On the Approximability of Robust Network Design
OOn the Approximability of Robust Network Design
Yacine Al-Najjar a,b, ∗ , Walid Ben-Ameur b , J´er´emie Leguay a a Huawei Technologies, Paris Research Center, France. b Samovar, Telecom SudParis, Institut Polytechnique de Paris, France.
Abstract
Considering the dynamic nature of traffic, the robust network design problemconsists in computing the capacity to be reserved on each network link suchthat any demand vector belonging to a polyhedral set can be routed. Theobjective is either to minimize congestion or a linear cost. And routing freelydepends on the demand.We first prove that the robust network design problem with minimumcongestion cannot be approximated within any constant factor. Then, usingthe ETH conjecture, we get a Ω( log n log log n ) lower bound for the approximabilityof this problem. This implies that the well-known O (log n ) approximationratio established by R¨acke in 2008 is tight.Using Lagrange relaxation, we obtain a new proof of the O (log n ) approx-imation. An important consequence of the Lagrange-based reduction and ourinapproximability results is that the robust network design problem with lin-ear reservation cost cannot be approximated within any constant ratio. Thisanswers a long-standing open question of Chekuri.Finally, we show that even if only two given paths are allowed for eachcommodity, the robust network design problem with minimum congestion orlinear costs is hard to approximate within some constant k . Keywords:
Approximability, PCP, ETH, Robust Network Design. ∗ Corresponding author. Full postal address:18 quai du point du jour, 92100 Boulogne-Billancourt, France.
Email addresses: [email protected] (Yacine Al-Najjar), [email protected] (Walid Ben-Ameur), [email protected] (J´er´emie Leguay)
Preprint submitted to Theoretical Computer Science September 29, 2020 a r X i v : . [ c s . CC ] S e p . Introduction Network optimization [1, 2] plays a crucial role for telecommunicationoperators since it permits to carefully invest in infrastructures, i.e. reducecapital expenditures. As Internet traffic is ever increasing, the network’scapacity needs to be expanded through careful investments every year oreven half-year. However, the dynamic nature of the traffic due to ordinarydaily fluctuations, long term evolution and unpredictable events requires toconsider uncertainty on the traffic demand when dimensioning network re-sources.Ideally, the network capacity should follow the demand. When the trafficdemand can be precisely known, several approaches have been proposed tosolve the capacitated network design problem using for instance decomposi-tion methods and cutting planes [3, 4, 5]. But in practice, perfect knowledgeof future traffic is not available at the time the decision needs to be taken.The dynamic nature of the traffic due to ordinary daily fluctuations, longterm evolution and unpredictable events requires to consider uncertainty ontraffic demands when dimensioning network resources. While overestimatedtraffic forecasts could be used to solve a deterministic optimization problem,it is likely to yield to a costly over-provisioning of the network capacities,which is not acceptable. Therefore, robust optimization under uncertaintysets is a must for the design of network capacities. In this context, our paperpresents new approximability results on two tightly related variants of therobust network design problem, the minimization of either the congestion ora linear cost.Let’s consider an undirected graph G = ( V ( G ) , E ( G )) representing acommunication network. The traffic is characterized by a set of commodities h ∈ H associated to different node pairs. And the routing of a commoditycan be represented by a flow f h ∈ R E ( G ) of intensity d h . To take into ac-count the changing nature of the demand, d is assumed to be uncertain andmore precisely to belong to a polyhedral set D . The polyhedral model wasintroduced in [6, 7] as an extension of the hose model [8, 9], where limits onthe total traffic going into (resp. out of) a node are considered.When solving a robust network design problem, several objective func-tions can be considered. Given a capacity c e for each edge e , one might beinterested in minimizing the congestion given by max e ∈ E ( G ) u e c e where u e is thereserved capacity on edge e . Another common objective function is given bythe linear reservation cost (cid:80) e ∈ E ( G ) λ e u e . This can also represent the average2ongestion by taking λ e = c e . The goal is to choose a reservation vector u so that the network is able to support any demand vector d ∈ D , i.e., thereexists a (fractional) routing serving every commodity such that the total flowon each edge e is less than the reservation u e .The robust network design problem where a linear reservation cost isminimized was proved to be co-NP hard in [10, 11] when the graph is di-rected. A stronger co-NP hardness result is given in [12] where the graph isundirected (this implies the directed case result). Some exact solution meth-ods for robust network design have been considered in [13, 14]. In the casewhere minimum congestion is considered a well-known O (log n ) approxima-tion ratio was presented in [15]. Robust network design is also referred to as dynamic routing in the literature since the network is optimized such thatany realization of traffic matrix in the uncertainty set has its own routing.Routing with uncertain demands has received a significant interest fromthe community. As opposed to dynamic routing, static routing or stablerouting was introduced in [6]: it consists in choosing a fixed flow x h of value1 for each commodity h . The actual flow f h ( d ) for the demand scenario d will then be scaled by the actual demand d h of commodity h , i.e. f h ( d ) = d h x h . Static routing is also called oblivious routing in [16, 17]. In this case,polynomial-time algorithms to compute optimal static routing (with respectto either congestion or linear reservation cost) have been proposed [6, 7, 16,17] based on either duality or cutting-plane algorithms.To further improve solutions of static routing and overcome complexity is-sues related to dynamic routing , a number of restrictions on routing have beenconsidered to design polynomial-time algorithms (see [18, 19] for a completesurvey). This includes, for example, the multi-static approach, introducedin [20], where the uncertainty set is partitioned using an hyperplane androuting is restricted to be static over each partition. This idea has been gen-eralized in [21] to unrestricted covers of the uncertainty set and an extensionto share the demand between routing templates, called volume routing, hasbeen proposed in [22]. [23] applied affine routing for robust network design,based on affine adjustable robust counterparts introduced in [24], restrictingthe recourse to be an affine function of the uncertainties. The performanceof this framework has been extensively compared to the static and dynamicrouting, both theoretically and empirically [25, 19]. In practice, affine rout-ing provides a good approximation of the dynamic routing while it can besolved in reasonable time thanks to polynomial-time algorithms. Finally, anapproach encompassing the previous approaches is the multipolar approach3roposed in [26, 27].In this work, we will only focus on the complexity of the robust networkdesign problem (i.e., under dynamic routing), while minimizing either con-gestion or some linear cost. To close this section, let us summarize the maincontributions of the paper and review some related work • We first prove that the robust network design problem with minimumcongestion cannot be approximated within any constant factor. Thereduction is based on the PCP theorem and some connections withthe Gap-3-SAT problem [28]. The same reduction also allows to showinapproximability within Ω(log n ∆ ) where ∆ is the maximum degree inthe graph and n is the number of vertices. • Using the ETH conjecture [29, 30], we prove a Ω( log n log log n ) lower boundfor the approximability of the robust network design problem with min-imum congestion. This implies that the well-known O (log n ) approxi-mation ratio that can be obtained using the result in [15] is tight. • We show that any α -approximation algorithm for the robust networkdesign problem with linear costs directly leads to an α -approximationfor the problem with minimum congestion. The proof is based on La-grange relaxation. We obtain that robust network design with mini-mum congestion can be approximated within O (log n ). This was al-ready proved in [15] in a different way. • An important consequence of the Lagrange-based reduction and our in-approximability results is that the robust network design problem withlinear reservation cost cannot be approximated within any constantratio. This answers a long-standing open question stated in [31]. • We show that even if only two given paths are allowed for each com-modity, there is a constant k such that the robust network design prob-lem with minimum congestion or linear costs cannot be approximatedwithin k . A fundamental tool in the design of approximation algorithms is the ap-proximation of finite metric by tree metric embedding. This theory culmi-4ated in [32] with the result that any n points metric space can be approx-imated with a distribution over dominating tree metric within a O (log n )distortion factor. As proved in [33, 31], this result leads to a O (log n ) ap-proximation algorithm for robust network design (dynamic routing and lin-ear reservation cost). [34] proved the existence of an oblivious routing witha competitive ratio of O (log n ) with respect to optimum routing of anytraffic matrix. [17, 6, 16] show how a routing achieving an optimal competi-tive ratio can be found in polynomial time. Then, [35] improved the boundto O (log n log log n ) and gave a polynomial-time algorithm to find such astatic routing. Finally, [15] described an O (log n ) approximation algorithmfor static routing with minimum congestion.Notice that the bound given by static routing cannot provide a betterbound than O (log n ) since a lower bound of Ω(log n ) is achieved by staticrouting for planar graphs [36, 37]. Several other approximation results areknown for single path routing and tree routing when some special types ofpolytopes are considered (such as the symmetric and the asymmetric hosemodels) (see, e.g., [10, 15, 38]). Using an approximate separation oracle forthe dual problem to obtain an approximate solution of the primal is a well-known technique already used in [39, 40, 41] at least in the context of packing-covering problems. Lagrangian relaxations are also used in [42, 43, 44] toproduce dual solutions that are near-optimal.
2. From Gap-3-SAT to robust network design with minimum con-gestion
Given an edge e , let s ( e ) and t ( e ) be the extremities of e . Similarlyto edges, for a commodity h ∈ H , let s ( h ) and t ( h ) denote the endpointsof h . And let U ( D ) be the set of u ∈ R E ( G ) such that each traffic vector d ∈ D can be routed on the network when a capacity c e is assigned to edge e . Since D is polyhedral, U ( D ) is also polyhedral (see, e.g, [31]). We areinterested in minimizing the congestion under polyhedral uncertainty anddynamic routing: min u ∈U ( D ) max e ∈ E ( G ) u e c e .Given a polytope represented by Ax ≤ b , the size of the polytope denotesthe total encoding size of the entries in A and b .Our first main result is related to the inapproximability of the minimumcongestion problem within a constant factor.5 heorem 2.1. Unless P = N P , the minimum congestion problem cannotbe approximated with a polynomial-time algorithm within any constant factoreven if D is given by { d : Ad + Bξ ≤ b } whose size is polynomially boundedby | V ( G ) | . Notice that it is important to consider polyhedral uncertainty sets that areeasy to describe (otherwise the inapproximability results would be a directconsequence of the difficulty to separate from the uncertainty set).To prove Theorem 2.1, we will need the PCP (Probabilistically CheckableProof) theorem [28] and an intermediate lemma. For a 3-SAT formula ϕ wenote val ( ϕ ) the maximum fraction of the clauses which are satisfiable atthe same time. In particular, val ( ϕ ) = 1 means that ϕ is satisfiable. PCPtheorem is recalled below. Theorem 2.2.
PCP (Probabilistically Checkable Proof ) theorem [28]: Thereis a constant < ρ < such that for any language L ∈ N P , there is afunction f from L to 3-SAT instances, computable in polynomial time, suchthat y ∈ L = ⇒ val ( f ( x )) = 1 while y (cid:54)∈ L = ⇒ val ( f ( x )) < ρ . The problem where we have to decide if val ( ϕ ) < ρ or val ( ϕ ) = 1 for a3-SAT formula ϕ is called Gap-3-SAT.To prove the theorem 2.1 we will use the following lemma (where cong denotes the optimal congestion of the corresponding instance). Lemma 2.1.
For every γ ∈ N there is a mapping f γ computable in poly-nomial time from 3-SAT instances to minimum congestion instances de-fined by an undirected graph G γ , a set of commodities H γ and a polytope D γ = { d : A γ d + B γ ψ γ ≤ b γ } such that | V ( G γ ) | = O ( m γ ) , | E ( G γ ) | = O ( m γ ) and the size of D γ is O ( m cγ ) where c is some positive constant and m is thenumber of clauses. The mapping satisfies the following: • val ( ϕ ) = 1 = ⇒ cong ( f γ ( ϕ )) ≥ γ (1 − ρ ) • val ( ϕ ) < ρ = ⇒ cong ( f γ ( ϕ )) ≤ .Proof. of Theorem 2.1 We are going to use Lemma 2.1 and Theorem 2.2for the proof. Suppose that congestion can be approximated in polynomialtime within a constant approximation factor α . We first choose γ such that α < γ (1 − ρ ). Starting from an instance y of an NP-Complete problem weconstruct in polynomial time ϕ as stated in Theorem 2 .
2. Then we construct6 igure 1: G and G f γ ( ϕ ) in polynomial time. Applying the α -approximation to f γ ( ϕ ), we geta congestion value ˜ β . If ˜ β < γ (1 − ρ ) holds we can deduce that theoptimal congestion is 1 and thus that val ( ϕ ) < ρ which implies that y is notaccepted. Otherwise we can deduce that y is accepted. Furthermore, as thesize of the polytope used in Lemma 2.1 is O ( m cγ ) while | V ( G γ ) | = O ( m γ ),its size is polynomially bounded in the number of vertices as announced inTheorem 2.1We are now going to prove Lemma 2.1 by first constructing instances ofthe congestion problem leading to some inapproximabilty factor. Then, thisfactor is increased by recursively building larger instances with higher valuesof γ . Proof. of Lemma 2.1, case γ = 1We start with a 3-SAT formula ϕ , with m clauses and r variables. Wenote L = { l , . . . , l r , ¬ l , . . . , ¬ l r } the set of the literals appearing in formula ϕ and l i,j the literal appearing in the i -th clause C i at the j -th position for i = 1 , ..., m and j = 1 , , l ∈ L a non-negative variable ξ l and for k = 1 , ..., r , we add theconstraint ξ l k + ξ ¬ l k = 1.We build as follows a graph G , a set of commodities H and a polyhedraluncertainty set D . For each i = 1 , ..., m , j = 1 , , e i,j (i.e. such that t ( e i, ) = s ( e i, ) and t ( e i, ) = s ( e i, )) and 3 commodi-ties h i,j with s ( h i,j ) = s ( e i,j ) and t ( h i,j ) = t ( e i,j ), and d h i,j = ξ l i,j . We imposethat all nodes s ( e i, ) (resp. t ( e i, )) for i = 1 , ..., m are equal to a single nodenoted s (resp. t ) (see Figure 1). We consider an additional commodity h between s and t whose value satisfies d h ≤ m (1 − ρ ). We also add non-negativity constraints ( d h ≥ h ∈ H ). The uncertainty polyhedron7 is then obtained by projecting Ξ on the space of d h variables. Finally, thecapacity c e of each edge e is here equal to 1 ( c e = 1).If val ( ϕ ) = 1, then there is a demand vector such that for each pathbetween s and t (there is one path corresponding to each clause), at leastone commodity whose endpoints are on the path is equal to 1 (a commoditycorresponding to a true literal). This implies that all paths are blocked andthus the optimal routing for commodity h is to equally spread m (1 − ρ )between the m paths leading to a congestion of 1 + (1 − ρ ).Let us now assume that val ( ϕ ) < ρ . Notice first that the extreme points ofthe polyhedron D are such that the d h i,j variables take their values in { , } .To see this, suppose that there is an extreme point d of D such that thereis some i , j such that 0 < d h i ,j <
1. Define d (cid:48) , ξ, ξ (cid:48)(cid:48) , d (cid:48)(cid:48) by ξ (cid:48) l i ,j = d (cid:48) h i,j = 1and ξ (cid:48)(cid:48) l i ,j = d (cid:48)(cid:48) h i,j = 0 if l i,j = l i ,j , ξ (cid:48)¬ l i ,j = d (cid:48) h i,j = 0 , ξ (cid:48)(cid:48)¬ l i ,j = d (cid:48)(cid:48) h i,j = 1if l i,j = ¬ l i ,j , d (cid:48) h i,j = d (cid:48)(cid:48) h i,j = d h i,j otherwise. We have ( ξ (cid:48) , d (cid:48) ) , ( ξ (cid:48)(cid:48) , d (cid:48)(cid:48) ) ∈ Ξ and d can be written as the convex combination d = αd (cid:48) + (1 − α ) d (cid:48)(cid:48) with α = d h i ,j contradicting the fact that d is an extreme point of D . For suchan extreme demand vectors d ∈ D there are at least m (1 − ρ ) free pathsto route the demand d h allowing a congestion less than or equal to 1. Thisimplies that all demands in D can also be routed with a congestion less thanor equal to 1Observe that | V ( G ) | = O ( m ), | E ( G ) | = O ( m ), D has the appropriateform ( D = { d : A d + B ψ ≤ b } ) and the size of D is O ( m c ) for someconstant c . Proof. of Lemma 2.1, case γ ≥ γ ≥
2, having constructed G γ − , H γ − , D γ − , we build G γ , H γ , D γ as follows. We will construct the graph G γ , by taking the graph G andreplacing each edge by a copy of the graph G γ − denoted by G i,jγ − . Eachcopy G i,jγ − contains a node s γ − that is identified with s ( e i,j ) and a node t γ − identified with t ( e i,j ) (see Figure 1). All commodities related to G i,jγ − (belonging to H γ − ) are also considered as commodities of H γ . Let us use d i,j ∈ R H γ − to denote the related demand vector. H γ also contains a non-negative commodity h ,γ constrained by d h ,γ ≤ m γ (1 − ρ ). Thus |H γ | =1 + 3 m × |H γ − | .Moreover, we build a polyhedron Ξ γ by considering auxiliary non-negativevariables ξ l for l ∈ L in addition to commodity variables and auxiliary non-8egative variables ψ i,j (a vector of variables for each i = 1 , ..., m and j =1 , , k = 1 , ..., r , we add the constraint ξ l k + ξ ¬ l k = 1. And for e i,j ∈ E ( G ),we add the constraints d i,j ∈ ξ l i,j D γ − . Let us explain how this can be done.By induction, we know that D γ − = { d : A γ − d + B γ − ψ γ − ≤ b γ − } andthis representation includes (among others) non-negativity constraints of allvariables in addition to constraints implying that all variables are upper-bounded. Then by writing A γ − d i,j + B γ − ψ i,j ≤ ξ l i,j b γ − , we can ensure that ξ l i,j = 0 implies d i,j = 0, while ξ l i,j > ξ li,j d i,j ∈ D γ − . In particularwhen ξ l i,j = 0, from outside, the whole subgraph corresponding to G i,jγ − actslike a single edge of capacity m γ − .We can observe, from the construction above, that D γ can be representedas the projection of a polytope Ξ γ = { A γ d + B γ ψ γ ≤ b γ } where ψ γ containsthe auxiliary variables ξ appearing in all levels. More precisely, Ξ γ is definedby: d h ,γ ≤ m γ (1 − ρ ); − d h ,γ ≤ − ξ l ≤ , ∀ l ∈ L ; ξ l k + ξ ¬ l k ≤ , − ξ l k − ξ ¬ l k ≤ − , ∀ k = 1 , ..., r ; A γ − d i,j + B γ − ψ i,jγ − − ξ l i,j b γ − ≤ , ∀ i = 1 , ..., m ; j = 1 , , . (1)By simple induction, we have | V ( G γ ) | = O ( m γ ), | E ( G γ ) | = O ( m γ ) andthe size of D γ is O ( m cγ ) where c is some positive constant.We observe that all extreme points of Ξ γ are such that ξ l ∈ { , } for l ∈ L . To verify that, we first recall that constraints (1) are equivalent to d i,j ∈ ξ l i,j D γ − (in this way, the vectors ψ i,jγ − can be ignored). Second, let L + be the set of literals appearing in positive form. We observe that variables ξ l for l ∈ L + are pairwise independent. Only variables d i,j such that either l i,j = l or l i,j = ¬ l depend on ξ l since d i,j ∈ ξ l D γ − in the first case and d i,j ∈ (1 − ξ l ) D γ − in the second case. This immediately implies that given some arbi-trary real vectors q i,j and f , minimizing (cid:80) i =1 ,..,m ; j =1 , , q Ti,j d i,j + (cid:80) l ∈L + f l ξ l is equiv-alent to minimizing (cid:80) l ∈L + ξ l (cid:32) f l + (cid:80) i,j : l i,j = l min d i,j ∈D γ − q Ti,j d i,j − (cid:80) i,j : l i,j = ¬ l min d i,j ∈D γ − q Ti,j d i,j (cid:33) .It is then clear that optimal ξ l values will be either 0 or 1. Since this holdsfor an arbitrary linear objective function, we get the wanted result aboutextreme points. 9et us now show that val ( ϕ ) < ρ = ⇒ cong ( f γ ( ϕ )) ≤
1. Assume that val ( ϕ ) < ρ . We prove by induction that the congestion of ( G γ , H γ , D γ ) is1. Suppose that this is true for some γ −
1. If ξ l i, = ξ l i, = ξ l i, = 0 forsome i , a flow of value m γ − can be routed between s γ and t γ by sending aflow of value 1 on each edge of G i,jγ − for j = 1 , ,
3. Since val ( ϕ ) < ρ , thereare necessarily at least m (1 − ρ ) such i , thus we can send the whole demand m γ − m (1 − ρ ) = m γ (1 − ρ ) this way. For the indices i, j such that ξ l i,j = 1,by the induction hypothesis ( cong ( f γ − ( ϕ )) ≤ G i,jγ − can be routed without sending more than one unit of flow on each edge of G i,jγ − .Notice that to show that all traffic vectors of D γ can be routed with con-gestion 1, we considered demand vectors corresponding with { , } ξ vari-ables. The result shown above about extreme points is useful here since itallows us to say that each extreme point of D γ can be routed with congestionless than or equal to 1 implying that each demand vector inside D γ can alsobe routed with congestion less than or equal to 1.Let us now show that val ( ϕ ) = 1 = ⇒ cong ( f γ ( ϕ )) ≥ γ (1 − ρ ). Weare going to use induction to build a cut δ ( C γ ) where C γ is set of verticesof V ( G γ ) containing s γ and not containing t γ . The number of edges of thecut will be m γ and each edge has a capacity equal to 1. We also show theexistence of a demand vector d ∈ D γ such that the sum of the demandstraversing the cut is greater than or equal to m γ (1 + γ (1 − ρ )). This wouldshow that there is at least one edge that carries at least 1 + γ (1 − ρ ) units offlow.Since ϕ is satisfiable, there is a truth assignment represented by ξ variables(the auxiliary variables) such that for each i = 1 , ..., m there is a j ( i ) suchthat ξ l i,j ( i ) = 1. By considering the graph G i,j ( i ) γ − and using the inductionhypothesis, we can build a cut δ ( C iγ − ) separating the node s ( e i,j ( i ) ) and t ( e i,j ( i ) ) and containing m γ − edges. We also build a demand vector d i,j ( i ) ∈D γ − such that the sum of demands traversing the cut is greater than or equalto m γ − (1 + ( γ − − ρ )) (still possible by induction). By taking the unionof these m disjoint cuts we get a cut δ ( C γ ) that is separating s γ and t γ havingthe required number of edges. A demand vector d can be built by combiningthe vectors d i,j ( i ) and the demand d h ,γ taken equal to m γ (1 − ρ ). Since thedemand from s γ to t γ is also traversing the cut, the total demand through δ ( C γ ) is greater than or equal to m γ (1 − ρ ) + m.m γ − (1 + ( γ − − ρ )) =10 γ (1 + γ (1 − ρ )).Lemma 2.1 can be further exploited in different ways since there are manypossible connections between the value 1 + γ (1 − ρ ) and the characteristics ofthe undirected graph built in the proof of the lemma. Observe, for example,that by a simple induction we get that the number of vertices | V ( G γ ) | =2 + 2 m (3 m ) γ − m − leading to | V ( G γ ) | (cid:39) × γ − m γ (when m goes to infinity).We also have ∆( G γ ) equal to m γ where ∆( . ) denotes the maximum degreein the graph. Consequently, log ( | V ( G γ ) | ∆( G γ ) ) (cid:39) γ log 3 + log 2 /
3. Then by takingany constant k such that k × log 3 < (1 − ρ ) where ρ is the constant in thePCP Theorem 2.2 we get a lower bound of the approximability ratio. Thisis stated in the following corollary. Corollary 2.1.
Under conditions of Theorem 2.1, for any constant k < − ρ log 3 , it is not possible to approximate the minimum congestion problem inpolynomial time within a ratio of k log | V ( G ) | ∆( G ) . Corollary 2.1 also implies that for any small constant (cid:15) , it is not possibleto approximate congestion within a ratio of log − (cid:15) | V ( G ) | ∆( G ) . A stronger inap-proximability result is shown in next section based on the stronger conjectureETH.
3. A Ω( log n log log n ) approximability lower boundConjecture 3.1 (Exponential Time Hypothesis) . [29, 30] There is a con-stant δ such that no algorithm can solve 3-SAT instances in time O (2 δm ) ,where m is the number of clauses. Let us use n to denote the number of vertices of the graph. Theorem 3.1.
Under Conjecture 3.1, there exists a constant k such that nopolynomial-time algorithm can solve the minimum congestion problem withthe approximation ratio k log n log log n .Proof. The combination of PCP Theorem 2.2 and ETH Conjecture 3.1 im-plies that distinguishing between 3-SAT instances such that val ( ϕ ) < ρ and val ( ϕ ) = 1 cannot be done in time O (2 m β ) for some constant β > O (2 m/ log c m ) for some constant c , but this will not help us toimprove the lower bound of Theorem 3.1).11uppose that there is an algorithm that solves the minimum congestionproblem with an approximation factor α ( n ) and a running time O ( n c ).Given a 3-SAT instance and a function γ : N −→ N we can construct aminimum congestion instance f γ ( m ) ( ϕ ) as in Lemma 2.1 in time O ( m c γ ( m ) )and where the number of vertices of the instance is m γ ( m ) . Then by runningthe approximation algorithm for minimum congestion we get a total time of O ( m c γ ( m ) ) where c = max { c , c } . Thus by choosing γ ( m ) = m β c log m we getan algorithm that runs in time O (2 m β ). And if the approximation factor α ( n )is small enough, that is if α ( m γ ( m ) ) < − ρ ) γ ( m ) for a big enough m ,we get an algorithm solving Gap-3-SAT and thus contradicting Conjecture3.1. This is the case for k log n log log n for some constant k . To see this, we canobserve that: − ρ ) γ ( m ) α ( m γ ( m ) ) = − ρ ) mβc m k mβ/c β log m − log c (cid:39) β (1 − ρ ) k . By taking k < β (1 − ρ ) we get thewanted inapproximability result.Notice that since log − (cid:15) n = o ( log n log log n ) for any small positive constant (cid:15) ,it is not possible to approximate minimum congestion within log − (cid:15) n .
4. From minimum congestion to linear costs
Given any λ ≥
0, the robust network design problem with linear costs issimply the following: min u ∈U ( D ) λ T u. (2)Assume that there exists a number α ≥ α . More precisely,we have a polynomial-time oracle that takes as input a non-negative linearcost λ ∈ R E ( G ) and outputs a u ap ( λ ) ∈ U ( D ) such that λ T u ( λ ) ≤ λ T u ap ( λ ) ≤ αλ T u ( λ ) where u ( λ ) ∈ U ( D ) is the optimal solution of (2).Recall that the congestion problem is given bymin ue ≤ ceβu ∈U ( D ) β (3)Let us consider a Lagrange relaxation of (3) by dualizing the capacityconstraints and using λ for the dual multipliers. The dual problem is thengiven by max λ ≥ min u ∈U ( D ) β + (cid:80) e ∈ E ( G ) λ e ( u e − βc e ), which is equivalent to:12ax λ ≥ (cid:80) e ∈ E ( G ) λece =1 min u ∈U ( D ) (cid:88) e ∈ E ( G ) λ e u e = max λ ≥ (cid:80) e ∈ E ( G ) λece =1 λ T u ( λ ) . (4)Since U ( D ) is polyhedral, all constraints and the objective function are linear,there is is no duality gap between (3) and (4).Observe that (4) can be expressed as follows:max β (5a) β ≤ (cid:88) e ∈ E ( G ) λ e u e ∀ u ∈ U ( D ) (5b) λ ≥ (cid:88) e ∈ E ( G ) λ e c e = 1 (5c)We are going to approximately solve (5) using a cutting-plane algorithmwhere inequalities (5b) are iteratively added by using the α -approximation or-acle. Let ( β (cid:48) , λ (cid:48) ) be a potential solution of (5), we can run the α -approximationof robust network design problem (2) with the cost vector λ (cid:48) to get a solution u ap ( λ (cid:48) ). If β (cid:48) > (cid:80) e ∈ E ( G ) λ (cid:48) e u ape ( λ (cid:48) ) we return the inequality β ≤ (cid:80) e ∈ E ( G ) λ e u ape ( λ (cid:48) ),otherwise the algorithm stops and returns ( β (cid:48) , λ (cid:48) ). We know from the separation-optimization equivalence theorem [45] that (5) can be solved by makinga polynomial number of calls to the separation oracle leading a globallypolynomial-time algorithm. Notice that this happens if the separation oracleis exact. In our case, the oracle is only an approximate one, implying that thecutting plane algorithm might be prematurely interrupted before obtainingthe true optimum of (5). Observe however that this implies that the com-puting time is polynomially bounded. Let ( ˜ β, ˜ λ ) be the solution returned bythe cutting-plane algorithm. Let ( β ∗ , λ ∗ ) be the true optimal solution of (5).The next lemma states that the returned solution is an α -approximation ofthe optimal solution. Lemma 4.1.
The cutting-plane algorithm computes in polynomial time asolution ˜ β satisfying: β ∗ ≤ ˜ β ≤ αβ ∗ . (6) Proof.
Observe that β ∗ = λ ∗ T u ( λ ∗ ). Moreover, since (5) is equivalent to(4), we get that λ ∗ T u ( λ ∗ ) = β ∗ ≥ ˜ λ T u (˜ λ ). From the approximation factor13f the oracle, one can write that ˜ λ T u ap (˜ λ ) ≤ α ˜ λ T u (˜ λ ) . Using the fact thatno inequalities can be added for ( ˜ β, ˜ λ ), we get that ˜ β ≤ ˜ λ T u ap (˜ λ ). Finally,since ( β ∗ , λ ∗ ) is feasible for (5), we obviously have ˜ β ≥ β ∗ . Combining the 4previous inequalities leads to (6).The above lemma has many consequences. Theorem 4.1.
The robust network design problem with linear costs cannotbe approximated within any constant ratio.Proof.
The result is an immediate consequence of Theorem 2.1 and Lemma4.1.The theorem above answers a long-standing open question of [31]. Allother inapproximability results proved for the congestion problem directlyhold for the robust network design problem with linear cost.Another important consequence is that the congestion problem can beapproximated within O (log n ). This result was already proved in [15] usingother techniques. In our case, the result is an immediate consequence ofthe O (log n )-approximation algorithm for the robust network design problemwith linear cost provided by [33, 46] and fully described in [31]. Theorem 4.2. [15] Congestion can be approximated within O (log n ) . Notice that Theorem 3.1 tells us that the ratio O (log n ) is tight.
5. Restriction to a constant number of given paths per commodity
First, observe that in the proof of Lemma 2.1, the minimum congestioninstances built there are such that some commodities can be routed alongmany paths. For example, in graph G (Figure 1), commodity h (between s and t ) can use up to m paths. Second, consider an instance of the mini-mum congestion problem where only one path is given for each commodity.Then computing the minimum congestion is easy since we only have to com-pute max d ∈D (cid:80) h ∈H e d h where H e denotes the set of commodities routed through e . The congestion is just given by max e ∈ E ( G ) 1 c e max d ∈D (cid:80) h ∈H e d h . Combining thesetwo observations, one can wonder whether the difficulty of the congestionproblem is simply due to the number of possible paths that can be used byeach commodity. We will show that the problem is still difficult even if eachcommodity can be routed along at most two fixed given paths.14 igure 2: G (cid:48) Theorem 5.1.
For some positive constant k , minimum congestion is difficultto approximate within a ratio k even if each commodity can be routed alongat most two given paths.Proof. The proof is a simple modification of the proof of Lemma 2.1 (case γ = 1). We are going to slightly modify graph G in such a way that at most2 paths are allowed for each commodity. Given a 3-SAT formula ϕ with m variables, we construct G (cid:48) , H (cid:48) , D (cid:48) as follows. We first create two nodes s and t and an edge e between s and t of capacity mρ ( ρ is the constantin PCP theorem). Then for each clause index i = 1 , ..., m , as in Lemma 2.1,we create 3 consecutive edges e i,j ( j = 1 , ,
3) such that t ( e i,j ) = s ( e i,j +1 )and a commodity h i,j between s ( e i,j ) and t ( e i,j ) that is allowed to be routedonly through e i,j . We also add one edge between s ( e i, ) and s and one edgeconnecting t and t ( e i, ) of infinite capacity and a commodity h i, between s ( e i, ) and t ( e i, ) with a demand d h i, = 1. h i, is allowed to be routed onlythrough the path P i containing the edges ( e i, , e i, , e i, ) and the path goingthrough s , e and t (See Figure 2). We consider auxiliary variables ξ l foreach literal l . We add constraints ξ l + ξ ¬ l = 1 and d h i,j = ξ l i,j .If val ( ϕ ) < ρ there are at least m (1 − ρ ) commodities h i, that can berouted on the paths P i and the the remaining mρ can be routed on the edge e . This implies that each extreme point of D (cid:48) can be routed with congestion ≤
1. Notice that the observation made in the proof of Lemma 2.1 aboutextreme points is still valid here: extreme points corresponds to 0 − ξ l .If val ( ϕ ) = 1, then there is a cut and a demand vector d (corresponding tothe truth assignment satisfying ϕ ) such that the capacity of the cut is mρ + m and the demand that needs to cross the cut is 2 m . There is consequently15t least one edge of congestion greater than or equal to m (1+ ρ ) m = ρ . Bytaking k < ρ we get the wanted result.Finally, observe that the result above can also be stated for the linearcost case using again the Lagrange based reduction of the previous section. Corollary 5.1.
For some positive constant k , robust network design withlinear costs is difficult to approximate within a ratio k even if each commoditycan be routed along at most two given paths. eferenceseferences