A Penalized Best-Response Algorithm for Non-Linear Single-Path Routing Problems

Transcription

1 A Pnalizd Bst-Rspons Algorithm for Non-Linar Singl-Path Routing Problms Olivir Brun, Balakrishna Prabhu, Josslin Vallt To cit this vrsion: Olivir Brun, Balakrishna Prabhu, Josslin Vallt. A Pnalizd Bst-Rspons Algorithm for Non-Linar Singl-Path Routing Problms. Ntworks, Wily, 2017, Static and Dynamic Optimization Modls for Ntwork Routing Problms 69 (1), pp < < /nt.21720>. <hal > HAL Id: hal Submittd on 8 Fb 2017 HAL is a multi-disciplinary opn accss archiv for th dposit and dissmination of scintific rsarch documnts, whthr thy ar publishd or not. Th documnts may com from taching and rsarch institutions in Franc or abroad, or from public or privat rsarch cntrs. L archiv ouvrt pluridisciplinair HAL, st dstiné au dépôt t à la diffusion d documnts scintifiqus d nivau rchrch, publiés ou non, émanant ds établissmnts d nsignmnt t d rchrch français ou étrangrs, ds laboratoirs publics ou privés.

2 A Pnalizd Bst-Rspons Algorithm for Non-Linar Singl-Path Routing Problms O. Brun 1,2, B.J. Prabhu 1,2, J. Vallt 3 1 CNRS, LAAS, 7 Av. du Colonl Roch, F Toulous, Franc 2 Univ. Toulous, LAAS, F Toulous, Franc 3 Vivris Tchnologis, 1 Avnu d l Europ, F Toulous, Franc Abstract This papr is dvotd to non-linar singl-path routing problms, which ar known to b NP-hard vn in th simplst cass. For solving ths problms, w propos an algorithm inspird from Gam Thory in which individual flows ar allowd to indpndntly slct thir path to minimiz thir own cost function. W dsign th cost function of th flows so that th rsulting Nash quilibrium of th gam provids an fficint approximation of th optimal solution. W stablish th convrgnc of th algorithm and show that vry optimal solution is a Nash quilibrium of th gam. W also prov that if th objctiv function is a polynomial of dgr d 1, thn th approximation ratio of th algorithm is ( 2 1/d 1 ) d. Exprimntal rsults show that th algorithm provids singl-path routings with modst rlativ rrors with rspct to optimal solutions, whil bing svral ordrs of magnitud fastr than xisting tchniqus. Kywords: bst rspons; singl-path routing; gam thory; non-linar programming; approximation algorithm; Nash quilibrium This is an xtndd vrsion of th papr "A gam-thortic algorithm for non-linar singl-path routing problms" that appard in th Procdings of INOC 2015 [38]. Th work of th third author was carrid out whil h was a doctoral studnt at LAAS. 1

3 1 Introduction Th singl-path routing problm, also known as th unsplittabl or non-bifurcatd multicommodity flow problm, naturally ariss in a varity of contxts and, in particular, plays a cntral rol in th traffic nginring of communication ntworks. It amounts to routing a givn st of fixd traffic dmands in a ntwork, allocating ach dmand to a singl path to minimiz th total ntwork cost, which is usually xprssd as th sum of link costs (s [25, 33, 39] and Chaptr 8 of [34] for th rlatd joint routing and congstion control problm). A common assumption is that th cost of a link is a linar function of th traffic flowing on that link. Undr this assumption, th singl-path routing problm [s p. 15] can b formulatd as a Mixd-Intgr Linar Programming (MILP) problm with binary variabls [4, 6, 7, 8]. Although this assumption might b appropriat in som contxts, it is no longr viabl whn th cost of a link is intrprtd as th dlay incurrd by a packt on that link sinc this dlay dpnds non-linarly on th amount of traffic flowing ovr that link. This papr is dvotd to singl-path routing problms with an additiv and non-linar objctiv function. Our intrst in ths problms originats from traffic nginring in communication ntworks basd on th Multi-Protocol Labl Switching (MPLS) tchnology [16]. MPLS changs th usual hop-by-hop paradigm by nabling ntwork flows to b routd along prdtrmind paths, which ar calld Labl Switchd Paths (LSPs). In an MPLS ntwork, ach flow is routd along a singl path, but two diffrnt flows with th sam sourc/dstination pair can us two diffrnt paths. Traffic nginring in MPLS ntworks mainly amounts to optimizing th quality of srvic of ntwork flows by tailoring th paths assignd to flows to th prvailing traffic conditions. As xplaind abov, whn th quality of srvic mtric is th ntwork dlay, th objctiv function dpnds non-linarly on th traffic flowing on ach link and th problm at hand bcoms a nonlinar singl-path routing problm. W aim to kp th discussion as gnral as possibl by proposing a solution stratgy that is also applicabl in othr application aras. W thrfor focus on th ssnc of th non-linar singlpath routing problm and omit on purpos many application-spcific dtails of traffic nginring in MPLS ntworks. Th problm w considr can b formulatd as follows. Givn som fixd traffic dmands, th goal is to find a singl-path routing stratgy that minimizs th sum of link costs. Th cost of a link dpnds on th amount of traffic y flowing on that link (masurd in packts/s) 2

4 and has th form y l (y ), whr th function l givs th dlay pr packt on link as a function of y. W spcifically assum that th link latncy function l is a non-linar function. A typical xampl is th M/M/1 dlay function l (x) = 1/(c x), whr c rprsnts th capacity of link. Not howvr that th rsults stablishd in this papr hold for a broad rang of cost functions. Th routing problm as dfind abov blongs to th class of non-linar mathmatical programs involving intgr variabls (actually, binary variabls). Ths mathmatical programs ar known to b xtrmly hard to solv, both from a thortical and from a practical point of viw (s Chaptr 15 of [27] for a survy as wll as [9] and rfrncs thrin for a thorough litratur rviw). Evn in th simplst cas with binary variabls, quadratic function and quality constraints, thy ar known to b NP-hard [14]. Svral gnral approachs hav bn proposd to solv non-linar intgr programming problms. Som transform th discrt problm into a continuous on (s for xampl [32]). Dspit thir qualitis, thos tchniqus do not scal vry wll with th siz of th problms. A rcnt altrnativ is th so-calld Global Smoothing Algorithm [30], which sms to scal bttr whil providing fairly good approximations (s Sction 5.1 for dtails). Huristics and mtahuristics hav also bn usd to find an approximat solution to non-linar intgr programming problms. Among othrs, ant-inspird optimization tchniqus ar known to b fficint for solving various routing problms [24, 36]. In this papr, w us th huristic mthod proposd in [24] for comparison purposs, as wll as xact NLP-basd branch-and-bound algorithms [12, 26]. W propos an approximation algorithm, inspird from Gam Thory [23], for solving nonlinar singl-path routing problms. W shall assum that individual dmands ar allowd to indpndntly slct thir path to minimiz thir own cost function. W dsign th cost function of th dmands so that th rsulting Nash quilibrium of th gam provids an fficint approximation of th optimal solution. W not that a similar algorithm was proposd in [1] for schduling of strictly priodic tasks. W stablish th convrgnc of th algorithm and show that vry optimal solution is a Nash quilibrium of th gam. W also prov that if th link latncy functions l ar polynomial of dgr d 0, thn th approximation ratio of th algorithm is ( 2 1/(d+1) 1 ) (d+1). As will b shown numrically, th main mrit of this algorithm is that it is svral ordrs of magnitud quickr than th mthod discussd abov whil providing good optimization rsults. Th rst of this papr is organizd as follows. W introduc our notation and formally stat th problm in Sction 2. W thn dscrib th proposd algorithm in Sction 3. In Sction 4, w show th convrgnc of th algorithm and obtain worst-cas guarants on th quality of th solutions 3

5 it producs. Sction 5 is dvotd to th mpirical prformanc valuation of th algorithm. Finally, som conclusions ar drawn in Sction 6. 2 Problm Statmnt Considr a ntwork rprsntd by a dirctd graph G = (V, E). To ach dg E is associatd a non-dcrasing latncy function l : IR + IR +. For any st π E, w dfin th constant δπ as 1 if π, and 0 othrwis. W ar givn a st K = {1, 2,..., K} of Origin-Dstination (OD) pairs. Lt s k and t k b th origin and dstination of OD pair k, and lt λ k IN b its traffic dmand. Each traffic dmand has to b routd in th ntwork ovr a singl path. W lt Π k b th st of all paths availabl for routing traffic btwn s k and t k. This st can contain all simpl paths from s k to t k, or only a subst of thos paths satisfying som constraints. W dfin a routing stratgy as a vctor π = (π k ) k K Π, whr π k is th path assignd to traffic dmand k and Π = Π 1 Π 2 Π K. Th goal is to find a routing stratgy that minimizs th cost of th ntwork F (π) = E y (π)l (y (π)), whr y (π) = k K δ π k λ k is th total traffic flowing on link in routing stratgy π. Formally, th problm is as follows: minimiz F (π) = E y (π) l (y (π)) (OPT) subjct to π k Π k k K. (1) W not that th problm can b formulatd as a 0-1 mathmatical programming problm by introducing th binary variabls 1 if dmand k is routd on path π x k,π = 0 othrwis. (2) Indd, w can rwrit problm (OPT) as follows: 4

6 minimiz E y l (y ) subjct to y = λ k δπx k,π k K π Π k E, (3) x k,π = 1 k K, (4) π Π k x k,π {0, 1} π Π k, k K. (5) Whn th objctiv function F is non-linar function of th variabls y, th abov problm blongs to th class of non-linar mathmatical programs involving intgr variabls (actually, binary variabls in our cas). Ths mathmatical programs ar known to b xtrmly hard to solv, thus motivating th dvlopmnt of fficint mthods for finding approximat solutions. 3 Bst-rspons algorithm Th bst-rspons algorithm w propos taks its inspiration from an algorithm of th sam nam in Gam Thory. In a gam, th bst-rspons of a playr is dfind as its optimal stratgy conditiond on th stratgis of th othr playrs. It is, as th nam suggsts, th bst rspons that th playr can giv for a givn stratgy of th othrs. Th bst-rspons algorithm thn consists of playrs taking turns in som ordr to adapt thir stratgy basd on th most rcnt known stratgy of th othrs until an quilibrium point is rachd. In this sction, w dsign th gam so that th Nash quilibrium of th gam provids an fficint approximation to problm (OPT). W follow th approach proposd in [15]. Lt us think of th traffic dmands as th playrs of th gam. Th stratgy of playr k is th path π k it chooss in th st Π k, and a stratgy profil of th gam is a vctor π Π. In othr words, a stratgy profil corrsponds to a fasibl solution of problm (OPT). Givn th stratgy of th othr playrs π i = (π 1, π 2,..., π i 1, π i+1,..., π K ), th valu f i (π, π i ) associatd to path π Π i by playr i rflcts th cost of this path, i.., 5

7 f i (π, π i ) = π λ i l (y (π, π i )), π Π i. (6) W not that F (π) = ( ) δπ k λ k l (y (π)) = k k λ k l (y (π)) = f k (π). (7) π k k Whn playr i minimizs th cost givn in (6), this gam is calld a wightd congstion gam (s [10] and rfrncs thrin). For ths gams, svral authors, including [10] and [2], hav invstigatd what is calld th Pric of Anarchy which is th ratio of th social cost (th function F in our cas) at th worst-cas Nash quilibrium and th social cost at th global optimum. For polynomial cost functions of dgr d, it was shown in [2] that th Pric of Anarchy for mixd quilibrium is Φ d+1 d, whr Φ d is th solution of (Φ d + 1) d = Φ d+1 d. Although on could dsign a bst-rspons algorithm basd on (6), that algorithm will hav two drawbacks: (i) a global optimum of F nd not b a Nash quilibrium of th wightd congstion gam as is shown in Exampl 1; and (ii) th convrgnc of th bst-rspons algorithm for th wightd congstion gam is known only in som spcial cass (linar latncy functions or whn th dmands ar th sam). Exampl 1. Considr a ntwork with two paralll links as shown in Figur 1. Thr ar two flows, ach of siz 1, which wish to go from O to D. Thr ar two possibl routs: (i) th top-rout with a latncy function of l(x) = x; and (ii) th bottom-rout with a latncy function of l(x) = 0.4x. 1 x 1 O D 0.4 x Figur 1: Exampl to show that th Nash quilibrium of a wightd congstion gam nd not b a global optimum. It can b vrifid that at th global optimum on of th flows will b routd through th top-rout and th othr will b routd through th bottom-rout. Th cost for th flow on th top-rout at 6

8 th optimal routing is 1 whil for th on on th bottom-rout it is 0.4. In ordr to chck that th optimum is not a Nash quilibrium for cost dfind in (6), w look at th bst-rspons of th flow on th top-rout. If this flow movs to th bottom-rout its cost will b 0.4(1 + 1) = 0.8 which is lss than its currnt cost of 1. Thus, at th global optimum th flow on th top-rout has an incntiv to mov to th bottom-rout. In ordr to driv th playrs towards a local minimum of F (π), w add a pnalty trm p i (π, π i ) to th cost of playr i. Th rol of th pnalty trm is that of an incntiv for playrs to tak into account th impact of thir actions on othr playrs. Whn playr i routs its traffic on path π, th incras in th cost of ach playr j i using link π is whr y i λ j [ l ( y i ) ( )] + λ i l y i, (8) = k i δ π k λ k rprsnts th total traffic flowing on link du to all playrs othr than i. As a consqunc, w dfin th pnalty trm as follows: p i (π, π i ) = π = π j i y i λ j δ π j [ l ( y i [ l ( y i ) ( )] + λ i l y i, ) ( )] + λ i l y i. (9) In summary, playr i computs its path to minimiz its own cost, that is, playr i solvs th following problm: minimiz π Πi c i (π, π i ) = f i (π, π i ) + p i (π, π i ). (OPT-i) In (OPT-i), th path π minimizing c i (π, π i ) is known as th bst rspons of playr i to th stratgy π i of th othr playrs. Th gam is in a Nash quilibrium if and only if th currnt stratgy of ach playr is its bst rspons to th stratgy of th othrs, implying that no playr has an incntiv to unilatrally dviat from its currnt stratgy. Formally, π is a Nash quilibrium if and only if c i (π) c i (π, π i ), π Π i, i K. 7

9 As w will show brifly, th bst-rspons algorithm that w propos as an approximation to problm (OPT) computs a Nash quilibrium. Th algorithm starts from an initial fasibl solution π (0). At ach itration, th playrs updat thir stratgis in a givn ordr by computing th optimal solution of problm (OPT-i) using any shortst path algorithm (.g., Dijkstra s algorithm). Not that a playr i dviats from its stratgy π (n) i at itration n to a nw stratgy π if and only if ( ) c i π, π (n) ( < c ) i π (n). Th algorithm stops whn no playr can dcras its cost by unilatrally i dviating from its stratgy, that is, π (n+1) = π (n). Th psudocod for th huristic is givn in Algorithm 1. Algorithm 1 Pnalizd bst-rspons Rquir: π (0) 1: n 0 2: rpat 3: for i = 1,..., K do 4: π (n+1) i argmin (OPT-i) 5: nd for 6: n n + 1 7: until π (n+1) π (n). 8: rturn π(n) In th following, w shall us th trm pnalizd bst-rspons to rfr to Algorithm 1. Th trm standard bst-rspons algorithm will rfr to th cas whn ach playr i dirctly optimizs f i (π, π i ) instad of th pnalizd cost function c i (π, π i ) = f i (π, π i ) + p i (π, π i ). 4 Proprtis of th pnalizd bst-rspons algorithm In this sction, w stablish svral proprtis of th pnalizd bst-rspons algorithm. Notably, w will show that (i) it has guarantd convrgnc; and (ii) th global optimum of F is an quilibrium of th algorithm. Ths two propritis ar not known to b tru for th standard bst-rspons algorithm. W first show in Sction 4.1 that th pnalizd bst-rspons algorithm convrgs in a finit numbr of stps to a Nash quilibrium. In contrast, th convrgnc of th standard bst-rspons algorithm in routing gams is still largly an opn issu. For th unsplittabl routing gams considrd in th prsnt papr, it has bn provn only (a) for linar latncy functions [22], and (b) whn 8

10 all flows hav th sam traffic dmands (that is, whn λ k = λ, k), in which cas th gam is a congstion gam [35]. It is also worth mntioning that for splittabl routing gams, th convrgnc of th standard bst-rspons algorithm has bn provn only in som spcial cass for ntworks of paralll links [3, 13, 29, 31]. In Sction 4.2, w obtain prformanc guarants for th solutions providd by th pnalizd bst-rspons algorithm in th spcial cas of polynomial link latncy functions. In this spcial cas, th algorithm is thrfor an approximation algorithm sinc it rturns a solution that is provably clos to optimal (in contrast to othr cass whr it is only a huristic that may or may not find a good solution). Our prformanc guarants tak th form of worst-cas bounds on th approximation ratio of our algorithm, that is, bounds on th ratio btwn th rsult obtaind by th algorithm and th optimal cost. W us tchniqus similar to thos usd in [5], whr th authors obtain bounds on th approximation ratio of th standard bst-rspons algorithm for affin and polynomial cost functions with non-ngativ cofficints. Th ida of th proof is similar in that it uss th Höldr inquality to bound th diffrnc btwn th Nash quilibrium and dviations from it. Sinc our problm is diffrnt from th on in [5], w gt a tight uppr bound. 4.1 Convrgnc of th pnalizd bst-rspons algorithm As provn in [15], th convrgnc of th pnalizd bst-rspons algorithm dirctly follows from th fact that F is a potntial function of th gam. W brifly dscrib th proof blow. Th crucial argumnt of this proof is statd in Lmma 1 blow. Lmma 1. Th objctiv function of playr i can b writtn as whr c i (π) = F (π) h(π i ), (10) h(π i ) = E y i l (y i ), (11) dos not dpnd on th path π i chosn by playr i. Proof. Not that y (π) = y i for π i, whil y (π) = y i + λ i for π i. Hnc 9

11 c i (π) = f i (π i, π i ) + p i (π i, π i ), = π i λ i l (y i = π i (y i = + λ i ) + π i y i + λ i ) l (y i [ l ( y i + λ i ) π i y i ) ( )] + λ i l y i, ( ) l y i, y (π) l (y (π)) + y (π) l (y (π)) πi πi = F (π) h(π i ). πi y i ( ) l y i + y i π i l ( y i ), Proposition 1. Th pnalizd bst-rspons algorithm convrgs in a finit numbr of stps to a Nash quilibrium. Proof. A dirct consqunc of Lmma 1 is that th function F is a potntial function of th gam, i.., c i (π) c i (π, π i ) = F (π) F (π, π i ), π Π i, i K, implying that any bst-rspons mov rsults in a dcras of th global cost F (π). Sinc F (π) can accpt a finit numbr of valus, th squnc will rach a local minimum in a finit numbr of stps. Anothr consqunc of Lmma 1 is that any global optimum is a Nash quilibrium, so that th pnalizd bst-rspons algorithm stops in an optimal point if it vr rachs it. Proposition 2. Any global optimum is a Nash quilibrium of th routing gam. Proof. Considr a global optimum π, i.., a point such that F (π ) F (π) for all π i Π i Assum to th contrary that π is not a Nash quilibrium. It follows that thr xists a playr i and a stratgy π i Π i such that c i (π i, π i ) < c i(π ). With Lmma 1, it yilds F (π i, π i ) h(π i ) < F (π ) h(π i ), from which w conclud that F (π i, π i ) < F (π ). This is clarly a contradiction to th global optimality of π. Hnc π is a Nash quilibrium. 10

12 Rmark 1. Th worst-cas computational complxity of finding a pur Nash quilibrium in a potnial gam can b xponntial in th numbr of playrs (th numbr of flows in our problm)[20]. Howvr, in [19] it is shown that on avrag th complxity is linar in th numbr of playrs. Thus, on random instancs, th bst-rspons algorithm can b xpctd to convrg much fastr than xact algorithms. 4.2 Prformanc guarants for polynomial link latncy functions An instanc I of our problm is dfind by th graph G = (V, E), by th st of traffic dmands K (including th dmand valu λ k and th st of candidat paths Π k for routing ach dmand k) and by th link latncy functions l : R + R +. Our goal in this sction is to stablish bounds on th approximation ratio of th pnalizd bst-rspons algorithm that hold uniformly ovr all instancs of th problm. Mor prcisly, givn an instanc I of th problm, lt π b any Nash quilibrium and lt π b an optimal routing stratgy for that instanc. W look for an uppr bound on th ratio F (π)/f (π ) that holds for all instancs I of th problm. In th following, w stablish such a bound whn th link latncy functions ar polynomial functions of th form l (x) = d j=0 a,jx j, whr d 0 and th a ar non-ngativ cofficints. Our main rsult is statd in Thorm 3. Thorm 3. If l (x) = d j=0 a,jx j, thn th approximation ratio of th pnalizd bst-rspons algorithm is boundd abov by ( 2 1/(d+1) 1 ) (d+1). Proof. S Appndix A. W can furthrmor prov that th uppr bound of Thorm 3 is tight, in th sns that thr xists an instanc of th singl-path routing problm for which it is rachd. Th lowr bound on th approximation ratio is statd in Proposition 4, whos proof is basd on an adaptation of th instanc givn in [10] (s Lowr Bound 2 and Lmma 4.5 in that papr). Proposition 4. If l (x) = d j=0 a,jx j, thn th approximation ratio of th pnalizd bst-rspons algorithm is boundd blow by ( 2 1/(d+1) 1 ) (d+1). Proof. W considr a problm instanc with a monomial cost function, that is, l (x) = a x d, and N origin-dstination pairs. In addition to ths nods, thr ar two othr nods R 0 and R 1 which act as routrs. Origin O i is connctd to nods O i 1 and O i+1, xcpt O 1 which is connctd to 11

13 R 0 and O 2, and O N which is connctd to O N 1 and R 1. Dstination D i is connctd to O i 1 and O i+1, xcpt D 1 which is connctd to R 0 and O 2, and D N which is connctd to O N 1 and R 1. In th following, w shall assum that links ar numbrd in such a way that link 0 is link R 0 O 1, link i is link O i O i+1 for i = 1,..., N 1 and link N is link O N R 1. For i = 0,..., N, link i has a cost of a i y d+1 whn it carris a traffic of y, whras for all othr links w hav a = 0. Th instanc has a total of 3N + 1 links of which 2N links hav a cost of 0 and th N + 1 othr links hav a non-zro cost. Figur 2 shows an instanc with N = 4 pairs. D 1 D 3 φ (d+1)0 φ (d+1)1 φ (d+1)2 φ (d+1)3 φ (d+1)3 R 0 O 1 O 2 O 3 O 4 R 1 D 2 D 4 Figur 2: Problm instanc with N = 4 origin-dstination pairs for th lowr bound on th approximation ratio. Lt λ i b th amount of traffic snt from origin O i to dstination D i. Each origin has two choics: ithr snd th traffic to th lft or to th right. In th global optimum (OPT) ach origin will snd its traffic to th right providd that a i+1 a i and a N = a N1. (12) Indd, in that cas, ach flow is routd ovr th link with th smallr a, and ovr anothr link of cost l (x) = 0. Similarly, w shall stablish th conditions undr which routing th traffic to th lft corrsponds to a Nash quilibrium (NE). It is nough, according to Lmma 1, to show that w cannot dcras th ntwork cost by dviating a singl flow from its quilibrium rout onto anothr rout. Providd that a N = a N 1, th cost dos not chang whn flow N is routd to th right on link O N R 1 instad of bing routd to th lft on link O N 1 O N. If flow i = 1,..., N 1 is routd to th right on link i instad of bing routd to th lft on link i 1, th traffic on link i incrass from λ i+1 to λ i + λ i+1, whras th traffic on link i bcoms 0. W dduc that th cost variation is 12

14 a i (λ i + λ i+1 ) d [ a i 1 λ d+1 i + a i λ d+1 i+1 ], so that routing to th lft corrsponds to a Nash quilibrium providd that a i 1 λ d+1 i + a i λ d+1 i+1 a i(λ i + λ i+1 ) d+1, i = 1,..., N 1, (13) and a N = a N 1. Lt us now assum that a i = φ (d+1)i for 0 i < N and that a N = φ (d+1)(n 1) y d+1, whr φ = 2 1/(d+1) 1. Not that th links O N 1 O N and O N R 1 hav th sam cost. In Figur 2, for th links with a non-zro cost, th corrsponding a is shown clos to th mid-point of th link. W furthr assum that λ i = φ N i. Sinc φ < 1, a N = a N 1 and a i+1 = φ (d+1) a i for i = 1,..., N 1 implis that th conditions (12) ar satisfid, so that routing to th right is indd an optimal stratgy. To s that routing to th lft corrsponds to a Nash quilibrium, obsrv that for i < N, a i 1 λ d+1 i + a i λ d+1 i+1 = 2 φ (d+1)(n 1) = (1 + φ) d+1 φ (d+1)(n 1) = a i (λ i + λ i+1 ) d+1, which provs that th ntwork cost cannot b dcrasd by dviating unilatrally any flow. It is intrsting to not that this instanc has bn dsignd in such a way that th corrsponding Nash quilibrium is a wak quilibrium, that is, for ach flow i, thr xists a path π diffrnt from its quilibrium path π i such that c i (π, π i ) = c i (π). For th instanc with N = 4, in Tabl 1, w giv th traffic on ach link and th associatd cost for ach of th two solutions (for th clarity of th prsntation, only links such that l (x) 0 ar prsntd). Summing th costs of th links in Tabl 1, w obsrv that th optimal cost is F 4 (π ) = 3φ 4(d+1) + φ 3(d+1), whras th cost at th Nash quilibrium is F 4 (π) = 4φ 3(d+1). In gnral, it can b sn that F N (π ) = (N 1)φ (d+1)n + φ (d+1)(n 1) and that F N (π) = 13

15 Tabl 1: Traffic on th links and associatd cost in th optimal solution and at th Nash quilibrium for th instanc shown in Figur 2. NE OPT Links Flow Traffic Cost Flow Traffic Cost R 0 O 1 1 φ 3 φ 3(d+1) 0 0 O 1 O 2 2 φ 2 φ 3(d+1) 1 φ 3 φ 4(d+1) O 2 O 3 3 φ 1 φ 3(d+1) 2 φ 2 φ 4(d+1) O 3 O 4 4 φ 0 φ 3(d+1) 3 φ 1 φ 4(d+1) O 4 R φ 0 φ 3(d+1) Nφ (d+1)(n 1). As a consqunc, th approximation ratio is g N (φ) = N (N 1)φ d (14) By taking N, w obtain an approximation ratio of φ (d+1) = (2 1/(d+1) 1) (d+1) as claimd. Rmark 2. For th sak of clarity, w hav shown th lowr bound on an undirctd graph. By adding links and intrmdiat nods, this instanc can b modifid to mak th graph dirctd as wll. Howvr, this will mak th figur and th notation cumbrsom. As an immdiat consqunc of Thorm 3 and Proposition 4, w obtain th following corollary. ( Corollary 5. For larg d, th approximation ratio is roughly d+1. d+1 log(2)) In Tabl 2, w giv th approximation ratios for diffrnt valus of d for th pnalizd bstrspons algorithm and compar it with that of th non-pnalizd vrsion, as stablishd in [5] and [10]. W mphasiz that it is not provn that th non-pnalizd vrsion convrgs for polynomial latncy functions, so that its approximation ratio provids only a guarant on th quality of th solution whn it convrgs. Intrstingly, in th cas d = 0, that is, whn th latncy function l (x) of ach link is a constant a 0, our algorithm is guarantd to provid an optimal solution. This is a dirct consqunc of Thorm 3, but can also b asily undrstood by noting that in this cas 14

16 Tabl 2: Approximation ratio for th pnalizd bst-rspons (BR) as wll as for th standard bst-rspons as a function of th dgr of th polynomial. Th valus hav bn roundd-off at two dcimal placs. dgr (d) Pnalizd BR Standard BR [2, 10] asymptotic Θ ( ( d+1 log(2) ) d+1 ) Θ ( ( d log(d) ) d+1 ) F (π) = E a y (π) = E a k K δ π k λ k = k K λ k ( π k a ) so that it is optimal to rout ach flow k on th path π k Π k minimizing π k a, which can b sn as th lngth of th path. Howvr, in that cas (9) yilds p i (π) = 0, from which it follows that c i (π, π i ) = f i (π, π i ) = λ i ( π i a ), so that our algorithm routs ach flow on a shortst path, xactly as dos th optimal solution. Whn d 1, th approximation ratio of th pnalizd bst-rspons algorithm is significantly wors than that of th non-pnalizd bst-rspons algorithm. Asymptotically, th non-pnalizd vrsion is a factor (log(d)) d+1 bttr than th pnalizd vrsion. On th downsid, th non-pnalizd vrsion dos not hav a guarantd convrgnc whras th pnalizd on is guarantd to convrg. It is howvr important to kp in mind that th abov prformanc guarants ar obtaind for worst-cas scnarios that ar not ncssarily rprsntativ of instancs mt in practic. W conjctur that th worst-cas prformanc of th algorithm occurs in vry asymmtric scnarios in trms of link latncy functions and in trms of traffic dmands. Considr for xampl th instanc usd in Proposition 4 to obtain a lowr bound on th approximation ratio. Assum that th link latncy functions ar monomials of dgr d such that a N = a N 1 and 1 x such that 1 x min i (λ i + λ i+1 ) (d+1) λ (d+1) i+1 λ d+1. i, ai a i+1 x for som fixd 15

17 Th abov conditions impos a crtain homognity btwn th cofficints a i. It is radily vrifid that quations (12) and (13) ar satisfid undr ths conditions, so that routing to th right is an optimal stratgy, and routing to th lft is a Nash quilibrium. Obsrv now that N 1 F (π) F (π ) = i=0 aiλ (d+1) i+1 N i=1 a iλ (d+1) i x N 1 i=0 a i+1λ (d+1) i+1 N i=1 a iλ (d+1) i so that in this xampl w can obtain a solution as clos as w want to an optimal solution by imposing a condition on th ratio = x, ai a i+1. Unfortunatly, w wr not abl to xplicitly charactriz th conditions undr which th approximation ratio will b lowr than a givn x in th gnral cas. Nvrthlss, th numrical xprimnts prsntd in Sction 5 rval that th pnalizd bst-rspons algorithm oftn provids good quality solutions, contrary to what is suggstd by th approximation ratio ( 2 1/(d+1) 1 ) (d+1). Ths xprimnts also show that th rsults of th pnalizd bst-rspons algorithm ar quit similar to thos of its non-pnalizd countrpart. 5 Empirical Prformanc Evaluation This sction is dvotd to th mpirical prformanc valuation of th bst-rspons algorithm. W first dscrib th bnchmark xprimnts that wr prformd, as wll as our mpirical algorithm comparison mthodology in Sction 5.1. Sction 5.2 rports th numrical rsults obtaind. 5.1 Bnchmark Exprimnts In our bnchmark xprimnts, w hav usd 800 randomly gnratd instancs obtaind using 8 standard ntwork topologis (s Tabl 3) collctd from th IEEE litratur and from th Rocktful projct [37]. For ach ntwork topology, w considr 100 random traffic matrics gnratd with uniform distributions in such a way that thr is positiv traffic dmand associatd to ach origin/dstination pair. Th traffic matrics ar computd so that thr xists a minimum-hop routing stratgy in which th ntwork congstion rat (that is, th maximum utilization rat of th ntwork links) is qual to γ, whr γ > 0 is a givn paramtr. Th mpirical valuation of th bst-rspons algorithm is don for two diffrnt typs of objctiv functions, as dscribd blow. 16

18 Tabl 3: Topologis : numbr of nods and links. Topology # nods # links ABOVENET ARPANET BHVAC EON METRO NSF 8 20 PACBELL VNSL Pic-wis Linar Objctiv Function Th main motivation for considring a pic-wis linar objctiv function is that in that cas w can compar th algorithm with th optimal solutions obtaind from a MILP solvr. Spcifically, w considr th following incrasing latncy function proposd in [21]: 1 if 0 y c < 1 3, c y if 1 3 y c < 2 3, c y 3 if 2 3 l (y) = y c < 9 10, c y 9 3 if 10 y c < 1, c y if 1 y c < 11 10, c y if y c, whr c rprsnts th capacity of link. This function xprsss that it is chap to snd flow ovr a link with a small utilization rat, whras, as th load approachs 100%, it bcoms mor xpnsiv. Not that th utilization rat can b gratr than 1, in which cas w gt havily pnalizd. For th abov latncy function, w assum that th st Π k contains all th possibl path btwn s k and t k for all flows k. Problm (OPT) can thn b formulatd as a Mixd-Intgr Linar Problm: 17

19 minimiz E Φ (OPT-PWL) subjct to 1 if n = s, x k, x k, = 1 if n = t, Out(n) 0 othrwis. In(n) n V, k K, (15) y = k K λ k x k, E, (16) Φ ay bc (a, b) C, E, (17) x k, {0, 1}, y 0, Φ 0 E, k K, (18) whr C = { (1, 0), (3, 2 3 ), (10, 16 3 ), (70, ), (500, 3 ), (5000, 3 ) } and In(n) (rsp. Out(n)) rprsnts th st of all ingoing (rsp. outgoing) links at nod n. In th abov nod/link formulation, th dcision variabl x k, is 1 if flow k is routd ovr link, and 0 othrwis. For ach of th 800 instancs, w hav computd th rlativ rror of th bst-rspons algorithm with rspct to th optimal solution of (OPT-PWL), which was computd using Gurobi 6.0 as th MILP solvr [26]. W hav limitd th total tim xpndd by th solvr to 15 minuts pr problm instanc. Th traffic matrics wr computd with γ = 1.2 as thrshold paramtr, so that w can guarant that thr xists a fasibl solution in which th utilization rat of th links is at most 120% Non-linar Objctiv Function W considr two diffrnt link latncy functions, ach on yilding a non-linar ntwork cost function. Th first on is a linar latncy function, l (y) = link, yilding th following quadratic objctiv function y (c ) 2, whr c rprsnts th capacity of ( ) 2 y (π). (19) F (π) = E c Th scond link latncy function corrsponds to th wll-known M/M/1 dlay function (a.k.a. 18

20 Klinrock function) [28], that is l (y) = 1 c y, so that F (π) = E y (π) c y (π). (20) This cost function is oftn usd whn on wants to minimiz th man packt dlay in th ntwork. For th abov two objctiv functions, it bcoms xtrmly hard to find an optimal solution to problm (OPT). W thrfor compar our pnalizd bst-rspons algorithm with a lowr bound on th optimal cost obtaind by rlaxing th 0-1 constraints in problm (OPT), thrby solving th following multi-path routing problm minimiz E y l (y ) (OPT-NL-MP) subjct to y = λ k δπx k,π k K π Π k E, (21) x k,π = 1 k K, (22) π Π k 0 x k,π 1 π Π k, k K, (23) which can asily b don with a projctd gradint algorithm. In addition to th comparison with th optimal multi-path solution, w compar our algorithm with four othr algorithms: Global Smoothing Algorithm (GSA): This algorithm was introducd in [30] for solving non-linar optimization problms with binary variabls. Its original approach is to solv a squnc of non-linar optimization problms without intgr variabls. In ach problm of th squnc, th goal is to minimiz th sum of th original objctiv function plus two pnalty trms: a logarithmic barrir pnalty (dsignd to smooth th function and kp th currnt point away from th intgr grid) and an xact" pnalty (dscribing how far th currnt point is from th intgr grid). Both trms ar wightd by on paramtr, which is updatd at ach itration in ordr to gradually bring th currnt point onto th intgr grid. At ach stp, a continuous optimization problm is solvd using a gradint algorithm 19

21 (w hav usd th modifid conjugat gradint algorithm prsntd in [11]). Th squnc of pnalizd problms should form a trajctory lading to a good approximation of th solution. W rfr to [30] for furthr dtails. Ant Colony Optimization (ACO): This huristic algorithm was proposd in [24]. It blongs to th family of ant colony algorithms, which ar known to b fficint for sarching an optimal path in a graph [17, 18, 36]. Th algorithm itrativly improvs an initial solution to th problm through a random xploration of th solution spac by agnts calld (artificial) ants. During on itration of th algorithm, ach ant builds a solution to th problm by randomly taking dcisions, that is by randomly assigning paths to flows. A wight, calld phromon trail, is assignd to ach dcision takn by ants and is updatd basd on th xprinc acquird by ants during problm solving. Ths wights, which rprsnt th larnd dsirability of th dcisions, ar usd to guid th random xploration of th solution spac. Gurobi quadratic-programming basd branch-and-bound algorithm: Th Gurobi MIP solvr can solv modls with a quadratic objctiv and/or quadratic constraints using a branch-and-bound algorithm [26]. This xact algorithm was usd to comput th optimal solution of th singl-path routing problm whn th objctiv function is givn by (19). As for th pic-wis linar objctiv function, w hav limitd th total tim xpndd by th solvr to 15 minuts pr problm instanc. Bonmin NLP-basd branch-and-bound algorithm: Bonmin is an opn-sourc softwar for solving gnral MINLP (Mixd-Intgr Non-Linar Programming) problms [12]. Bonmin faturs svral xact algorithms, including a NLP-basd branch-and-bound algorithm and an outr-approximation dcomposition algorithm. Th formr was usd to comput th optimal solution of th singl-path routing problm whn th objctiv function is givn by (20). As with Gurobi, w hav limitd th total tim xpndd by Bonmin to 15 minuts pr problm instanc. Givn th siz and complxity of th problm instancs, and in ordr to kp th running tims of ths algorithms blow a rasonabl amount, w rstrict ourslvs to two possibl paths for ach OD flow (that is, Π k = 2 for all k K). Ths paths ar obtaind by solving a 2-shortst-path problm in which th link wights ar qual to 1 [40]. Th traffic matrics wr computd with 20

22 γ = 1.0 as thrshold paramtr, so that w can guarant that thr xists a fasibl solution in which th capacity of no link is xcdd, which is particularly important for th M/M/1 latncy function. Th GSA and ACO algorithms hav bn implmntd in Matlab 2013b. For GSA, two diffrnt initial multi-path solutions ar considrd: th first on is randomly gnratd, whras th scond on is obtaind by adding a small random prturbation to th singl-path solution obtaind with th bst-rspons algorithm (so as to obtain a multi-path initial solution). Th rsults prsntd in th following for th GSA algorithms corrspond to th bst solution obtaind from ths two initial points. Rgarding ACO, w considr at most 50 itrations and 5 ants sinc our xprimnts rvald that incrasing ths valus dos not improv significantly th obtaind rsults whil ngativly impacting computing tims. 5.2 Numrical Rsults Pic-wis Linar ntwork cost function Th numrical rsults obtaind for this cost function ar rportd in Tabl 4. Intrstingly, th worst rsults ar obtaind for th smallst topologis, NSF and METRO. Excpt for ths topologis, th pnalizd BR algorithm prforms quit wll. Its avrag rror ovr th 800 instancs is only 3.31%, and it would b as low as 1.77% without th instancs gnratd with th NSF topology. Th avrag xcution tim of th algorithm is 0.51 sconds, and it nvr xcds 3.2 sconds. This is to b compard to th avrag xcution tim of 82.6 sconds of th MILP solvr (th tim limit of 15 minuts was rachd for 46 out of th 100 instancs gnratd with th EON topology) Quadratic ntwork cost function W first start by comparing th pnalizd BR algorithm with GSA and ACO for this cost function. Our numrical rsults ar rportd in Tabl 5. First of all, not that all thr algorithms prform quit wll sinc th avrag rlativ rror is blow 4.37% for ach on. Ovr th 800 instancs, th maximum rlativ rror of th bst-rspons algorithm is 20.16% (to b compard to th 13.32% of th othr algorithms). W also not that on avrag GSA prforms slightly bttr than ACO and th bst-rspons algorithm. Rmmbr howvr that GSA is initializd with th solution obtaind with th bst-rspons algorithm. If w now look at th computing tims prsntd in Tabl 9, w 21

23 Tabl 4: Rlativ gap (%) to th optimal solution for th pic-wis linar objctiv function. Topology min max avg ABOVENET ARPANET BHVAC EON METRO NSF PACKBEL VNSL Global obsrv that th bst-rspons algorithm is significantly fastr than all othr algorithms, sinc its worst computing tim is 6.8 sconds, whras th running tims of ACO and GSA can b as high as 50 and 300 sconds, rspctivly. It is also intrsting to not that th Gurobi solvr usually computs th optimal solution quickly (although always significantly slowr that that of th pnalizd BR algorithm). Givn ths fast computing tims of th Gurobi solvr, and in ordr to bttr assss th prformanc of th pnalizd BR algorithm, w ran onc again both of thm, but this tim with k = 6 candidat paths pr OD flow. Gurobi found th optimal solution for all instancs. Rsults ar rportd in Tabl 6. Excpt for th METRO topology, th rlativ gaps to th optimal solution ar vry modst and th avrag rror ovr th 800 instancs is only 1.07% (it would b as low as 0.2% without th instancs gnratd with th METRO topology) M/M/1 ntwork cost function Th rsults obtaind with this cost function ar prsntd in Tabl 7. In this cas, th pnalizd bstrspons and th ACO algorithms prform bttr than GSA, for which rlativ rrors up to 87.54% ar obtaind on som instancs. Th bst-rspons algorithm achivs th lowr avrag rlativ rror with only 1%, whras ACO has a lowr maximum rlativ rror with 14.99%. Howvr, as can b notd from Tabl 9, th bst-rspons algorithm provids th bst tradoff btwn optimization quality and computing tims. Ovr th 800 instancs, its worst xcution tim is only 4.7 sconds, whras th computing tims of GSA and ACO can b as high as 400 and 50 sconds, rspctivly. 22

24 Tabl 5: Rlativ gap (%) to th optimal multi-path solution for th quadratic cost function. Topology Pnalizd BR GSA ACO min max avg min max avg min max avg ABOVENET ARPANET BHVAC EON METRO NSF PACBELL VNSL Global Tabl 6: Rlativ gap (%) to th Gurobi solution for th quadratic cost function. Topology Pnalizd BR min max avg ABOVENET ARPANET BHVAC EON METRO NSF PACBELL VNSL Global W obsrv that th computing tims of Bonmin ar two ordrs of magnitud largr than that of th pnalizd BR algorithm. In practic, thy oftn rach th tim limit of 15 minuts for th largst instancs, vn though thr ar only k = 2 candidat paths pr OD flow (w had to incras th tim limit for th ARPANET ntwork in ordr to always obtain a fasibl solution). Tabl 8 shows th minimum, maximum and avrag rlativ gaps btwn th solution of th BR algorithm and th on computd by Bonmin. Hr again, dspit th fact that significant rrors ar obtaind for th smallst topologis (NSF and METRO), th avrag rror ovr th 800 instancs is only 23

25 Tabl 7: Rlativ gap (%) to th optimal multi-path solution for th M/M/1 cost function. Topology Pnalizd BR GSA ACO min max avg min max avg min max avg ABOVENET ARPANET BHVAC EON METRO NSF PACBELL VNSL Global %. Tabl 8: Rlativ gap (%) to th Bonmin solution for th M/M/1 cost function. Topology Pnalizd BR min max avg ABOVENET ARPANET BHVAC EON METRO NSF PACBELL VNSL Global W rmind th radr that all algorithms whr run with only 2 candidat paths for ach OD pair. In ordr to valuat how incrasing th numbr of paths affcts th quality of th solution and th computing tims, w also hav run th pnalizd BR algorithm with 4 and 6 paths for th ARPANET topology (th largst on) and th M/M/1 cost function. Ths paths wr chosn using Yn s algorithm for computing th k-shortst looplss paths in a graph [40]. Figur 3(a) prsnts th computing tims of th pnalizd BR algorithm with 4 and 6 paths (normalizd by 24

26 Scnario Tabl 9: Computing tims (sconds) for th 800 problm instancs. Quadratic Cost M/M/1 Cost BR GSA ACO Gurobi BR GSA ACO Bonmin Min Max Avrag th computing tims with 2 paths) for th first 25 instancs. Ovr th whol st of 100 instancs, th computing tims incras only by a factor 1.69 (rsp. 2.49) on avrag whn passing from 2 to 4 paths (rsp. 6). Similar rsults wr obtaind with th othr topologis and othr cost functions, which lads to th conclusion that running th pnalizd BR algorithm with 2n candidat paths yilds roughly an incras of computing tims by a factor n. This has usually a positiv impact on th quality of th solution, and Figur 3(b) shows th rlativ cost improvmnts whn using 4 and 6 candidat paths instad of 2 for th first 25 instancs. Although significant improvmnts can b obsrvd in som cass, th avrag rlativ improvmnt ovr th 100 instancs is only 2.1 % with 4 paths, and 2.34 % with 6 paths. W howvr obsrvd that, in contrast to th cas of computing tims, th bnfits of using mor candidat paths havily dpnd on th particular instanc considrd. Rmark 3. Th rsults obtaind with th standard bst-rspons algorithm ar comparabl to thos prsntd abov for th pnalizd bst-rspons algorithm (s Tabls 10 and 11). In practic, th rsults of th standard bst-rspons algorithm ar somtims bttr, and somtims wors, but th diffrnc is always in th ordr of a fw prcnts. On th othr hand, th computing tims of th standard bst-rspons algorithm ar clarly bttr sinc its worst xcution tim ovr th 800 instancs is 1.1 sconds for th M/M/1 cost. This is somthing xpctd sinc th pnalty trm implis an xtra computation with rspct to th standard bst-rspons algorithm. It is also intrsting to not that both vrsions of th BR algorithm rquir roughly th sam numbr of itrations. For instanc, for th quadratic objctiv function, th avrag numbr of itrations of th pnalizd BR algorithm ovr th 800 instancs was 4.63, whras it was 4.53 for th non-pnalizd BR algorithm. 25

27 !"#$%%$"!! " &' &' #$ #$ (a) Computing tims with 4 and 6 paths (normalizd by th computing tims with 2 paths). (b) rlativ cost improvmnts (%) with 4 and 6 paths instad of 2 paths. Figur 3: Impact of passing from 2 candidat paths to 4 or 6 candidat paths on th quality of th solution and on computing tims for th first 25 instancs of th ARPANET topology. Tabl 10: Rlativ gap (%) to th optimal multi-path solution for th quadratic cost function. Topology pnalizd BR standard BR min max avg min max avg ABOVENET ARPANET BHVAC EON METRO NSF PACBELL VNSL Global Conclusion In this papr, w studid non-linar singl-path routing problms, which ar known to b xtrmly hard to solv. Yt, ths problms naturally aris whn on sks to optimiz a quality of srvic 26

28 Tabl 11: Rlativ gap (%) to th optimal multi-path solution for th M/M/1 cost function. Topology pnalizd BR standard BR min max avg min max avg ABOVENET ARPANET BHVAC EON METRO NSF PACBELL VNSL Global mtric such as th dlay. For solving ths problms, w hav proposd an algorithm inspird from Gam Thory in which individual flows ar allowd to indpndntly slct thir path to minimiz thir own cost function. Givn th routing of th othr flows, ach flow sks to minimiz th sum of its own nd-to-nd dlay, plus a pnalty trm which plays th rol of an incntiv for playrs to tak into account th impact of thir actions on th othrs. W hav provn th convrgnc of this algorithm to a Nash quilibrium of th gam. In th cas of polynomial link latncy functions, w hav also stablishd a tight uppr bound on th approximation ratio of th algorithm. Th numrical rsults obtaind with this algorithm ar quit intrsting: th rlativ gap to th optimal multi-path solution rmains modst (quivalnt to that of GSA and ACO), whil th xcution tims ar substantially lowr than that of th othr algorithms. Ths rsults suggst that th bst-rspons algorithm is an appropriat solution for larg-scal singl-path routing problms with non-linar cost functions, whr othr tchniqus show thir limits. Acknowldgmnt This work was supportd by Frnch FUI projct NEC (AAP FUI 12). 27

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