MIT Sloan School of Management

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1 MIT Sloan School of Management Working Paper November 2003 Computational Complexity, Fairness, an the Price of Anarchy of the Maximum Latency Problem Jose R. Correa, Anreas S. Schulz, Nicolas E. Stier Moses 2003 by Jose R. Correa, Anreas S. Schulz, Nicolas E. Stier Moses. All rights reserve. Short sections of text, not to excee two paragraphs, may be quote without explicit permission, provie that full creit incluing notice is given to the source. This paper also can be ownloae without charge from the Social Science Research Network Electronic Paper Collection:

2 COMPUTATIONAL COMPLEXITY, FAIRNESS, AND THE PRICE OF ANARCHY OF THE MAXIMUM LATENCY PROBLEM JOSÉ R. CORREA, ANDREAS S. SCHULZ, AND NICOLÁS E. STIER MOSES Operations Research Center Massachusetts Institute of Technology 77 Massachusetts Avenue Cambrige, MA Abstract. We stuy the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-har, even when all arc latency functions are linear an there is a single source an sink. Still, one can prove that an optimal flow an an equilibrium flow share a esirable property in this situation: all flow-carrying paths have the same length; i.e., these solutions are fair, which is in general not true for the optimal flow in networks with nonlinear latency functions. In aition, the maximum latency of the Nash equilibrium, which can be compute efficiently, is within a constant factor of that of an optimal solution. That is, the so-calle price of anarchy is boune. In contrast, we present a family of instances that shows that the price of anarchy is unboune for instances with multiple sources an a single sink, even in networks with linear latencies. Finally, we show that an s-t-flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective, an it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency. 1. Introuction We stuy static network flow problems in which each arc possesses a latency function, which escribes the common elay experience by all flow on the arc as a function of the volume of the arc flow. Loa-epenent arc costs have a variety of applications in situations in which one wants to moel congestion effects, which are boun to appear, e.g., in communication networks, roa traffic, or evacuation problems. In this context, a unit of flow frequently enotes a huge number of users (or agents ), which might represent ata packages in the Internet, rivers on a highway system, or iniviuals fleeing from a builing. Depening on the concrete circumstances, the operators of these networks can pursue a variety of system objectives. For instance, they might elect to minimize the average latency, they might aim at minimizing the maximum latency, or they might try to ensure that users between the same origin-estination pair experience essentially the same latency. In fact, the ieal solution woul be simultaneously optimal or near optimal with respect to all three objectives. For linear latencies, we prove the existence of an s-t-flow that is at the same time optimal for two of the three objectives while its average latency is within a factor of 4/3 of that of an optimum. As attractive as this solution might be, we also show that it is NP-har to compute. Moreover, there is a surprising ifference between linear an nonlinear latency functions. Namely, this particular flow remains optimal with respect to the maximum latency an near optimal with respect to the average latency, but it oes in general not guarantee that ifferent users face the same latency. However, Key wors an phrases. System Optimum, User Equilibrium, Selfish Routing, Price of Anarchy, Approximation Algorithms, Multicriteria Optimization, Multicommoity Flows. 1

3 2 J.R. CORREA, A.S. SCHULZ, AND N.E. STIER MOSES an optimal s-t-flow for the average latency objective can be compute in polynomial time, an we show that the latency of any one user is within a constant factor of that of any other user. In particular, the maximum latency is within the same constant factor of the maximum latency of an optimal solution to the latter objective. This constant factor only epens on the class of allowable latency functions. For instance, its value is 2 for the case of linear latencies. Linear latencies are sufficient for certain congestion phenomena to occur. One interesting example is Braess paraox (1968), which refers to the fact that the aition of an arc can actually increase the (average an maximum) latency in a network in which users act selfishly an inepenently. This user behavior is capture by the Nash equilibrium of the unerlying game in which each user picks a minimal latency path, given the network congestion ue to other users (Warrop 1952). While the inefficiency of this so-calle user equilibrium an hence the severity of Braess paraox ha previously been boune in terms of the average latency, it turns out that it is also boune with respect to the maximum latency. Inee, the latencies encountere by ifferent users between the same origin-estination pair are the same. The user equilibrium therefore represents another flow that can be compute in polynomial time an that is optimal or close to optimal for all the three objectives introuce earlier. The Moel. We consier a irecte graph G = (N, A) together with a set of source-sink pairs K N N. For each terminal pair k = (s k, t k ) K, let P k be the set of irecte (simple) paths in G from s k to t k, an let k > 0 be the eman rate associate with commoity k. Let P := k K P k be the set of all paths between terminal pairs, an let := k K k be the total eman. A feasible flow f assigns a nonnegative value f P to every path P P such that P P k f P = k for all k K. In the context of single-source single-sink instances, we will rop the subinex k. Each arc a has a loa-epenent latency l a ( ). We assume that the functions l a : R 0 R 0 are nonnegative, nonecreasing, an ifferentiable. We efine the latency of a path P P uner a given flow f as l P (f) := a P l a( Q P:Q a f Q). The maximum latency of a feasible flow f is L(f) := max{l P (f) : P P, f P > 0}. We call a feasible flow that minimizes the maximum latency a min-max flow an enote it by ˆf. The maximum latency problem consists of fining a min-max flow. The average latency of a feasible flow f is efine as C(f) := P P l P (f)f P /. We refer to the optimal solution with respect to this objective function as the system optimum an enote it by f. A feasible flow is at Nash equilibrium (or is a user equilibrium) if for every k K an every two paths P 1, P 2 P k with f P1 > 0, l P1 (f) l P2 (f). In other wors, all flow-carrying s k -t k -paths have equal (an actually minimal) latency. In particular, equilibrium flows are fair, i.e., they have unfairness 1, if the unfairness of a feasible flow f is efine as max k K max{l P1 (f)/l P2 (f) : P 1, P 2 P k, f P1, f P2 > 0}. Main Results. While a user equilibrium can be compute in polynomial time (Beckmann, McGuire, an Winsten 1956), an so can a system optimum if x l a (x) is convex for all arcs a A, we show in Section 2 that it is an NP-har problem to compute a min-max flow. This result still hols if all latencies are linear an there is a single source-sink pair. Note that the flows that we are consiering are not require to be integer, neither on paths nor on arcs. As pointe out earlier, a Nash equilibrium has unfairness 1 by construction. In Section 3, we establish the somewhat surprising existence of a min-max flow that is fair too, when latencies are linear an there is a single source an a single sink. In aition, although it is well known that system optima are unfair, we provie a tight boun that quantifies the severity of this effect. This boun applies to general multicommoity flows an arbitrary latency functions. Finally, in Section 4, we show that in the single-source single-sink case uner arbitrary latency functions, there actually exist solutions that are simultaneously optimal or near optimal with respect to all three criteria (maximum latency, average latency, an unfairness). In fact, this property is share by the min-max flow, the system optimum an the user equilibrium, albeit with ifferent

4 COMPLEXITY, FAIRNESS, AND THE PRICE OF ANARCHY OF THE MAXIMUM LATENCY PROBLEM 3 maximum latency average latency unfairness min-max flow 1 4/3 Thm Thm. 5 system optimum 2 Thm Thm. 6 Nash equilibrium 4/3 Thm. 8 4/3 Thm. 7 1 Table 1. Summary of results for single-source single-sink networks with linear latency functions. The first entry in each cell represents a worst-case boun on the ratio of the value of the flow associate with the corresponing column to the value of an optimal flow for the objective function enote by the corresponing row. The secon entry refers to the theorem in this paper in which the respective result is prove. All bouns are tight, as examples provie after each theorem emonstrate. The boun of 4/3 on the ratio of the average latency of the user equilibrium to that of the system optimum was first prove by Roughgaren an Taros (2002); we give a simpler proof in Theorem 7. Weitz (2001) observe first that this boun carries forwar to the maximum latency objective for the case of only one source an sink; we present a generalization of this observation to multicommoity flows in Theorem 8. bouns. Table 1 presents the bouns obtaine for the three criteria in the single-source single-sink case with linear latencies. An important consequence of these results is that computing a user equilibrium or a system optimum constitutes a constant-factor approximation algorithm for the NP-har maximum latency problem. On the other han, alreay in networks with multiple sources an a single sink, the ratio of the maximum latency of a Nash equilibrium to that of the min-max flow is not boune by a constant, even with linear latency functions. Relate Work. Most papers on evacuation problems consier constant travel times; we refer the reaer to the surveys by Aronson (1989) an Powell, Jaillet, an Ooni (1995) for more etails. One notable exception is the work by Köhler an Skutella (2002). They consiere a ynamic quickest flow problem with loa-epenent transit times, for which they establishe strong NPharness. They also provie an approximation algorithm by consiering the average of a flow over time, which is a static flow. Köhler, Langkau, an Skutella (2002) propose to use time-expane networks to erive approximation algorithms for a similar problem. The concept of the price of anarchy, which is the ratio of the performance of a Nash equilibrium to that of an optimal solution, was introuce by Koutsoupias an Papaimitriou (1999) in the context of a game motivate by telecommunication networks. This inspire consierable subsequent work, incluing Mavronicolas an Spirakis 2001; Koutsoupias, Mavronicolas, an Spirakis 2002; Czumaj an Vöcking 2002; Czumaj, Krysta, an Vöcking These papers stuy the maximum latency of transmissions in two-noe networks consisting of multiple links connecting a single source with a single sink. Inee, uner certain assumptions, when users are selfish, the maximum latency is not too large compare to the best coorinate solution. Although these results are similar in nature to some of ours, their moel is not comparable to ours because they work with a finite number of players an consier mixe strategies. In contrast, in our setting, every player just controls an infinitesimal amount of flow, making mixe strategies irrelevant. Moreover, we work with arbitrary networks. For more etails on the various routing games, we refer the reaer to the excellent survey by Czumaj (2004). Roughgaren an Taros (2002), Roughgaren (2003), Schulz an Stier Moses (2003), an Correa, Schulz, an Stier Moses (2003) stuie the price of anarchy with respect to the average travel time in general networks an for ifferent classes of latency functions. In particular, if L is the set of allowable latency functions, the ratio of the average travel time of a user equilibrium to that of a system optimum is boune by α(l), where α(l) is a constant that only epens on L. For example, in case L only contains concave functions, α(l) = 4/3. We will later make use of this result (Section 4).

5 4 J.R. CORREA, A.S. SCHULZ, AND N.E. STIER MOSES For the maximum latency objective, Weitz (2001) was the first to observe that the price of anarchy is boune in single-source single-sink networks. He also presente a family of examples that showe that Nash equilibria can be arbitrarily ba in multiple commoity networks. Roughgaren (2004) gave a tight boun for the single-source single-sink case that epens on the size of the network. Game-theoretic concepts seem to offer an attractive way of computing approximate solutions to certain har problems. Inee, Anshelevich et al. (2003) approximate optimal solutions to a network esign problem that is NP-har with the help of Nash an approximate Nash equilibria. A relate iea was use by Fotakis et al. (2002) an Felmann et al. (2003) to show that although it is har to fin the best an worst equilibrium of the telecommunication game escribe before, there exists an approximation algorithm for computing a Nash equilibrium with minimal social cost. Correa et al. (2003) pursue the same iea by computing a provably goo Nash equilibrium in a setting with multiple equilibria in which computing the best equilibrium is har. In the context of Section 3, we shoul point out that there exist multiple (nonequivalent) efinitions of (un)fairness. The efinition we use here comes from the competition between ifferent agents in the routing game. Roughgaren (2002) efine unfairness as the ratio of the maximum latency of a system optimum to the latency of a user equilibrium; we later recover the bouns that he obtaine. Jahn et al. (2002) consiere the efinition of unfairness presente here; they looke for flows that minimize the total travel time among those with boune unfairness. 2. Computational Complexity In our moel, both the system optimum an the Nash equilibrium can be compute efficiently because they represent optimal solutions to certain convex programs. On the other han, it follows from the work of Köhler an Skutella (2002) on the quickest s-t-flow problem with loa-epenent transit times that the maximum latency problem consiere here is NP-har (though not necessarily in NP) when latencies inclue arbitrary nonlinear functions or when there are explicit arc capacities. Lemma 1 below implies that the general maximum latency problem is in NP, while Theorem 3 establishes its NP-harness, even in the case of linear latencies an a single source an a single sink. Note that the following result oes not follow from orinary flow ecomposition as it is not clear how to convert a flow on arcs into a path flow such that the latency of the resulting paths remains boune; in fact, it is a consequence of Theorem 3 that the latter problem is NP-har, too. Lemma 1. Let f be a feasible flow for a multicommoity flow network with loa-epenent arc latencies. Then there exists another feasible flow f such that L(f ) L(f), an f uses at most A paths for each source-sink pair. Proof. Consier an arbitrary commoity k K. Let P 1,..., P r be s k -t k -paths such that f Pi > 0 for i = 1,..., r, an r i=1 f P i = k. Slightly overloaing notation, we let P 1,..., P r also enote the arc incience vectors of these paths. Let s assume that r > A. (Otherwise we are one.) Hence, the vectors P 1,..., P r are linearly epenent an r i=1 λ ip i = 0 has a nonzero solution. Let s assume without loss of generality that λ r 0. We efine a new flow f (not necessarily feasible) by setting f P i := f Pi λ i λ r f Pr for i = 1,..., r, an f P := f P for all other paths P. Notice that uner f, the flow on arcs oes not change: r r 1 r 1 P i f P λ i r i = P i f Pi P i f Pr = P i f Pi. λ r i=1 i=1 i=1 Here, we use the linear epenency for the last equality. In particular, L(f ) L(f). Let us consier a convex combination f of f an f that is nonnegative an uses fewer paths than f. Note that such a flow always exists because f P r = 0, an the flow on some other paths P 1,..., P r 1 i=1

6 COMPLEXITY, FAIRNESS, AND THE PRICE OF ANARCHY OF THE MAXIMUM LATENCY PROBLEM 5 might be negative. Moreover, L(f ) L(f), too. If f still uses more than A paths between s k an t k, we can iterate this process so long as necessary to prove the claim. Corollary 2. The recognition version of the maximum latency problem is in NP. Proof. Lemma 1 shows the existence of a succinct certificate. Inee, there is a min-max flow using no more than K A paths. We are now reay to prove that the maximum latency problem is in fact NP-har. We present a reuction from Partition: Given: A set of n positive integer numbers q 1,..., q n. Question: Is there a subset I {1,..., n} such that i I q i = i I q i? Theorem 3. The recognition version of the maximum latency problem is NP-complete, even when all latencies are linear functions an the network has a single source-sink pair. Proof. Given an instance of Partition, we efine an instance of the maximum latency problem as follows. The network consists of noes 0, 1,..., n with 0 representing the source an n the sink. The eman is one. For i = 1,..., n, the noes i 1 an i are connecte with two arcs, namely a i with latency l ai (x) = q i x an ã i with latency lãi (x) = q i. Let L := 3 n 4 i=1 q i. Notice that the system optimum f has cost equal to L an fa = 1/2 for all a A. We claim that the given instance of Partition is a Yes-instance if an only if there is a solution to the maximum latency problem of maximum latency equal to L. Inee, if there is a partition I, the flow that routes half a unit of flow along the 0-n-path compose of arcs a i, i I, an ã i, i I, an the other half along the complementary path has maximum latency L. To prove the other irection, assume that we have a flow f of maximum latency equal to L. Therefore, C(f) L (there is unit eman), which implies that C(f) = L (it cannot be better than the optimal solution). As the arc flows of a system optimum are unique, this implies that f a = 1/2 for all a A. Take any path P such that f P > 0 an partition its arcs such that I contains the inices of the arcs a i P. Then, 3 n 4 i=1 q i = L = l P (f) = q i i I 2 + i I q i, an subtracting the left-han sie from the right-han sie yiels q i i I 4 = q i i I 4. Corollary 4. Let f be a (path) flow in an s-t-network with linear latencies. Let (f a : a A) be the associate flow on arcs. Given just (f a : a A) an L(f), it is NP-har to compute a ecomposition of this arc flow into a (path) flow f such that L(f ) L(f). In particular, it is NP-har to recover a min-max flow even though its arc values are given. Note that Corollary 4 neither hols for the system optimum nor the user equilibrium. In both cases any flow erive from an orinary flow ecomposition is inee an optimal flow respectively equilibrium flow. Let us finally mention that Theorem 4.3 in Köhler an Skutella (2002) implies that the maximum latency problem is APX-har when latencies can be arbitrary nonlinear functions or when there are explicit arc capacities. 3. Fairness User equilibria are fair by efinition. Inee, all flow-carrying paths between the same source an sink have equal latency. The next result establishes the same property for min-max s-t-flows in the case of linear latencies. Namely, a fair min-max flow always exists. Therefore, the ifference between a Nash equilibrium an a min-max flow is that the latter may leave paths unuse that are shorter than the ones carrying flow, a situation that cannot happen in equilibrium. This result is not true for nonlinear latencies, as we shall see later.

7 6 J.R. CORREA, A.S. SCHULZ, AND N.E. STIER MOSES Theorem 5. Every instance of the single-source single-sink maximum latency problem with linear latency functions has an optimal solution that is fair. Proof. Consier an instance with eman an latency functions l a (f a ) = q a f a + r a, for a A. Among all min-max flows, let ˆf be the one that uses the smallest number of paths. Let P 1, P 2,..., P k be these paths. Consier the following linear program: min s.t. z (1a) ( ) ) (q a f Ph + ra z for i = 1,..., k, (1b) a P i P h a k f Pi = i=1 (1c) f Pi 0 for i = 1,..., k. (1) Note that this linear program has k + 1 variables. Furthermore, by construction, it has a feasible solution with z = L( ˆf), an there is no solution with z < L( ˆf). Therefore, an optimal basic feasible solution gives a min-max flow that satisfies with equality k of the inequalities (1b) an (1). As f Pi > 0 for all i because of the minimality assumption, all inequalities (1b) have to be tight. A byprouct of this proof is that an arbitrary flow can be transforme into a fair one without increasing its maximum latency. In fact, just solve the corresponing linear program. An optimal basic feasible solution will either be fair or it will use fewer paths. In the latter case, eliminate all paths carrying zero flow an repeat until a fair solution is foun. Notice that the min-max flow may not be fair for nonlinear functions. Inee, the instance isplaye in Figure 1 features high unfairness with latencies that are polynomials of egree p. x p x p PSfrag replacements 1 1 ax p + b ax p + b Figure 1. Instance with nonlinear latencies illustrating that fair min-max flows may not exist. p 1 When a = (1+ε) p 1 an b = 2 ( 1+ε 2+ε) δ for some ε > 0 an δ > 0 such that b > 1, the minmax flow routes 1 2+ε units of flow along the top-bottom an bottom-top paths, respectively, an ε 2+ε units of flow along the top-top path. It is not har to see that this flow is optimal. Inee, the bottom-bottom path is too long to carry any flow. Moreover, by symmetry, the topbottom an bottom-top paths have to carry the same amount of flow. Therefore, the optimal solution can be compute by solving a one-imensional minimization problem, whose only variable is the amount x of flow on the top-top path. The unique optimal solution to this problem is x = ε 2+ε. Let us compute the unfairness of this solution. The top-top path has latency equal to 2 ( 1+ε 2+ε) p, which tens to ( 1 2) p 1 as ε 0. The latency of the other two paths use by the optimum is equal to 2 δ. Therefore, the unfairness of this min-max flow is arbitrarily close to 2 p. A typical argument against using the system optimum in the esign of route-guiance evices for traffic assignment is that, in general, it assigns some rivers to unacceptably long paths in orer

8 COMPLEXITY, FAIRNESS, AND THE PRICE OF ANARCHY OF THE MAXIMUM LATENCY PROBLEM 7 to use shorter paths for most other rivers; see, e.g., Beccaria an Bolelli (1992). The following theorem quantifies the severity of this effect by characterizing the unfairness of the system optimum. It turns out that there is a relation to earlier work by Roughgaren (2002), who compare the maximum latency of a system optimum in a single-sink single-source network to the latency of a user equilibrium. He showe that for a given class of latency functions L, this ratio is boune from above by γ(l), which is efine to be the smallest value that satisfies l a(x) γ(l)l a (x) for all l L an all x 0. Here, l a(x) := l a (x) + x l a(x) is the function that makes a system optimum for the original instance a user equilibrium of an instance in which the latencies are replace by l (Beckmann, McGuire, an Winsten 1956). For instance, γ(polynomials of egree p) = p+1. We prove that the unfairness of a system optimum is in fact boune by the same constant, even for general instances with multiple commoities. The same result was inepenently obtaine by Roughgaren (personal communication, October 2003). Theorem 6. Let f be a system optimum in a multicommoity flow network with arc latency functions rawn from a class L. Then, the unfairness of f is boune from above by γ(l). Proof. We will prove the result for the single-source single-sink case. The extension to the general case is straightforwar. As a system optimum is a user equilibrium with respect to latencies l, there exists L such that l P (f ) = L for all paths P P with fp > 0. From the efinitions of l an γ(l), we have that l a (x) l a(x) γ(l)l a (x) for all x. Let P 1, P 2 P be two arbitrary paths with fp 1, fp 2 > 0. Hence, l P1 (f ) L an l P2 (f ) L /γ(l). It follows that l P1 (f )/l P2 (f ) γ(l). Notice that Theorem 6 implies Roughgaren s earlier boun for the single-source single-sink case. Inee, for a Nash equilibrium f, min{l P (f ) : P P, fp > 0} min{l P (f) : P P, f P > 0}. Otherwise, C(f ) > C(f), which contraicts the optimality of f. In aition, the example shown in Figure 2 proves that the boun given in Theorem 6 is tight. Inee, it is easy to see that the system optimum routes half of the eman along each arc, implying that the unfairness is l (/2)/l(/2). Taking the supremum of that ratio over 0 an l L, we get γ(l). l (/2) PSfrag replacements l(x) Figure 2. Instance showing that Theorem 6 is tight. 4. Price of Anarchy an Relate Approximation Results Nash equilibria in general an user equilibria in particular are known to be inefficient, as evience by Braess paraox (1968). Koutsoupias an Papaimitriou (1999) suggeste measuring this egraation in performance, which results from the lack of central coorination, by the worst-case ratio of the value of an equilibrium to that of an optimum. This ratio has become known as the price of anarchy, a phrase coine by Papaimitriou (2001). It is quite appealing (especially for evacuation situations) that in the routing game consiere here, the price of anarchy is small; i.e., the selfishness of users actually rives the solution close to optimality. Recall that the user equilibrium results from everyboy choosing a shortest path uner the prevailing congestion conitions. Since a user equilibrium can be compute in polynomial time, this also leas to an approximation algorithm for the maximum latency problem.

9 8 J.R. CORREA, A.S. SCHULZ, AND N.E. STIER MOSES In orer to erive a boun on the price of anarchy for the maximum latency objective, we use a corresponing boun for the average latency of Nash equilibria, which was first prove for linear latency functions by Roughgaren an Taros (2002) an then extene to ifferent classes of latency functions by Roughgaren (2003) an Correa, Schulz, an Stier Moses (2003). For the sake of completeness, let us inclue a simpler proof of Roughgaren an Taros result (see also Correa et al. 2003). Theorem 7 (Roughgaren an Taros 2002). Let f be a user equilibrium an let f be a system optimum in a multicommoity flow network with linear latency functions. Then C(f) 4 3 C(f ). Proof. Let l a (x) = q a x + r a with q a, r a 0 for all a A. Then, C(f) = (q a f a + r a )f a (q a f a + r a )fa (q a fa + r a )fa + 1 q a fa 2 C(f ) C(f). a A a A a A a A The first inequality hols since the equilibrium flow f uses shortest paths with respect to the arc latencies cause by itself. The secon inequality follows from (fa f a /2) 2 0. In general, ( { ( ) x l() l(x) }) 1 C(f) α(l)c(f ), where α(l) := 1 sup, (2) l L, 0 x l() an the proof is similar to the one given above for Theorem 7; see Roughgaren (2003) an Correa et al. (2003) for etails. For polynomials with nonnegative coefficients of egree 2, α(l) equals 1.626; for those with egree 3, α(l) = 1.896; in general, α(l) = Θ(p/ ln p) for polynomials of egree p. It was first note by Weitz (2001) that in networks with only one source an one sink, any upper boun on the price of anarchy for the average latency is an upper boun on the price of anarchy for the maximum latency. We inclue a multicommoity version of this result. Theorem 8. Consier a multicommoity flow network with latency functions in L. Let f be a Nash equilibrium an ˆf a min-max flow. For each commoity k K, L k (f) k α(l)l( ˆf), where L k is the maximum latency incurre by commoity k, k is its eman rate, an is the total eman. Proof. Let f be the system optimum. Then, k L k (f) C(f) α(l)c(f ) α(l)c( ˆf) α(l)l( ˆf). Here, the first inequality hols because f is a Nash equilibrium, the secon inequality is exactly Equation (2), the thir one comes from the optimality of f, an the last one just says that the average latency is less than the maximum latency. This implies that, for the single-source single-sink case, computing a Nash equilibrium is an α(l)-approximation algorithm for the maximum latency problem. Notice that this guarantee is PSfrag replacements l(x)/2 l()/2 0 l()/2 l(x)/2 Figure 3. Instance showing that Theorem 8 is tight for single-commoity networks. tight as shown by the example given in Figure 3, which goes back to Braess (1968). Inee, the

10 COMPLEXITY, FAIRNESS, AND THE PRICE OF ANARCHY OF THE MAXIMUM LATENCY PROBLEM 9 latency of a Nash equilibrium is l() while the maximum latency of a min-max flow, which coincies with the system optimum, is { x ( ) } PSfrag replacements l() max l() l(x). 0 x Taking the supremum over 0 an l L, the ratio of the latency of the Nash equilibrium to that of the min-max flow is arbitrarily close to α(l). For instances with multiple sources an a single sink, the maximum latency of a user equilibrium is unboune with respect to that of an optimal solution, even with linear latencies. In fact, we will show that the price of anarchy cannot be better than Ω(n), where n is the number of noes in the network. Note that this also implies that the price of anarchy is unboune in single-source single-sink networks with explicit arc capacities. Weitz (2001) showe that the price of anarchy is unboune in the case of multiple commoities, an Roughgaren (2004) prove that it is boune by n 1 if there is a common source an sink. Theorem 9. The price of anarchy in a single-commoity network with multiple sources an a single sink is Ω(n), even if all latencies are linear functions a n a 2 a 1 ε 2 ε n ε n 1 ε n n ã n x/ε n x/ε 2 x/ε ã 2 ã 1 ε + + ε n Figure 4. Instance showing that Nash equilibria can be arbitrarily ba when multiple sources are present. Proof. Fix a constant ε > 0 an consier the instance presente in Figure 4. Noes n, n 1,..., 1 are the sources while noe 0 is the sink. Noes i an i 1 are connecte with two arcs: a i with constant latency equal to 1 an ã i with latency equal to x/ε i. Let the eman entering noe i > 0 be ε i. The user equilibrium of this instance routes the flow along paths of the form ã i, a i 1,..., a 1 an has maximum latency n. To show the claim, it suffices to exhibit a goo solution. For instance, for origin i, let its eman flow along the path a i, ã i 1,..., ã 1. Uner this flow, the loa of ã i is equal to ε i ε n an its traversal time is (ε i ε n )/ε i = ε ε n i. Hence, we can boun the maximum latency from above by 1 + nε 1 ε, which tens to 1 when ε 0. In the single-source single-sink case, not only Nash equilibria represent goo approximations to the maximum latency problem; an immeiate corollary of Theorem 6 is that system optima are also close to optimality with respect to the maximum latency objective. Theorem 10. For single-source single-sink instances with latency functions rawn from L, computing a system optimum is a γ(l)-approximation algorithm for the maximum latency problem. Proof. Theorem 6 states that the length of a longest path use by the system optimum f is at most γ(l) times the length of a shortest flow-carrying path. The latter value cannot be bigger than the maximum latency of a path use by the min-max flow because f is optimal for the average latency; the result follows. The boun given in Theorem 10 is best possible. To see this, consier the instance epicte in Figure 5. The min-max flow routes the entire eman along the lower arc, for a small enough ε > 0. On the other han, the unique system optimum has to satisfy l (f ) = l () ε, where f is

11 10 J.R. CORREA, A.S. SCHULZ, AND N.E. STIER MOSES l () ε PSfrag replacements l(x) Figure 5. Instance showing that Theorem 10 is tight. the flow along the lower arc. Therefore, the upper arc has positive flow an the maximum latency is l () ε. The ratio between the maximum latencies of the two solutions is arbitrarily close to l ()/l(). Taking the supremum over 0 an l L shows that the boun in Theorem 10 is tight. To complete Table 1, let us prove that the average latency of the min-max flow is not too far from that of the system optimum. Theorem 11. Let ˆf be a min-max flow an let f be a system optimum for an instance with a single source, a single sink an latencies rawn from L. Then, C( ˆf) α(l)c(f ). Proof. Note that C( ˆf) L( ˆf) L(f) = C(f) α(l)c(f ), where f is the Nash equilibrium of the instance. l() + ε PSfrag replacements l(x) Figure 6. Instance showing that Theorem 11 is tight. Again, the guarantee given in the previous theorem is tight. To show this, it is enough to note that the equilibrium flow an the min-max flow coincie in the example of Figure 6, an their average latency is l(). Moreover, the average latency of the system optimum is arbitrary close to l() max 0 x { x (l() l(x)) } Taking the supremum of the ratio of these two values over 0 an l L completes the argument. In Table 2, we summarize the finings for single-source single-sink networks with latencies rawn from a given class L of allowable latency functions. maximum latency average latency unfairness min-max flow 1 α(l)? system optimum γ(l) 1 γ(l) user equilibrium α(l) α(l) 1 Table 2. Overview of approximation guarantees for single-source singlesink networks when latencies belong to a given set L. All bouns are tight. The? inicates that no upper boun is known; recall from the example epicte in Figure 1 that 2 p is a lower boun for polynomials of egree p..

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