, S. On open hore pah fir relaed nework opimiaion problem. Journal of Performance Evaluaion 48, 1-4 (May 2002), Inria

18 On he Hardne of Equal Shore Pah Rouing 15 Conen 1 Inroducion 3 2 Problem Definiion 5 3 Inapproximabiliy reul 6 4 Poiive reul 9 5 Concluion & Open problem 13 RR n 8175

19 RESEARCH CENTRE SOPHIA ANTIPOLIS MÉDITERRANÉE 2004 roue de Luciole - BP Sophia Anipoli Cedex Publiher Inria Domaine de Voluceau - Rocquencour BP Le Chenay Cedex inria.fr ISSN