Online Multicommodity Routing with Time Windows


 Nathaniel Edwards
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1 KonrdZuseZentrum für Informtionstechnik Berlin Tkustrße 7 D BerlinDhlem Germny TOBIAS HARKS 1 STEFAN HEINZ MARC E. PFETSCH TJARK VREDEVELD 2 Online Multicommodity Routing with Time Windows 1 Institute of Mthemtics, Technicl University Berlin, Strße des 17. Juni 136, 1623 Berlin, Germny. 2 Mstricht University, Deprtment of Quntittive Economics, P.O. Box 616, 62 MD Mstricht, The Netherlnds. ZIBReport 722 August 27)
2 ONLINE MULTICOMMODITY ROUTING WITH TIME WINDOWS TOBIAS HARKS, STEFAN HEINZ, MARC E. PFETSCH, AND TJARK VREDEVELD Abstrct. We consider multicommodity routing problem, where demnds re relesed online nd hve to be routed in network during specified time windows. The objective is to minimize time nd lod dependent convex cost function of the ggregte rc flow. First, we study the frctionl routing vrint. We present two online lgorithms, clled Seq nd Seq 2. Our first min result sttes tht, for cost functions defined by polynomil price functions with nonnegtive coefficients nd mximum degree d, the competitive rtio of Seq nd Seq 2 is t most d+1) d+1, which is tight. We lso present lower bounds of.265 d + 1)) d+1 for ny online lgorithm. In the cse of network with two nodes nd prllel rcs, we prove lower bound of ) on the competitive rtio for Seq nd Seq 2, even for ffine liner price functions. Furthermore, we study resource ugmenttion, where the online lgorithm hs to route less demnd thn the offline dversry. Second, we consider unsplittble routings. For this setting, we present two online lgorithms, clled USeq nd USeq 2. We prove tht for polynomil price functions with nonnegtive coefficients nd mximum degree d, the competitive rtio of USeq nd USeq 2 is bounded by O1.77 d d d+1 ). We present lower bounds of.537 d + 1)) d+1 for ny online lgorithm nd d + 1) d+1 for our lgorithms. Third, we consider specil cse of our frmework: online lod blncing in the l pnorm. For the frctionl nd unsplittble vrint of this problem, we show tht our online lgorithms re nd Op) competitive, respectively. Such results where previously known only for scheduling jobs on restricted un)relted prllel mchines. 1. Introduction In this pper, we consider multicommodity routing problem, in which sets of commodities rrive over time nd hve to be routed in network. Ech commodity is only live during specified time window. We study two vrints: i) the demnd of commodity cn be split long severl pths; ii) single pth unsplittble) routing. In ll cses, once demnd is routed, no rerouting is llowed. The cost of routing the next smll unit of flow on n rc is determined by nondecresing nd continuous price function of the flow tht hs lredy been routed on this rc. For fixed point in time, the routing cost on n rc is defined by the integrl over the rc flow t tht time with respect to its price function. Given certin time spn, the totl routing cost on n rc is obtined by integrting the routing cost over Dte: December 28, 27. Supported by the DFG Reserch Center Mtheon Mthemtics for key technologies in Berlin. 1
3 2 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD time. The gol is to find routing with miniml cost. Since the demnds rrive over time, both problems splittble nd unsplittble) hve nturl online vrint, which we study in this pper. An online lgorithm lerns bout the existence of routing request only t its relese time, i.e., the lower limit of its time window. We evlute the qulity of online lgorithms by competitive nlysis [24, 21], which hs become stndrd yrdstick for mesuring performnce. An lgorithm Alg is sid to be ccompetitive, if for ny instnce the cost of the solution produced by Alg is not more thn c times the optiml solution vlue. The infimum over ll such constnts c is the competitive rtio of Alg. For both vrints of the problem, we define two greedy online lgorithms, which we cll Seq nd Seq 2 for the splittble vrint nd USeq nd U Seq 2 for the unsplittble vrint. Algorithms Seq nd USeq greedily compute routing decision for ll commodities relesed t the sme time by minimizing the cost ssocited with these requests. The other two lgorithms Seq 2 nd USeq 2 consider the commodities relesed t the sme time one by one, computing minimum cost routing for ech individul commodity. We investigte cses in which Seq USeq) nd Seq 2 USeq 2 ) re ccompetitive for some constnt c. The problem under investigtion rises, for instnce, in n interdomin Qulity of Service QoS) mrket, in which multiple service providers offer network resources cpcity) to enble Internet trffic with specific QoS constrints, see for exmple Yhy nd Sud [27] nd Yhy, Hrks, nd Sud [26]. In such mrket, ech service provider dvertises prices for resources tht he wnts to sell. Buying providers reserve cpcity long pths to route demnd coming from own customers) from source to destintion vi domins of other providers. The routing of demnd long pths is fixed by estblishing binding contrct between the source domin nd ll domins long the pths. The price for unit bndwidth chnges dynmiclly with the totl bndwidth tht is currently in use, tht is, routing flow of unit size prompts n updte of rc prices. It is ssumed tht price updtes on n rc re determined by nondecresing nd continuous price function. Hrks et l. [17, 18] considered the problem without time windows. In prctice, demnd requests between service providers come with lifetime nd service providers my hve severl demnd requests t the sme time, which llows for coordinted routing of these demnds. In this pper, we explicitly ddress these issues nd generlize the model of Hrks et l. [17, 18], see Section 6 for comprison. Another ppliction of our frmework is the lod blncing problem in intrdomin networks. In this problem, trffic demnds hve to be routed from entry nodes to exit nodes of the domin. A nturl gol for domin opertor is to distribute the lod cross the network, i.e., minimize the l p  norm of the vector of the rc lods in the network, see for instnce Fortz nd Thorup [14]. Since trffic demnds re usully not known beforehnd, this problem hs nturl online vrint, which we study in this pper Relted Work Yhy et l. [27, 26] empiriclly studied the performnce of greedy singlepth routing protocol for fixed network topology. This problem ws first
4 ONLINE ROUTING WITH TIME WINDOWS 3 Tble 1: Lower bounds LB) nd upper bounds UB) on the competitive rtio of ny deterministic online lgorithm nd on Seq 2 for ffine liner price functions nd without time windows s discussed in Hrks et l. [18]; here n is the number of commodities. splittble unsplittble ny lg. Seq, Seq 2 ny lg. USeq, USeq 2 LB 4 3 mx{ 2n 1 n, 4} UB 4n 2 n+1) modeled s n online multicommodity flow problem in Hrks et l. [17, 18]. They nlyzed greedy online lgorithm which is closely relted to Seq 2 ), nd proved tht this lgorithm is symptoticlly 4competitive for ffine price functions; n overview of the results discussed in [18] is given in Tble 1. Moreover, they showed some generl lower bounds on the competitive rtio of ny online lgorithm. In the context of trffic engineering, multicommodity routing problems hve been studied by e.g. Fortz nd Thorup [13, 14]. There, the gol is to route given demnd subject to cpcity constrints in order to blnce the totl lod in the network. In this setting, centrl plner hs full knowledge of ll demnds, which is not the cse in our pproch. Whenever there is no centrl plner, routing decisions hve to be mde by selfish users. This hs been extensively studied using gme theoretic concepts, cf. Roughgrden nd Trdos [23], Corre, Schulz, nd Stier Moses [1], Altmn, Bsr, Jimenez, nd Shimkin [2], nd the references therein. These works study the efficiency of Nsh equilibri. The difference to our model is the notion of n equilibrium, which llows for rerouting of demnds upon relese of new demnd. In our model, once routing decision hs been mde, it remins unchnged. In ddition, in this pper, the totl cost of multicommodity flow is time dependent, tht is, the totl cost depends on the time windows of the commodities. Frzd et l. [12] consider selfish routing problems, where the trvel times of gents on n rc re defined by priority order, see lso the thesis of Olver [22]. The totl cost on n rc is defined s the integrl of the rc flow with respect to given ltency functions. They proved tight upper bounds of d + 1) d+1 on the price of nrchy for polynomil ltency functions with nonnegtive coefficients. For unsplittble flows, they showed tight upper bound of for ffine ltencies nd n upper bound of O2 d d d+1 ) for polynomil ltency functions. Their model, however, does not cpture time vrying trffic demnds. The results of Frzd et l. [12] re tilored to polynomil ltency price) functions, wheres our results re bsed on generl technique, which is lso pplicble to generl continuous nd nondecresing price functions. Furthermore, our upper bounds for polynomil price functions in the unsplittble vrint improve upon the bounds derived in [12]. The lower bounds presented in Frzd et l. [12] crry over to lower bounds for our lgorithms. Hrks nd Végh [19] study n online version of the selfish routing problem. Demnds re relesed in n online fshion, nd for every relesed demnd
5 4 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD Tble 2: Known lower bounds LB) nd upper bounds UB) on the competitive rtio of ny deterministic online lgorithm nd on Seq 2 for polynomil price functions of degree d. splittble unsplittble ny lg. Seq, Seq 2 ny lg. USeq, USeq 2 LB.265 d + 1)) d+1 d + 1) d d + 1)) d+1 d + 1) d+1 UB d + 1) d+1 O1.77 d d d+1 ) Nsh solution is constructed. Their work is restricted to splittble flows nd does not cover time vrying demnds. Online lod blncing problems on prllel mchines cn lso be seen s multicommodity routing problem; see Albers [1] for n overview of online lgorithms nd lod blncing in prticulr. In the simplest form, lod blncing cn be seen s routing n unsplittble demnds from the sme source to the sme destintion over m prllel links. More sophisticted topologies re considered in Section 4.4. Awerbuch et l. [4] considered greedy online lgorithm, where the gol is to minimize the l p norm of the ggregted server lods. In prticulr, they proved n upper bound on the competitive rtio of for the l 2 norm nd Op) for the l p norm. In the sme context, Avidor et l. [3], Azr et l. [6], Crginnis et l. [8], nd Suri et l. [25] strengthened the nlysis of [4] for specil cses e.g., identicl mchines, integrl jobs, temporry jobs). Their setting, however, is restricted to m prllel rcs nd ll relesed jobs hve to be ssigned to exctly one mchine rc). In the context of online routing, Awerbuch et l. [5] present online routing lgorithms tht mximize throughput under the ssumption tht routings re irrevocble. They present competitive bounds tht depend on the number of nodes in the network Our results The contributions of this pper cn be summrized s follows. First, we study vrint, where demnds cn be split. For this setting, we investigte the two online lgorithms Seq nd Seq 2 by mens of competitive nlysis. We show tht every upper bound of Seq crries over to Seq 2, nd every lower bound of Seq 2 crries over to Seq. Since Hrks et l. [17, 18] showed tht for generl price functions no online lgorithm cn be competitive, we consider clsses of restricted price functions. Our min result sttes tht for polynomil price functions with nonnegtive coefficients nd degree d, the competitive rtio of both lgorithms is bounded by d + 1) d+1. In prticulr, we prove n upper bound of 4 for generl continuous, nondecresing, nd concve price functions. Using construction presented by Frzd et l. [12], it cn be shown tht the upper bounds for our lgorithms re tight. We lso present lower bounds of.265 d + 1)) d+1 for ny online lgorithm, mtching our upper bound up to constnt fctor. If we restrict the input grph to two nodes connected by prllel rcs, we stte lower bound of ) for Seq nd for Seq 2, even for ffine liner price functions.
6 ONLINE ROUTING WITH TIME WINDOWS 5 Furthermore, we consider form of resource ugmenttion, in which Seq hs less demnd to route thus wekening the power of the offline dversry. When the price functions re polynomils nd the dversry hs to route demnds which re incresed by fctor γ 1, we present n upper bound on the competitive rtio for Seq nd for Seq 2 of d+1)d+1 d+1) γ d. Second, we study unsplittble routings, where demnds must be routed on single pth. For this setting, we nlyze the competitiveness of the online lgorithms USeq nd USeq 2. We gin show tht every upper bound of USeq crries over to USeq 2 nd every lower bound of USeq 2 crries over to USeq. Our min result sttes tht for polynomil price functions with nonnegtive coefficients nd degree d, the competitive rtio of both lgorithms is bounded by O1.77 d d d+1 ). In prticulr, we prove n upper bound of 1 + 2) 2 for ffine liner price functions nd n upper bound of for continuous, nondecresing, nd concve price functions. Bsed on reduction from online lod blncing problems in the context of scheduling prllel relted mchines see Awerbuch et l. [4], Suri, Toth, nd Zhou [25] nd Crginnis et l. [8]), we prove lower bound on the competitive rtio of.537 d + 1)) d+1 for ny online lgorithm. Finlly, we study online lod blncing problem in generl networks. Here, the gol is to route commodities online so s to minimize the l p norm of the rc lods. We show tht this problem is specil cse of our model nd investigte the online lgorithms Seq nd USeq for the splittble nd unsplittble vrint in generl networks. Our min results in this setting re upper bounds on the competitive rtio of nd Op) for Seq nd USeq, respectively. To the best of our knowledge, such bounds were previously known only in the context of scheduling problems, see Awerbuch et l. [4], Crginnis et l. [8], nd Suri, Toth, nd Zhou [25]. An overview of some lower nd upper bounds discussed in this pper is presented in Tble Problem Description We consider the following multicommodity routing problem. Given is directed network D = V, A) nd nondecresing nd continuous price functions : R + R + for ech rc A. Furthermore, set K = {1,..., n} of commodities is given. Ech commodity j K is specified by time window [ τ j, T j ), where τ j is the relese time nd T j is the expiring time, nd bndwidth request s j, t j, d j ). Here s j, t j V re the source nd destintion of the request, respectively, nd d j > corresponds to the required bndwidth or cpcity tht needs to be reserved on pths from s j to t j in the grph D during the time window [ τ j, T j ). We ssume tht τ j T j for ll j K. Let the excess flow in vertex v for commodity j K be denoted by γ j) v = d j if v = s j, d j if v = t j, otherwise. 1)
7 6 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD Then, for commodity j K, fesible flow, or routing decision, is nonnegtive vector f j) R A +, stisfying, f j) f j) = γ v j), δ + v) δ v) where δ + v) nd δ v) re the rcs leving nd entering v, respectively. Note tht, we ssume tht the demnd cn be split rbitrrily, tht is, ech infinitesiml mount of the demnd cn be routed over its own pth from s j to t j in the network. The routing of the demnd for commodity j stys fixed from the relese time τ j until the expiring time T j. When routing n infinitesiml mount of demnd on n rc A, which hs totl reserved bndwidth, or lod, l, this mount of bndwidth incurs mrginl cost of l ) per time unit for this rc. The totl cost on rc is given by the integrl of over the totl flow using. As the bndwidth for commodity only needs to be reserved during the corresponding time window, the lod on n rc vries over time nd thus lso the costs per rc. Given fesible flow f = f 1),..., f n) ), the cost ssocited t time t with rc is Pj Lt) f j) z) dz, where Lt) = {j K : τ j t < T j } is the set of commodities live t time t. The totl cost of the fesible flow is defined s Cf) = A Pj Lt) f j) z) dz dt. This expression of the totl cost cn be chieved by single pth routing for rbitrrily smll demnds, see Hrks et l. [18] for the cse without time windows. We cll the problem of finding fesible flow f tht minimizes Cf) the Multicommodity Routing problem with Time Windows MRTW). Note tht we do not hve ny cpcity restrictions for the rcs Problem Reformultion Since n online lgorithm hs to route ll requests with the sme relese time immeditely, one cn view the commodities s prtitioned into rounds. This viewpoint yields n equivlent representtion of the cost function see Lemm 2.1 below) tht we develop in the following nd is needed for the description of the online lgorithms tht we study in this pper. We define the number of different relese times s R := { τ 1,..., τ n } nd write {τ 1,..., τ R } = { τ 1,..., τ n } with τ 1 < τ 2 < < τ R. The set of commodities for ech round r R := {1,..., R} is denoted by K r := {j K : τ j = τ r } with n r := K r. The mximum expiring dte of round r is denoted by T r,mx := mx{t j : j K r }. Every commodity belongs to exctly one round, tht is, K 1 K R = K, nd for two rounds r nd r r r ) we hve K r K r =.
8 ONLINE ROUTING WITH TIME WINDOWS 7 For round r R nd time point t, let J r t) = {j K r : τ r t < T j } be the set of commodities belonging to round r tht re still live t time t, nd let I r t) = {j J 1 t) J r 1 t) : τ r t < T j } be the set of commodities tht hve been relesed before round r nd re still live t time t. Note tht for t < τ r we hve I r t) = J r t) =. The set Lt) of commodities live t time t cn be rewritten s Lt) = J 1 t) J R t). For nottionl convenience we define K r t) = I r t) J r t) s the set of commodities of rounds 1,..., r tht re still live t time t. Consider round r R nd fesible flow f. The totl cpcity used on rc t time t [τ r, T r,mx ) due to the requests relesed before round r is denoted by F,r f, t) = j I rt) f j), nd by G,r f, t) = j J rt) f j) we denote the mount of flow on rc A t time t [τ r, T r,mx ) due to the requests of this round. To ech round r R we ssocite costs of fesible routing decision f. We denote by f r = f j) : j K r ) the flow vectors of round r, nd we write f r = f j) : j K r ) for the vector of ll flow vlues on rc for round r. The cost C f r r, t; f 1,..., f r 1 ) ssocited to round r t time t on rc A is given by Pj Jrt) f j) j I rt) f j) ) + z dz = G,rf,t) F,r f, t) + z ) dz. The totl cost ssocited to round r is obtined by integrting the cost on rc over the time intervl corresponding to r nd summing this vlue for ll rcs: C r f r ; f 1,..., f r 1 ) := A T r,mx Cr f r, t; f 1,..., f r 1 ) dt. τ r The reltion between the cost ssocited to the rounds 1,..., R nd the totl cost of the full routing decision is given in the following lemm. Lemm 2.1. Let f be fesible flow. Then the cost of this fesible flow is equl to the sum of the costs ssocited to ech round, i.e., Cf) = R C r f r ; f 1,..., f r 1 ). r=1
9 8 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD Proof. Consider fesible routing decision f. Then the corresponding cost of this flow is Cf) = A = A P f j) j Lt) R r=1 z) dz dt = A rp P f j) i=1 j J i t) r 1 P P i=1 j J i t) f j) RP P f j) r=1 j Jrt) z) dz dt. z) dz dt For t [τ r, T r,mx ), we hve J r t) =. Hence, the lower nd upper bounds of the inner integrl re equl, implying tht it evlutes to. Therefore, for round r, we my restrict the time integrl to [τ r, T r,mx ). Furthermore, using tht I r t) = J 1 t)... J r 1 t), we cn write = A = = R r=1 R r=1 A T r,mx τ r T r,mx τ r P j Krt) P j Irt) f j) f j) z) dz dt F,rf,t)+G,rf,t) F,rf,t) R C r f r ; f 1,..., f r 1 ), r=1 which completes the proof. z) dz dt Observe tht we cn prtition the time horizon into intervls on which the flows re constnt: Consider the set of ll events E := { τ j, T j : j K}, nd let ˆτ 1 < < ˆτ E be the sorted sequence of ll events, i.e., E = {ˆτ 1,..., ˆτ E }. Then, the sets I r t), J r t), nd the functions F,r f, t) nd G,r f, t) re constnt on the segments [ˆτ i, ˆτ i+1 ) i = 1,..., E 1). Hence, E 1 C r f r ; f 1,..., f r 1 ) = ˆτ i+1 ˆτ i ) C f r r, ˆτ i ; f 1,..., f r 1 ). 2) A i=1 This reltion llows to replce the time integrl by sum nd will be used severl times in this pper. Remrk 2.2. In the following we consider slightly extended version of the MRTW model in which the rounds re further refined, i.e., we llow for R { τ 1,..., τ n } rounds with corresponding relese times τ 1,..., τ R tht re wekly) sorted s τ 1 τ 2 τ R, i.e., rounds with equl relese time re now possible. Here, it is understood tht the order in which the rounds rrive, is given by their numbering 1,..., R. Hence, only the online
10 ONLINE ROUTING WITH TIME WINDOWS 9 version of MRTW is ffected. In fct, the online lgorithms discussed in this pper nturlly work in this generlized model. This more generl model cn be pproximted by the originl model within rbitrry precision in the following sense. We use smll ε > nd for ll rounds 1 r,..., r + k R with τ r 1 < τ r = = τ r+k < τ r+k+1 we use modified relese times τ r = τ r, τ r+1 = τ r +ε,..., τ r+k = τ r +k 1) ε for convenience, we set τ = nd τ R+1 = ). The usge of ε gurntees tht the rounds re routed sequentilly in the order 1,..., R. Then for ny fesible flow f the cost of f in the pproximted model tends to the cost of f in the extended model for ε tending to note tht f is fesible for both models). Remrk 2.3. MRTW s just described generlizes the work of Hrks et l. [17, 18] in two wys. First, in ech round more thn one commodity is llowed to be routed. Second, the commodities now come with given lifetime. More precisely, the setting of [17, 18] is the specil cse with n r = 1, τ r = for ll rounds r R, nd T j = 1 for ll j K. In this cse, 2) evlutes to C r f r ) = C A r f r, ), nd hence the integrl over time cn be skipped. 3. Online Algorithms We study two online lgorithms, which we cll Seq nd Seq 2. For commodities in the sme round, Seq greedily computes routing decision, minimizing the cost for these demnds. More formlly, for ech round r R nd given the flows f 1,..., f r 1, fixed in the previous rounds, Seq solves the following mthemticl progrm min C r f r ; f 1,..., f r 1 ) s.t. f j) δ + v) δ v) f j) = γ j) v, v V, j K r 3) f j) A, j K r, to obtin the routing decision f r = f j) : j K r ). Here γ v j) is defined s in 1). Note tht in this definition of Seq, the rounds do not hve to rrive in chronologicl order, nor need the commodities of one round hve to hve the sme relese time. For the online problem, however, we of course need these conditions. The second lgorithm Seq 2 behves in similr wy s Seq. Wheres Seq routes ll commodities in round t the sme time, Seq 2 rbitrrily orders the commodities of round nd routes them sequentilly. Formlly, Seq 2 constructs new instnce of MRTW with n rounds, ech consisting of single commodity, nd then pplies Seq on the new instnce. The order of the commodities cn be obtined s in Remrk 2.2. Note tht Seq 2 cn construct the new rounds in n online mnner. Remrk 3.1. Due to the fct tht the functions ) A) re nonnegtive nd nondecresing, the cost functions C r ) re convex. As the
11 1 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD flowconserving constrints in 3) re liner constrints, the mthemticl progrm 3) is convex progrm nd cn be solved within rbitrry precision in polynomil time, see e.g., Grötschel et l. [15] Chrcteriztion of Seq nd Seq 2 Since the cost function is convex, we know tht ny locl minimum is lso globl minimum nd vice vers. To show tht certin flow is optiml for C r, we need to compute the grdient. Strightforwrd clcultions show tht for r R, j K r, nd A: C r f r ; f 1,..., f r 1 ) = f j) Tj τ r F,r f, t) + G,r f, t) ) dt. Here nd in the following, we use the vector f of ll flows on the right hnd side, lthough the expression does not depend on the flows of rounds lrger thn r. The necessry nd sufficient conditions for the flow of round r to minimize the cost C r, given the routing decisions of the previous rounds re s in the following lemm. Lemm 3.2 [11]). Let f 1,..., f r 1 be fesible flows for the commodities of the first r 1 rounds. Then fesible flow f r = f j) : j K r ) for round r R solves the convex progrm 3) if nd only if for ech commodity j K r nd ny fesible flow x j) for this commodity, the following inequlity holds [ T j F,r f, t) + G,r f, t) ) ] f j) dt x j) ). 4) A τ r Inequlity 4) is clled vritionl inequlity nd certifies tht the fesible flow f j) is locl optimum, which is lso globl optimum, becuse C r is convex. An optiml offline flow for MRTW is n optiml solution f of the following convex optimiztion problem: min s.t. Cf) δ + v) f j) δ v) f j) = γ j) v, v V, j K r, r R f j) A, j K r, r R. As in Remrk 3.1, n optiml offline solution cn be computed in polynomil time within rbitrry precision. To shorten nottion we use the following convention throughout the pper: When we spek of sequence σ = 1,..., R of rounds, we refer to the full specifiction d j, s j, t j, τ j, T j ) of the commodities j K r for r = 1,..., R. For sequence σ, we denote by Optσ) the vlue of n optiml offline flow, nd for n online lgorithm Alg we denote by Algσ) the cost Cf) of the solution f produced by Alg for sequence σ.
12 ONLINE ROUTING WITH TIME WINDOWS Figure 1: Grph construction for the proof of Propositions Comprison of Online Algorithms Seq nd Seq 2 Before we investigte the competitiveness of Seq nd Seq 2, we compre their reltive strengths. Intuitively, lgorithm Seq 2 is disdvntged compred to Seq, since it does not use ll vilble informtion of given round r when determining the routing. If there is only one round, Seq computes n optiml solution, nd obviously we hve tht Seqσ) Seq 2 σ) in this cse. This rises the question whether Seq lwys outperforms Seq 2? To study this question, we define the concept of domintion: We cll n lgorithm Alg 2 dominted by nother lgorithm Alg 1, if we hve Alg 1 σ) Alg 2 σ) for ny sequence σ. With this nottion, we cn nswer the bove question to the negtive: Proposition 3.3. Seq 2 is not dominted by Seq. Proof. Consider the network in Figure 1. We ssume tht the two rcs leving node 1 hve price function z nd the other two rcs hve price function. We consider two rounds. The first hs two commodities, nd the second contins one commodity. All commodities hve the sme time window of [, 1). The first commodity of the first round hs demnd of 1 tht hs to be routed from node 1 to node 4. There re two possible pths for routing this demnd. Since the cost on both pths re equl, Seq 2 splits the demnd eqully. The resulting cost is [ 1 z dz dt = z2] 2 = 1 4. The second commodity hs demnd of 2, which hs to be routed from node 1 to node 3, leving no routing choice. The cost of Seq 2 for this demnd is z) dz dt = [ 1 2 z z2] 2 = 3. Finlly, the demnd of the second round hs size of 4 nd must be routed from node 1 to node 2, gin leving no routing choice. Here, the cost of Seq 2 is ) 4 = 1. Hence, the totl cost for Seq2 is given by Seq 2 σ) = = 53 4 ; the cost for round 1 is In contrst, Seq coordintes the routings of the first round in order to minimize the cost. An optiml routing for the two commodities of the first round is to route the first entirely long the lower pth 1, 2, 4) nd the second on rc 1, 3). This incurs cost of = 5 2, which is indeed
13 12 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD smller thn the cost of Seq 2 for this round. For the second round there is no routing choice, nd the cost of Seq for this round is ) 4 = 12. Therefore, the totl cost of Seq is = 29 2, which is lrger thn the cost of 53 4 obtined by Seq2. The close reltionship between Seq nd Seq 2 is expressed by the following proposition, which is immedite from the definition of Seq 2. Proposition 3.4. For every instnce σ, there exists sequence σ such tht Corollry 3.5. Seqσ ) = Seq 2 σ) nd Optσ ) = Optσ). 1) Ech lower bound on the competitive rtio for Seq 2 is lso lower bound on the competitive rtio for Seq. 2) Ech upper bound on the competitive rtio for Seq is lso n upper bound on the competitive rtio for Seq 2. Concluding, Seq nd Seq 2 do not dominte ech other, nd if Seq is ccompetitive then Seq 2 is ccompetitive s well. See lso the discussion in the conclusions. 4. Competitive Anlysis In Hrks et l. [17, 18] it is shown tht there exists no competitive deterministic lgorithm if the price functions re rbitrry; the used price functions re monomils with degree tending towrds infinity. However, when degrees of the polynomils re bounded, we we derive in the bounded upper bounds on the competitive rtio A Generl Upper Bound In this section, we derive quite generl upper bound on the competitive rtio of the online lgorithms Seq nd Seq 2. In the next section we pply this result to polynomils with given fixed degree. The generl bound is obtined by using the vritionl inequlity in Lemm 3.2 nd bounding rcwise the cost of flow produced by Seq with respect to the optiml flow. Then the mximum of these bounds over ll rcs is bound on the competitive rtio. Similr techniques hve previously been pplied to bounding the price of nrchy in selfish routing problems, see Roughgrden [23] nd Corre, Schulz, nd StierMoses [1]. The bounds on rcs A depend on their price functions ). We will use the following vlues for ech A nd λ 1: x) x x δ ) := sup x p if x z) dz z) dz > 5) otherwise. p f) λ x) ) x if p ω ; λ) := sup f) f > p x,f f) f 6) if f) f =.
14 ONLINE ROUTING WITH TIME WINDOWS 13 f) λ x) x) ) x f Figure 2: Illustrtion of the vlue ω; λ) in 6) with 1 < λ < pf) x). The gryshded re corresponds to the numertor of ω; λ) nd the whole re to the denomintor. Figure 2 gives n illustrtion of ω ; λ). For clss C of nondecresing nd continuous price functions, we further define δc) := sup δ ) nd ωc; λ) := sup ω ; λ). C C The generlized vlue ωc; λ) ws first introduced by Hrks [16] for investigting selfish routing problems involving tomic plyers. In the sme context, similr vlue βc), with βc) = ωc; 1), ws defined in Corre, Schulz, nd StierMoses [1]. Furthermore, Roughgrden [23] presented the soclled nrchy vlue αc), with αc) = 1 ωc; 1) ) 1. We define the fesible scling set for C s ΛC) := {λ 1 : 1 δc) ωc; λ) > }. Theorem 4.1. If ΛC), then the online lgorithms Seq nd Seq 2 re ccompetitive, for [ ] λ δc) c = inf. λ ΛC) 1 δc) ωc; λ) Proof. Let f be the flow constructed by Seq nd x be n rbitrry fesible flow. We wnt to bound the cost Cf) with respect to Cx). To this end, we obtin the following sequence of inequlities. We strt by pplying Lemm 2.1. Cf) = R r=1 A T r,mx τ r G,rf,t) F,r f, t) + z ) dz dt. Becuse the price functions re nondecresing nd nonnegtive, we hve: = R r=1 A R T r,mx τ r E 1 r=1 A i=1 F,r f, t) + G,r f, t) ) G,r f, t) dt ˆτi+1 ˆτ i F,r f, t) + G,r f, t) ) j J rt) ) f j) dt,
15 14 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD where we used decomposition similr to 2) nd the definition of G,r f, t). Becuse J r t) is constnt on ech piece, we cn reorder terms: = R E 1 r=1 A i=1 j J rˆτ i ) ˆτi+1 f j) F,r f, t) + G,r f, t) ) dt. ˆτ i We now use the definition of J r t) nd undo the decomposition into time segments, to get: = R r=1 A f j) j K r Tj τ r F,r f, t) + G,r f, t) ) dt. Using Lemm 3.2 yields: R r=1 A x j) j K r Tj τ r F,r f, t) + G,r f, t) ) dt. Arguing in the reverse wy s bove by using the definition of J r t)), we cn move x j) into the integrl: = R r=1 A T r,mx τ r F,r f, t) + G,r f, t) ) j J rt) ) x j) dt. We now use the definitions of F,r f, t) nd G,r f, t) nd extend the bounds of the integrl we use tht ech is nonnegtive): R r=1 A l K rt) f l ) j J rt) ) x j) dt. As J r t), K r t) Lt) nd ) is nonnegtive nd nondecresing, we cn bound this by A j Lt) For λ 1, we now dd nd obtin: = λ A + A j Lt) [ ) ) f j) x j) dt. j Lt) j Lt) ) x j) f j) j Lt) ) x j) dt ) λ j Lt) For fixed A nd t we use f = f j) nd x = j Lt) )] ) x j) x j) dt. j Lt) j Lt) x j)
16 ONLINE ROUTING WITH TIME WINDOWS 15 in the definition of δ ) nd ω ; λ) to get: λ δ ) A + A ω ; λ) Pj Lt) xj) j Lt) z) dz dt ) ) f j) f j) dt. j Lt) The bound involving ω ; λ) holds becuse of the following: If f) = then trivilly f) λ x)) x nd the bound is true. If f =, then f) λ x)) since λ 1 nd ) is nondecresing. The cse f) f > follows from the definition. Using the definition of δc) nd pplying δ ) to the second term yields: λ δc) Cx) + A ω ; λ) δ ) λ δc) Cx) + δc) ωc; λ) Cf). Pj Lt) f j) z) dz dt Applying this inequlity to the optiml offline solution x, rewriting, nd tking the infimum over λ ΛC) yields the desired result for Seq. The result for Seq 2 follows from Corollry 3.5 2) Polynomil Price Functions To fcilitte the result of Theorem 4.1, one needs to bound δc) nd ωc; λ). In this section we consider the clss C d of polynomils with nonnegtive coefficients nd degree t most d N: C d := {c d x d + + c 1 x + c : c s, s =,..., d}. Note tht polynomils in C d re nonnegtive for nonnegtive rguments, continuous, nondecresing, nd convex. Theorem 4.2. For price functions in C d, the online lgorithms Seq nd Seq 2 re d + 1) d+1 competitive. Before we prove Theorem 4.2, we first derive bound on the vlue δc d ) nd ωc d ; λ). Lemm 4.3. For the clss C d of polynomils with mximl degree d, the vlue δc d ) is t most d + 1. Proof. Let x) = c d x d + + c 1 x + c C d. We pply the definition of the vlue δ ): δ ) = sup x x d c i x i i= sup d c i i+1 xi+1 x i= d c i x i+1 i= d i= = c i d+1 xi+1 where the inequlity follows since c i nd x. 1 1 d+1 = d + 1,
17 16 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD + 1 z + 2 z 2 1 z 2 z 2 s t s v w t Figure 3: Reduction of polynomils to monomils. By introducing the two nodes v nd w, the rc s, t) is prtitioned into three seprte rcs with monomil price functions. We now observe tht the cost function Cf) is liner in ech of the price functions ). We cn therefore reduce the nlysis to monomil price functions. For this we subdivide ech rc into d rcs 1,..., d with monomil price functions s x) = c s x s for ny s = 1,..., d, see Figure 3. The following lemm cn lso be found in Hrks [16]. We present proof for completeness. Lemm 4.4. For the clss C of monomils c s x s of degree s {1,..., d} with c s nd for λ 1, we hve ωc; λ) mx µ λ µ d+1 ). µ Proof. For ) C, we cn ssume tht f) f >, since otherwise ωc; λ) = nd the clim is trivilly true. By definition, we hve p f) λ x) ) x ω ; λ) = sup. x,f f) f Defining µ := x f recll tht f > ), we obtin p f) λ µ f) ) µ f ω ; λ) = sup. µ f) f Consider the monomil price function x) = c s x s of degree s {1,..., d}. To bound the vlue ω ; λ) from bove, we hve to compute: c s f s λ c s µ s f s ) µ f ω ; λ) = sup µ c s f s+1 = mx µ λ µ s+1 ). 7) µ Becuse of the ssumption λ 1, the mximum is ttined t point with µ 1. It follows tht mx µ λ µ s+1 ) mx µ λ µ d+1 ). µ µ This shows the clim. For polynomils in C d nd n pproprite choice of λ, we cn prove the following bound on ωc d ; λ). Proposition 4.5. For λ := d + 1) d 1) 1, we hve ωc d ; λ) Proof. Using Lemm 4.4 we get ωc d ; λ) mx µ µ λ µ d+1 ) = mx µ µ d + 1) d 1) µ d+1). d d+1) 2. The unique solution turns out to be µ = 1 d+1. Evluting the objective leds to: ωc d ; λ) 1 d + 1 d + 1 ) d+1 d 1)d 1) = d + 1 d + 1) 2.
18 ONLINE ROUTING WITH TIME WINDOWS 17 This proves the clim. With these prerequisites we cn prove Theorem 4.2. Proof of Theorem 4.2. Let the flow f be produced by the online lgorithm Seq nd let x be n rbitrry fesible flow for the given instnce. We define λ := d + 1) d 1) nd pply Proposition 4.5, which yields ωc d ; λ) d. d+1) 2 In order to pply Theorem 4.1, we hve to verify tht λ ΛC d ). Wht remins to be shown is tht d 1 d + 1) d + 1) 2 > holds, where δc d ) d + 1 by Lemm 4.3. This inequlity is equivlent to >, which is trivilly true. Applying Theorem 4.1 yields 1 d+1 Cf) d + 1)d 1) d + 1) 1 d + 1) d d+1) 2 ) Cx) = d + 1) d+1 Cx). Tking x s the optiml offline solution proves the clim. By Corollry 3.5 2) the result for Seq 2 follows s well. Corollry 4.6. For ffine liner price functions ) with nonnegtive coefficients, i.e., ) C 1, Seq nd Seq 2 re 4competitive. For ffine liner price functions, Hrks et l. [18] proved tht Seq is 4n 2 )/1 + n) 2 competitive when there re no time windows nd there is only one commodity per round. The nlysis, however, is tilored to ffine liner price functions nd does not provide bounds for higher degree polynomils. We do not know whether this result cn be generlized to improve the bound of Corollry 4.6 mde commodity dependent). Corollry 4.7. For concve price functions, the competitive rtio of Seq nd Seq 2 is bounded by 4. Proof. The sttement follows from geometricl rgument similr to Corre et l. [9], see Figure 2 for n illustrtion. The worst cse rtio between the two res in the figure occurs exctly in the cse in which the price function is liner Resource Augmenttion As noted bove, in generl, no online lgorithm is competitive if the set of llowble price functions is not restricted. This my be sign tht the dversry offline optimum) is too powerful. To limit the power of the dversry, we cn give her the disdvntge tht she hs to route demnd tht is incresed by fctor γ 1. This cn be viewed s wy of resource ugmenttion, see e.g., Klynsundrm nd Pruhs [2] or Roughgrden nd Trdos [23]. Before we stte n upper bound in this context, we generlize the definition of ΛC) s follows. ΛC; γ) := {λ 1 : γ δc) ωc; λ) > }.
19 18 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD Theorem 4.8. Let the price functions on the rcs be from clss of nondecresing nd continuous functions C, for which ΛC; γ), nd suppose tht the dversry Adv needs to route demnd incresed by fctor γ 1. Then for ny request sequence σ, we hve tht Seqσ) c Advσ), where c = inf λ ΛC;γ) [ λ δc) γ δc) ωc; λ) Proof. Let x be the optiml flow with respect to γ. Using Lemm 3.2 we obtin for commodity j: [ T j A τ r ]. F,r f, t) + G,r f, t) ) ] f j) dt xj) ), γ since xj) γ is fesible flow for commodity j. Following the proof of Theorem 4.1, we get: Rewriting proves the clim. γ Cf) λ δc) Cx s tr) + δc) ωc; λ) Cf). If we restrict the set of llowble price functions to polynomils in C d, we cn show the following: Corollry 4.9. Let the price functions on the rcs be in C d, nd suppose tht the dversry Adv needs to route fctor γ 1 s much demnd per request thn the online lgorithm Seq. Then, for ny request sequence σ, we hve tht d + 1)d+1 Seqσ) d + 1) γ d Advσ). Proof. Defining λ := d + 1) d 1 nd pplying Lemm 4.5, Lemm 4.3, nd Theorem 4.8 we get: Rewriting proves the clim. γ Cf) d + 1) d Cx) + d d + 1 Cf). Note tht for γ = d+1) d + d d+1 we hve Seqσ) Advσ). In prticulr, this implies tht for ffine liner price functions, the cost of Seq for given sequence σ does not exceed the cost of Adv with respect to γ = 2.5. Using Corollry 3.5 2), the sttement of Theorem 4.8 nd Corollry 4.9 hold for Seq 2 s well Reltion to the Lod Blncing Problem In the following, we will obtin lower bounds for the online MRTW by trnsfering lower bounds for the online lod blncing problem OLB). In this scheduling problem, jobs rrive in n online fshion nd hve to be scheduled on mchines in order to minimize the l d Norm of the vector of the server lods. Ech job cn only be scheduled on given subset of permissible mchines. It turns out tht our setting generlizes the specil cse of relted prllel) mchines: ech job induces work of w j when it is scheduled on ny permissble mchine. This problem hs received much ttention, see
20 ONLINE ROUTING WITH TIME WINDOWS 19 jobs perm. mchines 1 1, , 4 s 1 s 2 s 3 u 1 u 2 u 3 t u 4 Figure 4: Exmple of the grph construction of Section 4.4. Awerbuch et l. [4], Crginnis et l. [8], nd Suri et l. [25]. In the following we will describe the reltion between this problem nd our setting. Given re J independent jobs, ech of which needs to be scheduled on one of M prllel mchines. The jobs re relesed in n online mnner, such tht job j rrives before job j+1, for j = 1,..., J 1. Ech job j cn be scheduled on subset of permissible mchines on which it induces work w j. Tht is, ech job j induces work of w ij {w j, } on mchine i {1,..., M}. Ech mchine i hs speed q i nd the lod l i of mchine i is the sum of weights w j of jobs ssigned to it divided by its speed q i. A schedule is determined by mtrix S = S ij ) with S ij [, w j ], for i {1,..., M}, j {1,..., J}. A schedule is fesible if M S ij = w j, i=1 nd S ij = if w ij =. For the unsplittble vrint it is required tht S ij {, w ij }. The cost of schedule S is defined s LS) := M ) 1/d M 1 = l d i i=1 i=1 q i J ) ) d 1/d, S ij j=1 for d N, which is the the l d norm of the server lods. We cll the tuple σ = M, q, J, w, d) n instnce of OLB. Suppose we re given n instnce σ for OLB. Then we construct n equivlent instnce Gσ) consisting of directed grph D with commodities nd price functions) for MRTW. The grph D contins nodes s j for ech job j. Furthermore, for ech mchine i it contins nodes u i nd one dditionl node t. There re rcs u i, t) for ll mchines i. For ech job j, the set of permissible mchines define set of rcs A j := {s j, u i ) : w ij }. We define A := A 1 A J. See Figure 4 for n exmple. For every rc = u i, t) the price function is x) = d x/q i ) d 1. All remining rcs hve constnt price functions. We define sequence of rounds σ = 1,..., J, where ech round j contins one commodity with demnd w j tht hs to be routed from s j to t. We do not consider time windows, i.e., we set τ j =, T j = 1, see Remrks 2.2 nd 2.3. This completes the specifiction of the instnce Gσ).
21 2 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD By flow conservtion, fesible flow f for Gσ) is completely determined by the flow vlues f, j = s j, u i ) A. We let { f j for = s j, u i ) A S ij f) := otherwise be the schedule Sf) induced by flow f. Conversely, we sy tht flow fs) is induced by the schedule S if f j S) := S ij, for s j, u i ) A, nd extended by flow conservtion to the remining rcs. Note tht this trnsformtion works in n online fshion since it cn be done commodity job) wise. We obtin the following strightforwrd observtion. Lemm 4.1. Let σ be n instnce for OLB. Then, every fesible schedule S for σ induces fesible flow fs) for Gσ), nd every fesible flow f for Gσ) induces fesible schedule Sf) for σ. The reltion of the corresponding objective function vlues is s follows. Lemm Let σ be n instnce for OLB. Then, every fesible schedule S for σ nd fesible flow f for Gσ) stisfy LS) = CfS)) 1/d nd Cf) = LSf)) d. Proof. Let S be ny fesible schedule for σ with cost LS). flow fs) evlutes s follows: CfS)) = = = = M P J j=1 f j u i,t) i=1 M P J j=1 f j s j,u i ) i=1 M P J j=1 S ij i=1 M 1 i=1 q i j=1 The reverse direction follows similrly. d z/q i ) d 1 dz d z/q i ) d 1 dz d z/q i ) d 1 dz J ) d S ij = LS) d. The cost of Corollry Let σ be n instnce for OLB. Then, every optiml solution S for σ induces n optiml solution fs ) for Gσ). Conversely, every optiml solution f for Gσ) induces n optiml solution Sf S)) for σ. Proof. Let S be n optiml schedule for σ. By Lemm 4.11, we know tht S induces fesible flow fs ) with LS ) = CfS )) 1/d. Assume fs ) is not optiml for Gσ). Then, there exists fesible flow x with Cx) < CfS )). Then, x induces fesible schedule Sx) with cost LSx)) = Cx) 1/d < CfS )) 1/d = LS ), contrdicting the optimlity of S. The reverse direction follows similrly. For polynomil price functions, we obtin:
22 ONLINE ROUTING WITH TIME WINDOWS 21 Lemm If Alg is ccompetitive deterministic online lgorithm for the online MRTW with price functions in C d 1, then there exist c 1/d  competitive deterministic online lgorithm Alg for OLB. Proof. Suppose Alg is ccompetitive deterministic online lgorithm for MRTW with price functions in C d 1. Then, for ny given instnce σ of OLB we construct the instnce Gσ), which hs price functions in C d 1. Let f be the flow computed by Alg on Gσ). Algorithm Alg is the deterministic online lgorithm tht returns the induced schedule Sf) for σ. By Lemm 4.11, it follows tht LSf)) = Cf) 1/d. Combining Corollry 4.12 nd Lemm 4.11 we know tht LS ) = Cf ) 1/d for optiml solutions S of σ nd f of Gσ). By ssumption we hve tht Cf) c Cf ). Thus, it follows tht LSf)) c 1/d LS ). Since the input instnce σ ws rbitrry, the clim is proven. Theorem A lower bound c for the competitive rtio of ny deterministic online lgorithm for OLB with l d norm crries over to lower bound of c d for the competitive rtio of ny deterministic online lgorithm for the online MRTW with price functions in C d 1. Proof. Let c be lower bound for the competitive rtio of ny deterministic online lgorithm for OLB. Assume there exist ĉcompetitive deterministic online lgorithm for the online MRTW with ĉ < c d. Then, Lemm 4.13 implies tht we cn construct n online lgorithm Alg, which is ĉ 1/d  competitive for OLB. Monotonicity of the dth root implies ĉ 1/d < c contrdicting the first sttement. We now generlize the bove setting in two wys. First, we cn esily include time windows. Second, we cn consider online lod blncing problems in the l d norm for generl networks. Hence, the cost of flow f generlized schedule) is [ ) d ] 1/d. Lf) := f j) dt A j Lt) Corollry For the online lod blncing problem in generl networks, time windows, nd the l d norm, Seq nd Seq 2 re dcompetitive. Furthermore, there exist n instnce σ with Seq 2 σ) d Optσ). Proof. Using Theorem 4.1 for price functions in C d 1, we get Lf) d = A P j Lt) f j) d z d 1 dz dt = Cf) d d Cx) = d d Lx) d. A lower bound cn be constructed bsed on construction in Frzd et l. [12], see Theorem 4.18 below Lower Bounds on the Competitive Rtio Before we give lower bounds on the competitive rtio of the online lgorithms Seq nd Seq 2, we present generl lower bound for ll online lgorithms. We gin consider the clss of nonnegtive nd nondecresing polynomil price functions C d for d N nd use n instnce of the OLB to
23 22 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD derive lower bound see Theorem 4.14). The construction of this bound is bsed on ides of Awerbuch et l. [4]. The work in [4], however, only dels with unsplittble jobs, wheres the following bound holds for the splittble cse. Theorem For price functions in C d, the competitive rtio of ny deterministic online lgorithm for the online MRTW is t lest.265 d+1)) d+1. Proof. Let Alg be deterministic online lgorithm for the OLB. We re given M = 2 k prllel mchines nd construct the input sequence σ of jobs depending on the decisions of the online lgorithm. The mchines will be either lbeled s ctive or inctive t ech step of the sequence. Initilly ll mchines re ctive. There re k rounds. In ech round, the ctive mchines re grouped into pirs. For ech pir of mchines i, i ), the sequence σ contins job j of weight w j = 1, whose permissible mchines re i nd i. Assume tht Alg produces schedule S by plcing α [, 1] units on mchine i nd 1 α on mchine i. The sequence is constructed in such wy tht for every job the mchine with the smller frction is lbeled inctive ties re broken rbitrrily). For bounding the cost of Alg from below we cn ssume α = 1 2. In ech new round only hlf of the mchines of the previous round re ctive. We construct schedule S by plcing job entirely on the mchine tht becomes inctive fter the corresponding round. The cost with respect to the l d norm of S is LS ) d = 2 k k = 2 k 1. The cost of the schedule S of Alg is given by k i ) d LS) d = 2 k i. 2 The rtio of the cost of S nd S is bounded from below by LS) d k LS ) d = i=1 i 2 )d 2 k i ) log2 e) d d 2 k.265 d) d. 1 2 e i=1 As Theorem 4.14 considers price functions in C d 1, we cn replce d by d+1, obtining the desired result. Remrk For d = 1, Hrks et l. [17, 18] proved lower bound of 4 3 for ny deterministic online lgorithm, which in this cse is better thn the bove bound. Theorem The online lgorithms Seq nd Seq 2 hve competitive rtio of t lest d + 1) d+1, even without time windows nd only one commodity per round. Proof. Frzd et l. [12] presented n exmple tht, when trnslted to the online MRTW without time windows nd only one commodity per round), provides the lower bound. The networks used in the proofs of the previous theorems hve the property tht the commodities hve different sources nd destintions. In Hrks et l. [17, 18], it is shown tht Seq or Seq 2 ) compute n optiml routing for
24 ONLINE ROUTING WITH TIME WINDOWS Figure 5: Grph construction for the proof of Proposition networks consisting of two nodes connected by prllel rcs, time windows of size [, 1], nd rbitrry nondecresing price functions. This rises the question whether Seq or Seq 2 ) returns n optiml solution for prllel rcs nd rbitrry time windows. The following theorem shows tht this is not true. Theorem If the input grph consists of two nodes connected by prllel rcs, Seq nd Seq 2 cnnot be better thn )competitive, even for ffine liner price functions. Proof. Consider the network depicted in Figure 5. Ech commodity hs node 1 s source nd node 2 s destintion. Further, rc 1 hs constnt price function of 1 z) = 1 nd 2 liner price function of 2 z) = z. The request sequence consists of two rounds. The first round contins one commodity with demnd 1 nd time window [, 1). The second round contins one commodity with demnd 1 nd time window [, T ), for T = The cost for routing the first commodity completely over rc 1 is 1 nd over rc 2 it is 1 2. Therefore, Seq2 ssigns the demnd of the first commodity completely to rc 2. Suppose we route n mount of α [, 1] of the demnd of the second commodity on rc 2 nd 1 α over rc 1. Then the totl cost for the second commodity is T 1 α 1 α T α dz dt+ 1+z) dz dt+ z dz dt = 1 2 T α2 T 1) α+t. 1 8) By definition, Seq nd Seq 2 ) computes fesible flow for the second commodity tht minimizes 8). Tht is, it sends flow of α = T 1 T on rc 2 nd the rest over rc 1. The totl cost for this solution is 1 T 1)2 T 1)2 + + T = T T 2 T T. On the other hnd, if we route the first commodity completely over 1 nd the second commodity fully over 2, we obtin solution with cost 1 1 dz dt + T 1 z dz dt = T. Hence, the competitive rtio cnnot be better thn T T T = 2 1 3, 2 for T = 1 + 3, which concludes the proof.
25 24 T. HARKS, S. HEINZ, M. E. PFETSCH, AND T. VREDEVELD The bove theorem shows tht the introduction of time windows hs n effect on the performnce of the online lgorithms Seq nd Seq Unsplittble Routings In this section, we study the unsplittble MRTW: the vrint of MRTW in which the demnd of ech commodity hs to be routed long single pth. For this vrint, it will turn out tht the obtined bounds re lrger thn those for the splittble version. Furthermore, wheres in the splittble version the solutions to the offline problem nd Seq cn be computed in polynomil time within rbitrry precision), this is not true for the corresponding unsplittble counterprts unless P = NP. In the following, we study the unsplittble versions of Seq nd Seq 2, which we cll USeq nd USeq 2, respectively. For ech round, USeq greedily computes n unsplittble routing, minimizing the cost for the corresponding demnds. More formlly, for ech round r R nd given unsplittble flows f 1,..., f r 1 of the previous rounds, USeq minimizes C r f r ; f 1,..., f r 1 ) to obtin the unsplittble flow f r. The online lgorithm USeq 2 routes the demnds of round one by one long the chepest pth with respect to ll previously routed demnds. Remrk 5.1. As shown in Hrks et l. [17], the offline problem of MRTW is NPhrd, even without time windows. It follows tht the problem to minimize C r f r ), which USeq hs to solve, is lso NPhrd. However, the online lgorithm USeq 2 hs polynomil running time, since it only hs to solve shortest pth problem for every commodity of given round. The sme rgumenttion of Corollry 3.5 pplies to USeq nd USeq 2 nd yields: Corollry ) Ech lower bound on the competitive rtio for USeq 2 is lso lower bound on the competitive rtio for USeq. 2) Ech upper bound on the competitive rtio for USeq is lso n upper bound on the competitive rtio for USeq 2. In the following lemm, we express the cost minimiztion of USeq ssocited with every round r, given the routing decisions of the previous rounds. Lemm 5.3. Let f 1,..., f r 1 be fesible flows for the commodities of the first r 1 rounds. If fesible flow f r = f j) : j K r ) for round r is computed by USeq, then the following inequlity holds for ny other fesible flow x r = x j) : j K r ) for round r. A A T r,mx τ r T r,mx τ r G,rf,t) G,rx,t) F,r f, t) + z ) dz dt 9) F,r f, t) + z ) dz dt. 1) Proof. The inequlity simply follows from the definition of USeq routing the demnds of round r with minimum cost.
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