Wavelength assignment in optical networks with fixed fiber capacity

Wavelength Assignment in Optical Networkswith Fixed Fiber CapacityMatthew Andrews and Lisa ZhangBell Laboratories,600Mountain Avenue,Murray Hill,NJ07974{andrews,ylz}@Abstract.We consider the problem of assigning wavelengths to de-mands in an optical network of m links.We assume that the route ofeach demand isfixed and the number of wavelengths available on afiberis some parameterµ.Our aim is to minimize the maximum ratio be-tween the number offibers deployed on a link e and the number offibersrequired on the same link e when wavelength assignment is allowed tobe fractional.Our main results are negative ones.We show that there is no constant-factor approximation algorithm unless NP⊆ZPP.No such negative re-sult is known if the routes are notfixed.In addition,unless all lan-guages in NP have randomized algorithms with expected running timeO(n polylog(n)),we show that there is no logγµapproximation for anyγ∈(0,1)and no logγm approximation for anyγ∈(0,0.5).Our anal-ysis is based on hardness results for the problem of approximating thechromatic number in a graph.On the positive side,we present algorithms with approximation ratiosO(log m+logµ),O(log D max+logµ)and O(D max)respectively.HereD max is the length of the longest path.We conclude by presenting two variants of the problem and discussingwhich of our results still apply.Keywords:Optical networking,wavelength assignment,fixed capacityfiber, inapproximability.1IntroductionWe consider the problem of achieving transparency in optical networks.A path is said to be routed transparently if it is assigned the same wavelength from its source to its destination.Transparency is desirable since wavelength conversion is expensive and defeats the advantage of all optical transmission.More formally,we consider an optical network consisting of vertices and optical links and a set of demands each of which needs to be routed from a source vertex to a destination vertex on a single wavelength.Each optical link has one or multiple parallelfibers deployed.The fundamental constraint is that for each wavelengthλ,eachfiber can carry at most one demand that is assigned wavelengthλ.A common problem is to minimize the number of wavelengthsrequired so that all demands can be routed assuming onefiber per link.However, in reality a more pertinent problem is that the number of wavelengths that each fiber can carry isfixed to some valueµ,i.e.the total number of wavelengths is fixed.(For example,[11]lists thefiber capacities from different vendors.)The problem now is to minimize the number offibers required.For most service providers,the cost of afiber on a link can be divided into two components.First,there is the cost of renting thefiber from a“dark-fiber”provider.Second,there is the cost of purchasing optical equipment to“light”the fiber.When networks are being designed,the exact form of these costs are of-ten not well known.For example,the dark-fiber providers may regularly update their rental rates and the cost of optical equipment may be subject to negotia-tion.Moreover,the service providers may have to rent from different dark-fiber providers in different parts of the country and each may have different pricing strategies.Therefore,over time thefiber cost may vary nonuniformly from link to link.Despite this,we do know that the number offibers we use on a link must be at least the total number of demands routed through the link divided by the number of wavelengths perfiber.One robust way to ensure our network cost is low regardless of the exact cost structure is to minimize the ratio between the number offibers actually used on the link and this lower bound.In this paper we assume that the path followed by each demand is already fixed.Wavelength assignment is therefore the only problem.In an alternative formulation,routing and wavelength assignment could be performed simultane-ously.However,in many practical situations arising in optical network design, routing is determined by some higher-level specifications(e.g.carriers may re-quire min-hop routing,see[10]).Hence,it is important to consider the wave-length assignment problem in isolation.We also remark that once a demand is assigned a wavelength,whichfiber on each link actually carries the demand is not an issue.This is because modern optical devices such as mesh optical add-drop multiplexers allow distinct wavelengths from differentfibers to be multiplexed into a newfiber.Fiber minimization withfixed routing is NP-hard on networks with general topology(by a simple reduction from graph coloring).In this paper we focus on upper and lower bounds for approximating the problem.1.1Problem definition and preliminariesWe now describe the basic version of our problem.We consider a network and a set of demands D where each demand i is routed on a given path P i.We require that each demand is assigned a wavelengthλfrom the set{0,1,...,µ−1}.For each link e,if at most r e demands passing through link e are assigned wavelength λfor eachλ,then the number offibers required on link e is r e.Ifℓe is the number of paths that pass through e,then f e=ℓe/µis clearly a lower bound on r e.There are a number of distinct ways to define the objective function.For the reasons mentioned earlier we focus on a variant in which our goal is to minimize the maximum ratio between the number offibers deployed on a linke and the corresponding lower boundf e.(We mention some other variants in Section5.)The problem may be formulated as an integer program.Let variable C i,λindicate whether or not demand i uses wavelengthλ.Our problem,which we call Min-Fiber,can be written as follows for binary C i,λ.min zsubject toi:e∈P i C i,λ≤z·f e∀e,λ(1)λC i,λ=1∀i(2)We note that the linear relaxation of the above IP always has an optimal solution z=1and C i,λ=1/µfor all demands i and wavelengthsλ.1.2Our Results–We begin in Section2by presenting a negative result.We show that unless NP⊆ZPP there is no polynomial-time constant-factor approximation algo-rithm for the Min-Fiber problem.ZPP is the class of languages that can be recognized using a randomized algorithm that always gives the correct answer and whose expected running time is polynomial in the size of the input.Our result is based on the hardness result for graph coloring of Feige and Kilian[9].–In Section3we further improve the lower bound.Unless all languages in NP have randomized algorithms with running time O(n polylog(n)),we show that there is no logγµ-approximation for anyγ∈(0,1)and no logγm-approximation for anyγ∈(0,0.5)where m is the number of links in the network.–In Section4we turn our attention to positive results.In Section4.1we show that using randomized rounding we can obtain a solution in which the number offibers required on each link e is at most2f e+6(log m+ logµ).(All logarithms are to the base e.)This gives us an O(log m+logµ) approximation algorithm.We note that this algorithm can be derandomized using the standard method of conditional expectations.In Section4.2we apply the path-length rounding scheme of[12]to createa solution in which the number offibers required on each link e is at mostf e+D max,where D max is the length of the longest path in the network.Thisgives us an O(D max)approximation algorithm which is an improvement over the randomized rounding method when the paths are short.In the full version of the paper[2]we apply a constructive version of the Lov´a sz Local Lemma to obtain a randomized algorithm with approximation ratio O(log D max+logµ)and polynomial expected running time.–In Section5we conclude by presenting two variants of the Min-Fiber prob-lem and indicating which of our results still apply.1.3Previous WorkFor the case in which the number of available wavelengths is notfixed,the problem of minimizing the number of wavelengths used has been much stud-ied,e.g.[1,3,4,21].Some papers focus on common special topologies such as rings[14,24]and trees[16,15,7].The work listed here is by no means complete.A good survey on the subject can be found in[13].Our problem offiber minimization with afixedfiber capacity has been in-troduced more recently.In[25,18]the authors prove that coloring demands on a line only requires the minimum number offibers per link,i.e.⌈f e⌉fibers on link e.This generalizes the well-known algorithm for coloring interval graphs. In addition,[18]shows that the problem becomes NP-hard once the network topology is more complicated.The authors provide2-approximation algorithms for rings and stars.Recent work on trees include[6,8]and the results in[6] imply a4-approximation.For a general network topology,[23]uses randomized rounding to obtain an approximation algorithm for the variant of the problem in which the aim is to minimize the total amount offiber deployed.2Basic Lower BoundIn this section we show that there is no constant factor approximation to the Min-Fiber problem unless NP⊆ZPP.Our construction is based on hardness of approximation results for graph coloring.For any graph G we useχ(G)to denote the chromatic number of G andα(G)to denote the size of the maximum independent set of G.Throughout this section we shall use the terms“color”and“wavelength”interchangeably.Feige and Kilian[9]construct a randomized reduction from3SAT to graph coloring with the following properties.Given a3CNF formulaϕand a constant ε,they randomly construct an n-node graph G(where n is polynomial in the size ofϕ)such that,–Ifϕis satisfiable then with probability1,G can be colored with nεcolors,i.e.χ(G)≤nε.–Ifϕis not satisfiable then with high probability the maximum independent set in G has at most nεnodes,i.e.α(G)≤nεwith high probability.Note that sinceα(G)·χ(G)≥n this immediately implies thatχ(G)≥n1−ε. Feige and Kilian use this reduction to show that there is no n1−εapproximation for graph coloring unless NP⊆ZPP.We shall use it to show that for any constantc there is no c-approximation for Min-Fiber unless NP⊆ZPP.2.1Constructing an instance of Min-FiberWe now demonstrate how to take a graph G and create an instance of Min-Fiber on a network N.For each node v in G we have a demand d v.The links in N consist of two sets E1and E2.All links in E1are non-adjacent,i.e.no2links in E 1have a vertex in common.The links in E 2are used to connect up the links in E 1.More precisely,for each clique Q in G with c +1nodes we create a link e Q in N and these links form the link set E 1.The demand d v passes through e Q for all v ∈Q .If demand d v has to pass through links e Q 0,...,e Q z −1then there also exists a link f v,j in E 2that connects the head of e Q j with the tail of e Q j +1.The full path of d v is e Q 0,f v,0,e Q 1,...,e Q z −2,f v,z −2,e Q z −1.We illustrate the construction of the network N from a graph G in Figure 1.The number of colors in our instance of Min-Fiber is µ=n ε.31267541d 2d 3d 4d d 2d 37d 1d d 5d 6Fig.1.An example of the construction for c =2.(Left)Graph G with 4cliques of size3.(Upper right)Demands and routes created from G .(Lower right)Network N ,solid lines represent links in E 1and dotted lines represent those in E 2.2.2Reduction from 3SAT to Min-FiberGiven a 3CNF formula ϕwe first choose a constant εsuch that ε<1c +1.We then construct a random n -node graph G according to the method of Feige andKilian [9]for this parameter ε.Finally,we convert the graph G into an instance of Min-Fiber on a network N according to the method of the previous section.Note that since c is a constant,the number of demands and links in N are both polynomial in n which is in turn polynomial in the size of ϕ.Lemma 1.If ϕis satisfiable then with probability 1the demands in N can be colored such that at most one fiber is required on each link.If ϕis not satisfiable then with high probability,for any coloring of the demands in N ,some link requires c +1fibers.Proof.Suppose that ϕis satisfiable.Then with probability 1the graph G is colorable with µ=n εcolors.For any such coloring,we color the demands in N such that demand d v receives the same color as node v .Clearly,for any clique Q in G and any color λ,there is at most one node in Q that receives color λ.Hence for any link e Q in E 1,there is at most one demand passing through linke Q that receives colorλ.Therefore each link in E1requires only onefiber in order to carry all its demands.The links in E2have only one demand and so they trivially require onefiber only.Hence at most onefiber is required on any link in N.To prove the other direction,suppose thatϕis unsatisfiable.Then with high probabilityα(G)≤nε.Suppose for the purpose of contradiction that we can color the demands in N withµ=nεcolors such that each link requires at most cfibers.This implies that for any link e Q in E1,not all the demands passing through e Q receive the same color.Consider now the corresponding coloring of the nodes in G.1By the construction of our network N,for any clique Q with c+1nodes,not every node in Q receives the same color.Let X be the induced subgraph of G on the set of nodes that constitutes the largest color class.We have just shown that X does not contain a clique of size c+1.Moreover,since X is contained in G,α(X)≤α(G)≤nε.Ramsey’s theorem(see e.g.[19])immediately implies that,|X|≤ α(G)+(c+1)−2(c+1)−1 ≤α(G)c.(3) Since X constitutes the largest color class and there are nεcolors,|X|nε≥n. Hence,|X|≥n1−ε⇒α(G)c≥n1−ε⇒α(G)≥n1−εc>nε,.This contradicts the fact thatα(G)≤nε.sinceε<1c+1Theorem1.There is no c-approximation to Min-Fiber for any constant c unless NP⊆ZPP.Proof.Suppose for the purpose of contradiction that C is a polynomial time c-approximation algorithm.We use this to construct a randomized algorithm B for3SAT.For each instanceϕ,algorithm B creates a random graph G and then converts it to an instance of Min-Fiber on a network N as described above.It then runs algorithm C on the instance of Min-Fiber.If the solution returned by algorithm C is at most c then algorithm B returns“satisfiable”,otherwise algorithm B returns“unsatisfiable”.Lemma1implies that,–Ifϕis satisfiable then the optimal solution to the instance of Min-Fiber is1.Since algorithm C is a c-approximation algorithm,it returns a value of atmost c.Therefore algorithm B outputs“satisfiable”.–Ifϕis unsatisfiable then with high probability the optimal solution to the instance of Min-Fiber is c+1.Therefore algorithm C returns a solution of at least c+1.Therefore algorithm B outputs“unsatisfiable”.1Note that this is not necessarily a proper coloring.Some edges in G may have both endpoints assigned the same color.Note that algorithm B has one-sided error.Hence3SAT∈coRP and so NP⊆coRP.This implies RP⊆NP⊆coRP⊆coNP which in turn implies NP= coNP=RP=coRP=RP∩coRP=ZPP.3Improved Lower BoundIn this section we derive more general hardness results by examining the con-struction of Feige and Kilian in more detail.In particular,given a3CNF formula ϕand a constantε,they construct a random graph G on n nodes with parame-ters a,ρ,A and k.(As an aside,the parameters a,ρand A are associated with a randomized Probabilistically Checkable Proof for NP and k is associated with a random graph product on a graph generated from the PCP.However,these interpretations are not important for our purposes.)The parameters are chosen so that the following relationships hold.More specifically,the parameters a and ρarefixed to some constants such that(5)holds.The parameter A is polyno-mial in the size ofϕand k is polylogarithmic in the size ofϕ.In particular,k is chosen sufficiently large such that Lemma2holds.n=a k(4)1≥log aρlog a≥1−ε(5)A=poly(|ϕ|)(6)k=Θ(log1/δ|ϕ|)for anyδ∈(0,1)of our choice(7) Feige and Kilian show that Graph G has the following properties.1.Ifϕis satisfiable then with probability1,G can be colored with(1+log n)/ρkcolors,i.e.χ(G)≤(1+log n)/ρk.2.Ifϕis not satisfiable thenα(G)≤kA with high probability,which impliesχ(G)≥n/(kA).From the graph G we construct an instance of Min-Fiber in the same manner as in the previous section.The number of links m in the newly constructed network N is O(n c).We set,µ=(1+log n)/ρk(8)c=log1−δn for anyδ∈(0,1)of our choice(9) Lemma2.We can choose k=Θ(log1/δ|ϕ|)such that(1+log n)(kA)cρk<n. Proof.Immediate from the parameter definitions.The following is analogous to Lemma1.Lemma3.Ifϕis satisfiable then with probability1the demands in N can be colored withµcolors such that at most1fiber is required on each link.Ifϕis not satisfiable then with high probability,for any coloring of the demands in N, some link requires c+1fibers.Proof.For the case in whichϕis satisfiable,the proof is identical to Lemma1.For the other direction,suppose thatϕis unsatisfiable but we color the demands in N withµcolors such that each link requires at most cfibers.Consider the corresponding coloring of G and let X be the induced subgraph of G on the set of nodes that constitutes the largest color class.As in inequality(3)in the proof of Lemma1,|X|≤α(G)c.By the construction of G,with high probability α(G)≤kA,which implies|X|≤(kA)c.Since X constitutes the largest color class and there areµ=(1+log n)/ρk colors,|X|≥n/µ=nρk/(1+log n).These inequalities imply that(kA)c≥nρk/(1+log n)which contradicts Lemma(2).Note that since we have a link in the network N for each subset of c+1nodes in G,the size of the instance of Min-Fiber is polynomial in n c.The following is analogous to Theorem1.Theorem2.Unless3SAT has a randomized algorithm with expected running time O(|ϕ|polylog(|ϕ|)),there is no logγµ-approximation to Min-Fiber for any γ∈(0,1),and there is noΘ(logγm)-approximation for anyγ∈(0,0.5).Here,µis the number of colors perfiber and m is the number of links in Min-Fiber. Proof.As in Theorem1we assume for the purpose of contradiction that C is a polynomial time c-approximation algorithm where c is defined in(9).From C we can construct a randomized algorithm B for3SAT such that ifϕis satisfiable then B outputs“satisfiable”;ifϕis unsatisfiable then with high probability B outputs“unsatisfiable”.The correctness of B is identical to Theorem1.The running time of B is O(|ϕ|polylog(|ϕ|))since both k and c are polylogarithmic in|ϕ|.Sinceµ≤n and m=O(n c)we can show that c>(logµ)1−δand c=Ω((log m)1−1/(2−δ)).We note that B can give an incorrect answer with low probability.However,in the same way that NP⊆coRP implies NP⊆ZPP we can convert B into a randomized algorithm that always gives the correct answer and whose expected running time is O(|ϕ|polylog(|ϕ|)).4Upper Bounds4.1Randomized RoundingRecall that the linear relaxation of the our Min-Fiber problem always has an optimal solution z=1and C i,λ=1/µfor all demands i and wavelengthsλ. We adopt the technique of randomized rounding introduced in[20].For each demand i we choose a number x i uniformly at random in the range[0,1].If x i∈[k/µ,(k+1)/µ)then we round C i,λto1forλ=k and round C i,λto0 forλ=k.After rounding the constraint(2)still holds.We use the ChernoffBound[17]to see how much constraint(1)is violated.LetˆC i,λdenote the rounded solution.[ChernoffBound]If X 1,...,X n are independent binary random variables where the expectation x =E [ i X i ],then it holds for all δ>0that,Pr [i X i ≥(1+δ)x ]≤e −min(δ2,δ)·x/3.Lemma 4.For a particular link e and wavelength λ,Pr i :e ∈P i ˆC i,λ≥2f e ≤m −2µ−2if f e ≥6(log m +log µ),Pr i :e ∈P i ˆC i,λ≥f e +6(log m +log µ)≤m −2µ−2if f e <6(log m +log µ).Proof.By definition,the expected value of E [ˆC i,λ]is 1/µ.Hence,E i :e ∈P i ˆC i,λ =f e .Note that for a fixed link e and wavelength λ,the rounding of variables C i,λfor demands i that go through e are independent events.We can therefore apply the ChernoffBound.If f e ≥6(log m +log µ),thenPr i :e ∈P i ˆC i,λ≥(1+1)f e ≤e−f e /3≤e −2log m −2log µ=m −2µ−2.If f e <6(log m +log µ),thenPr i :e ∈P i ˆC i,λ≥ 1+1f e ·6(log m +log µ) f e ≤e−2log m −2log µ=m −2µ−2.By applying the union bound over all links and wavelengths,we obtain the following.Theorem 3.We can round the fractional optimal solution such that with high probability the number of fibers deployed on each link e is at most 2f e +O (log m +log µ).This implies an O (log m +log µ)approximation algorithm.We note that this algorithm can be derandomized by the standard method of conditional probabilities.We also note that for large values of f e the approxi-mation ratio approaches stly,we remark that by using the slightly tighter Chernoffbound P r [ i X i ≥(1+δ)x ]≤(e δ/(1+δ)1+δ)x ,the approximationratio can be marginally improved to O (log m log log m +log µlog log µ).However,for ease ofexposition we typically ignore “log log”factors in this paper.In the full version of the paper [2]we apply a constructive version of the Lov´a sz Local Lemma (see for example Theorem 3.84of [22])to obtain a ran-domized algorithm with polynomial expected running time and approximation ratio O (log D max +log µ),where D max is the maximum number of links along any demand path.In many optical networks D max is significantly smaller than m .We omit the proof from this version in the interests of space.Theorem 4.We can round the fractional optimal solution such that the number of fibers deployed on each link e is at most 2f e +6(log D max +log µ).4.2Path Length RoundingThe following rounding theorem is due to Karp,Leighton,Rivest,Thompson, Vazirani and Vazirani[12].[KLRTVV Rounding Theorem]Let A be a real-valued r×s matrix and let x be a real-valued s-vector,let b be a real-valued r-vector such that Ax=b and let∆be a positive real number such that in every column of A,1.the column sum of the positive elements is at most∆,and2.the column sum of the negative elements is at least−∆.Then we can compute an integral s-vectorˆx such that,1.ˆx is a rounded version of x,i.e.ˆx i=⌊x i⌋orˆx i=⌈x i⌉for1≤i≤s,and2.Aˆx=ˆb whereˆb i−b i≤∆for all1≤i≤r.In the case that all entries in Aand b are integers,then a stronger bound applies:ˆb i−b i≤∆−1.It is easy to see that matrix A in the LP formulation of Min-Fiber has0/1 entries and its column sum is upper bounded by the longest path length plus1, i.e.max i|P i|+1.By applying the KLRTVV Rounding Theorem,we obtain, Theorem5.We can round the fractional optimal solution such that the number offibers deployed on each link e is at most f e+D max.5ConclusionsIn this paper we have presented positive and negative results for approximating the Min-Fiber problem.We conclude by briefly discussing two variants of Min-Fiber with different objective functions and seeing how our results apply.In the basic Min-Fiber problem the objective is to minimize the ratio between the number offibers deployed on link e and the lower bound f e.For the sake of comparison,we restate the integer program.Basic Versionmin zsubject to i:e∈P i C i,λ≤z·f e∀e,λλC i,λ=1∀iIn thefirst variant the new objective is to minimize the maximum,over all links e,of the number offibers used on link e.As an integer program,this variant may be written as,Variant1min zsubject to i:e∈P i C i,λ≤z∀e,λλC i,λ=1∀iWe note that the hardness results of Sections2and3follow through,e.g.there is no constant-factor approximation for Variant1of Min-Fiber unless NP⊆ZPP. This is because for the case in which the3CNF formulaφis satisfiable,all links in the network require exactly1fiber.We also note that a lower bound on the optimal value of this problem is max e f e.By concentrating on the link e with the maximum value of f e,it is not hard to see that the approximation ratios proved in Section4also hold for this variant.In the second variant we assume that we somehow know the cost perfiber on link e.We denote this cost by L e.Our objective is to minimize the total cost offiber needed to carry all the demands.We can formulate this variant as the following integer program.Variant2min e z e L esubject to i:e∈P i C i,λ≤z∀e,λλC i,λ=1∀iOnce again,the approximation ratios proved in Sections4apply to this vari-ant.Furthermore,it can be shown that randomized rounding actually gives an O(logµ)approximation for this variant[5].However,our hardness results of Sections2and3no longer apply.Indeed,for the instances constructed in our reductions,randomized rounding gives a constant factor approximation for the problem of minimizing the totalfiber length.AcknowledgementThe authors wish to thank Chandra Chekuri,Bruce Shepherd and the anony-mous referees for many helpful comments.References1. 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