The Price of Atomic Selfish Ring Routing∗

Bo Chen† Xujin Chen‡ Xiaodong Hu§

May 2008

Abstract

We study selfish routing in ring networks with respect to minimizing the maximum la-tency. Our main result is an establishement of constant bounds on the price of stability(PoS) for routing unsplittable flows with linear latency. We show that the PoS is at most6.83, which reduces to 4.57 when the linear latency functions are homogeneous. We alsoshow the existence of a (54,1)-approximate Nash equilibrium. Additionally we address somealgorithmic issues for computing an approximate Nash equilibrium.

1 Introduction

A major component of large-scale network systems is the routing mechanism, namely choosing

communication paths between sources and destinations of traffic. In choosing a routing path,

a typical objective is to minimize the maximum latency. In many of these network systems

it is impossible to maintain a central authority that imposes efficient routing strategies on the

network traffic. As a result, users act independently and “selfishly”: each user tries to minimize

his own traffic latency based on current network traffic.

This problem can be mathematically formalized in terms of classical game theory as follows.

The network users are viewed as independent players participating in a non-cooperative game.

(In this paper we use terms “user” and “player” exchangeably.) Each player, with his own

pair of source and destination in the network, wishes to establish a communication between

the source and the destination using one or more paths with latency as low as possible, given

the link congestion caused by other players. We are interested in situations where the system

has reached some kind of stable state. The most popular notion of stability in non-cooperative

game theory is the Nash equilibrium: a “stable point” for the players, from which no player has∗To appear in Journal of Combinatorial Optimization†Corresponding author: Warwick Business School (WBS) and Centre for Discrete Mathematics and its Appli-

cations (DIMAP), University of Warwick, Coventry, CV4 7AL, United Kingdom, b.chen@warwick.ac.uk‡Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China, xchen@amss.ac.cn§Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China, xdhu@amss.ac.cn

1

the incentive to deviate unilaterally. It is well known that Nash equilibria do not in general

optimize the social welfare and can be far from the global optimum. In their seminal paper

[14], Koutsoupias and Papadimitriou proposed to analyze the performance degradation due to

lack of players’ coordination from a worst-case perspective; this leads to the notion of price of

anarchy (PoA), which is the ratio between the worst social cost of a Nash equilibrium to the

cost of social optimum. A more recent measure, the price of stability (PoS), was defined in [2] to

capture the gap between the best possible Nash equilibrium and the globally optimal solution.

This measure becomes more and more prevalent as it measures the minimum penalty in network

performance required to ensure a stable (equilibrium) outcome, and thus it is crucial from the

network designer’s perspective, who would like to propose (rather than let the players end up

in) a Nash equilibrium that is as close to the global optimum as possible.

The ring topology is a fundamental topology that is frequently encountered in communication

networks, and thus it has attracted considerable attention in research community [1, 20, 22, 4].

However, even in a ring, the simplest 2-connected network, the problem of choosing routes in

response to communication requests is not trivial. Additionally, in many real networks that

involve ring topology the traffic demand from a source to a destination must be satisfied by

choosing a single path between the source and the destination. For example, splitting the

traffic causes the problem of packet reassembly at the receiver and thus is generally avoided [3].

Motivated by the wide practical applications, we study the unsplittable model (unless otherwise

noted explicitly) in this paper, and investigate the deterioration of ring network performance

measured in maximum traffic latency under the selfish user behaviors.

The general model

Consider a network G = (V, E) with node set V , link set E, and source-destination node pairs

(si, ti),i = 1, . . . , k, where G is called a single-commodity network if k = 1 and a multi-commodity

network if k ≥ 2. For each i, the nonempty set Pi consists of paths with ends si and ti, called

si-ti paths, in G; in addition there is one unit of traffic to be routed from si to ti through path(s)

in Pi. A (feasible) flow f for this network is a nonnegative real function on P = ∪kj=1Pj with∑

P∈PifP = 1 for every i = 1, 2, . . . , k. Each link e ∈ E bears a load with respect to f defined

as fe =∑

P∈P:e∈E(P ) fP , the sum of flow along e; and each e ∈ E is associated with a load-

dependent latency (function) le(·), meaning that every path P in G with e ∈ E(P ) experiences

latency le(fe) along e. The latency of a path P in G with respect to flow f is thus defined as

lP (f) =∑

e∈E(P ) le(fe). For flow f , we use Mi(f) = maxP∈Pi:fP >0 lP (f) to denote the maximum

2

latency experienced by traffic from si to ti (i.e., by user i), and use

M(f) ≡ max1≤i≤k

Mi(f) = maxP∈P:fP >0

lP (f) (1.1)

to denote the maximum latency experienced by all network traffic. We call Mi(f) the maximum

latency of user i (w.r.t. f) and M(f) the maximum latency of the network or overall traffic. A

selfish routing model is then specified by a triple (G, (si, ti)ki=1, l), and it captures the setting

where each user wishes to minimize his own maximum latency while the network designer (for

social welfare) aims at minimizing the maximum latency of the network. Note that selfish

routing is naturally generalized to weighted selfish routing which requires that the amount of

flow routed from si to ti be a given integer di, i = 1, 2, . . . , k, in stead of just one unit as in

(unweighted) selfish routing.

A network game/routing is said atomic if there are finitely many players, each controlling

a non-negligible amount of flow (in unweighted setting, that is one unit). An atomic routing is

unsplittable if every player must route his flow along a single path [3, 6, 12, 19]; it is splittable if

players are permitted to route their flow fractionally [7]. In contrast, a network game/routing

is said nonatomic if every player controls a negligible portion of the overall traffic so that the

actions of a single individual have negligible impact on the latency caused by others.

Related results

The selfish routing model falls within the general framework of congestion game [17], which has

the fundamental property that a Nash equilibrium always exists in pure strategies. On the other

hand, it has been shown that finding a Nash equilibrium for multi-commodity congestion games

is PLS -complete [10], though a pseudo-polynomial-time algorithm is available for computing a

Nash equilibrium in any atomic congestion game with linear latency functions [12].

When the maximum latency is to be minimized, the PoA of atomic congestion games with

linear latency is 2.5 in single-commodity networks, but it explodes to Θ(√

k) in multi-commodity

networks [6]. Analogously, for nonatomic weighted selfish routing with linear latency, recent work

by Correa et al. [8] has shown the existence of an optimal flow in single-commodity network

that is “fair”. (Remark: “fairness” does not necessarily imply “equilibrium” though they do

bear much similarity). They also proved that it is NP-hard to find an optimal flow in single-

commodity networks, and that the PoS can be unbounded in multi-source-single-sink networks.

The PoA and PoS depend not only on the game itself, but also on the definition of the social

(or system) objective. Previous works in [19, 18, 3, 6] have quantified how much the average

latency of traffic at a Nash equilibrium can exceed that of an optimal solution. Roughgarden

3

[18] proved that, as far as average latency is concerned with nonatomic players, it is actually the

class of allowable latency functions, not the specific topology of the network, that determines

the PoA.

Recently, Busch and Magdon-Ismail [5] studied atomic unsplittable network game/routing

from a bottleneck point of view, where players choose a path with the objective of minimizing the

maximum congestion along the edges of their path; and the social cost is the global maximum

congestion over all links in the network. They showed that the PoS = 1 and PoA = O(`+log n),

where ` is the length of the longest path in the player strategy sets, and n is the size of the

network. The bottleneck objective for nonatomic routing was discussed in [8] with emphasis on

its difference from maximum/average latency objective.

Our contributions and their significance

We focus on the problem of selfish unsplittable routing in ring networks with atomic players and

linear latency, which we refer to as Selfish Ring Routing (SRR). We prove that the PoS of the

SRR problem is at most 6.83 and is at most 4.57 if the linear latency functions are homogenous.

On the other hand, we show that there exists an optimal solution which is a 54-approximate

Nash equilibrium. Our theoretical results lead to pseudo-polynomial time algorithms for finding

solutions of good balance between efficiency and stability.

The vast majority of the work on bounding the PoA and PoS in routing games has been

focused on the criterion of the average latency of all players and on that of the maximum latency

for single-commodity networks, with very few results for multi-commodity networks [3, 16], which

we study in this paper. Our work on ring topology breaks previous restriction to parallel-link

networks or layered networks [14, 9, 12].

Our bounds on the PoA and PoS for ring routing are constants, independent of the network

size and the number of players, which stands in contrast to the unbounded PoA for general

networks [21, 8] with nonatomic players. Based on an elegant example in [21], we further exhibit

below a complementary example, which shows unbounded PoS in general atomic unsplittable

routing with linear latency.

In the selfish routing instance (G, (si, ti)ki=1, l) depicted in Figure 1, the underlying directed

network G = (V,E) has 2h + 4 (h ≥ 4) nodes s1, t1, s, t, u1, v1, u2, v2, . . . , uh, vh and k = 1 + h3

source-destination pairs with s2 = s3 = · · · = sk = s and t2 = t3 = · · · = tk = t. The latency

function on the top link ej (resp. bottom link e′j) from uj to vj , j = 1, 2, . . . , h, is lej (fej ) = hfej

(resp. le′j (fe′j ) = fe′j ). All the other links have zero latency. Evidently, the minimum maximum

latency h2 is realized by the optimal flow f∗ in which one unit of flow between s1 and t1 is

4

routed along the top links, and h3 units of flow between s and t is divided evenly between h

paths, sujvjt, going through the bottom link e′j , j = 1, 2, . . . , h. Let f be any Nash flow. It is

easily seen that the flow between s and t does not go through the link from vj to uj+1 to avoid

unnecessary additional latency and hence fvjuj+1 = 1 for any j = 1, 2, . . . , h − 1. Suppose first

that fsuj ≤ h2 − 2 for some j ∈ 1, 2, . . . , h. Then flow conservation implies that (i) either

fej < h− 1 or fe′j < h(h− 1) and (ii) fsuj′ ≥ h2 + 2 for some j′ ∈ 1, 2, . . . , h − j. From (ii)

we conclude that, between s and t, either at least 1h+1(h2 + 2) > h − 1 units of flow is routed

through ej′ , or at least hh+1(h2 + 2) > h(h− 1) units of flow is routed through e′j′ , which implies

that Mi(f) > h(h−1) for some i ∈ 2, 3, . . . , k. It follows from (i) that player i could reduce his

own latency to a value no more than h(h− 1) by unilaterally changing his strategy to the path

sujvjt, which goes through ej if fej < h− 1 and through e′j if fe′j < h(h− 1). This contradicts

the fact that f is a Nash flow. Therefore, fsuj ≥ h2 − 1 for all j ∈ 1, 2, . . . , h. This together

with flow conservation makes impossible, for any j′, either fej′ < h− 1 or fe′j′

< h(h− 1), since

otherwise some player would benefit by switching his own flow from ej′ to e′j′) or vise versa.

Therefore, fej ≥ h − 1 and fe′j ≥ h(h − 1) for all j = 1, 2, . . . , h, which implies that PoS =

M(f)/M(f∗) ≥ M1(f)/h2 ≥ h2(h− 1)/h2 = h− 1 has lower bound Ω( 3√

k).

Figure 1. Selfish routing instance with unbounded PoS.

Our results demonstrate salient difference between the selfish routing for minimum maximum

latency and that for minimum average/total latency in that network topology does play an

important role for the former, while makes almost no difference in the latter.

Paper organization

After preliminaries in Section 2, we present in Sections 3 and 4 some upper bounds on PoS

and PoA of the SRR problem and the extent to which an optimal solution can be close to the

Nash equilibrium. We then discuss in Section 5 algorithmic issues of finding efficient and stable

solutions of the SRR problem. We conclude the paper in Section 6 with remarks on future

research.

5

2 Preliminaries

Selfish ring routing with linear latency. As the name suggested, the underlying network

of the selfish ring routing (SRR) model is a ring R = (V,E) which is a (undirected) cycle in

terminology of graph theory (see Figure 2(a) for an illustration, where V = v1, v2, . . . , vnand E = ei = vivi+1 : i = 1, 2, . . . , n). The ring size is defined as |V | = n. For any two

ordered nodes u, v ∈ V , we denote by R[u, v] the clockwise path in R between u, v, and set

R(u, v] = R[u, v]\u, R[u, v) = R[u, v]\v and R(u, v) = R[u, v]\u, v. Given a path P in R,

let x be a node or edge on P , we write x ∈ P , or x ∈ V (P ) or x ∈ E(P ) to avoid confusion.

( ) =( , )a R V E

v1 v2

vn

vn-1

v3

v4

e1

e2

e3en-1

en

vn+1=s1

s2

l xe1( )=

t2

t1

x

l xe3( )= x

l xe2( )=x

4-l xe4( )=x

4-

( ) > , =b PoA 2 PoS 1 ( ) = = /C PoS 8 7PoA

s1 s2

l xe1( )=

t2 t1

x

l xe2( )=2l xe4( )= x5-7

l xe3( )= x2-3

e1

e2

e3

e4

e1

e2

e3

e4

Figure 2. Undirected ring and SRR instances with two players

There are k users 1, 2, . . . , k in the SRR instance (R, (si, ti)ki=1, l), where each user i has commu-

nication request for routing one unit of flow from his source si ∈ V to his destination ti ∈ V , and

his strategy set Pi consists of two link-disjoint si-ti paths Pi and Pi in R, satisfying Pi∪ Pi = R.

For convenience, let ¯Pi = Pi, i = 1, 2, . . . , k. Given feasible (unsplittable) flow f ∈ 0, 1P with

P = ∪ki=1Pi, Pi, in view of the correspondence between f ∈ 0, 1P and user strategies adopted

for the SRR on (R, (si, ti)ki=1, l), we abuse the notation slightly by writing f = Q1, Q2, . . . , Qk

with the understanding that, for each i = 1, 2, . . . , k, Qi ∈ Pi, Pi, f(Qi) = 1 and the one unit

of flow requested by user i is routed along Qi.

The latency le(·) of each link e ∈ E is always assumed to be linear in the link load fe which

equals the number of paths in Q1, Q2, . . . , Qk each going through e. More precisely, for all

e ∈ E, le(fe) = aefe + be, where ae and be are nonnegative reals. The latency l is said to be

homogenous if all be = 0. For any path P in R, possibly P = R, define

laP (f) =∑

e∈E(P )

aefe, ||P ||a =∑

e∈E(P )

ae, ||P ||b =∑

e∈E(P )

be, ||P || = ||P ||a + ||P ||b.

We write Pi = Pi, Pi such that ||Pi|| ≤ ||Pi|| for every i = 1, 2, . . . , k. Note that

||Pi||+ ||Pi|| = ||R||, for every i = 1, 2, . . . , k. (2.1)

6

We say that user i adopts the short path strategy if ||Qi||a ≤ ||Qi||a and the long path strategy

otherwise.

Recall that, in the SRR, users are non-cooperative and each user i wishes to minimize his

own maximum latency Mi(f) in feasible flow f with no regard to the global optimum. We

denote by 〈f〉 the set of users i ∈ 1, 2, . . . , k with Mi(f) = M(f).

Nash equilibria. A Nash equilibrium is characterized by the property that no user has the

incentive to change his strategy unilaterally. Recall that, as a congestion game, selfish ring

routing always allows for a Nash equilibrium in pure strategies. We say that a feasible flow

f = Q1, Q2, . . . , Qk for SRR is at Nash equilibrium or simply call it a Nash (flow) if the

following inequality holds true for every i ∈ 1, 2, . . . , k

lQi(f) ≤∑

e∈E(Qi)

le(fe + 1). (2.2)

For real number α ≥ 1, flow f is called an α-approximate Nash flow if the following inequality

holds true for every i ∈ 1, 2, . . . , k and P ∈ Pi with fP > 0

lP (f) ≤ α∑

e∈E(P )

le(fe + 1). (2.3)

If, in addition, for real number β ≥ 1, the maximum latency M(f) of f is at most β times the

maximum latency OPT achieved by an optimum flow/routing, i.e., M(f) ≤ βOPT, then f is called

an (α, β)-approximate Nash (flow).

Let F be the set of Nash flows for the SRR on (R, (si, ti)ki=1, l). The price of anarchy (PoA)

and the price of stability (PoS) of the SRR instance is given respectively by

ρA(R, (si, ti)ki=1, l) = max

f∈FM(f)OPT

and ρS(R, (si, ti)ki=1, l) = min

f∈FM(f)OPT

. (2.4)

Correspondingly, the PoA (resp. PoS) of the SRR problem is set to be the maximum of

ρA(R, (si, ti)ki=1, l) (resp. ρS(R, (si, ti)k

i=1, l) ) over all SRR instances (R, (si, ti)ki=1, l).

As an example, consider the ring R = (V, E) in Figure 2(b) with V = v1, v2, v3, v4 and

E = e1, e2, e3, e4. The SRR instance (R, (si, ti)i=1,2, l) is given by s1 = v1, t1 = v2, s2 = v3,

t2 = v4, and le1(x) = le3(x) = x, le2(x) = le4(x) = x4 , where x ∈ R+. For i = 1, 2, let

Qi be the clockwise path in R from si to ti. It is easily checked that both f∗ = Q1, Q2and f = Q1, Q2 are Nash flows, and f∗ is an optimum flow for the SRR instance. Hence

ρA(R, (si, ti)i=1,2, l) ≥ M(f)/M(f∗) = 2/1 = 2 and ρS(R, (si, ti)i=1,2, l) = 1. For the SRR

7

instance depicted in Figure 2(c), via enumeration of all four feasible flows, we see that its

optimal flow f∗ = s1s2t1, s2t1t2 has maximum latency M(f∗) = 3.5 while its unique Nash

flow f = s1s2t1, s2s1t2 has M(f) = 4. Hence the PoA and PoS of this instance both equal to

8/7.

The main purpose of the next two sections is to bound from above α and β on any (α, β)-

approximate Nash flow for the SRR problem. Our analysis leads to constant bounds of (1,6.83)

and (54,1), providing indication of the worst-case performance of inefficiency of Nash equilibria

and instability of the optimal solutions, respectively.

3 Inefficiency of Nash equilibria

The main result of this section is the following upper bounds on the PoS. In terms of (α, β)

approximation, we bound β while keeping α = 1.

Theorem 1 The price of stability of the SRR is at most 6.83 and is at most 4.57 if the linear

latency functions are homogenous.

To avoid triviality, we assume in our analysis that there exists an optimal flow f∗ =

Q1, Q2, . . . , Qk for the SRR instance (R, (si, ti)ki=1, l) that is not a Nash flow. Therefore

some user i′ ∈ 1, 2, . . . , k can benefit from unilaterally changing his strategy provided strate-

gies of other users remain the same, which implies that the SRR has a feasible flow f ′ =

Q1, . . . , Qi′−1, Qi′ , Qi′+1, . . . , Qk satisfying lQi′(f∗) + ||Qi′ ||a = lQi′

(f ′) < lQi′ (f∗) ≤ M(f∗),

where the equality is directly based on the definition of new flow f ′. Therefore,

lR(f∗) = lQi′ (f∗) + lQi′

(f∗) < 2M(f∗)− ||Qi′ ||a. (3.1)

Since lQi′(f ′) ≥ ||Qi′ || and lQi′ (f

∗) ≥ ||Qi′ ||, it follows from (2.1) that

||Qi||+ ||Qi|| = ||R|| = ||R||a + ||R||b = ||Qi′ ||+ ||Qi′ || < 2M(f∗) for i = 1, 2, . . . , k. (3.2)

Let f be an arbitrary Nash flow. Suppose, without loss of generality, that users 1, 2, . . . , k are

ordered such that for some 1 ≤ j ≤ k, f = Q1, Q2, . . . , Qj , Qj+1, . . . , Qk. Notice that

lQi(f) + lQi(f) = lR(f) for i = 1, 2, . . . , k. (3.3)

By (2.2), we have

lQi(f) ≤

∑

e∈E(Qi)

le(fe + 1) =∑

e∈E(Qi)

(aefe + ae + be) = lQi(f) + ||Qi||a for i = 1, 2, . . . , j. (3.4)

8

Similarly,

lQi(f) ≤ lQi(f) + ||Qi||a for i = j + 1, j + 2, . . . , k. (3.5)

It is instant from (3.3)–(3.5) and (1.1) that

M(f) ≤ maxQ∈f

lQ(f) + lQ(f) + ||Q||a2

≤ lR(f) + ||R||a2

, (3.6)

and from which, along with (3.2), we obtain

M(f) ≤ lQi(f) +||Qi||a

2+||R||a

2, for i = 1, 2, . . . , j. (3.7)

In what follows, we break the proof of Theorem 1 into two lemmas: the first deals with

homogeneous latency case, whose proof technique is then extended to establishing the second

lemma dealing with the general linear latency case.

Lemma 1 Given any SRR instance with homogeneous linear latency functions, either every

optimal flow is a Nash flow, or the price of anarchy of the instance is at most γ0 = 5+√

172 ≤ 4.57.

Proof. For homogeneous linear latency functions, we have ||·|| = ||·||a. Without loss of generality

assume that γ = ||Q1||||Q1|| = maxj

i=1||Qi||||Qi|| ≥ 1. (Note that if γ < 1, then M(f) ≤ lR(f) < lR(f∗) <

2M(f∗) by (3.1) and we are done.) By (3.2), ||Q1|| = γ||Q1|| implies

||Q1|| = ||R||γ + 1

. (3.8)

Due to homogeneous linearity of the latency function, we have lR(f) = lR(f∗) +∑j

i=1(||Qi|| −||Qi||), from which we can bound lR(f) as follows depending on the value of i′ in inequality

(3.1). If i′ ≤ j, then

lR(f) ≤ 2M(f∗)− ||Qi′ ||+j∑

i=1

(||Qi|| − ||Qi||)

≤ 2M(f∗)− ||Qi′ ||+ (γ − 1)j∑

i6=i′,i=1

||Qi||

≤ 2M(f∗)− γ||Qi′ ||+ (γ − 1)lR(f∗)

≤ 2M(f∗)− γ||Qi′ ||+ (γ − 1)(2M(f∗)− ||Qi′ ||)= 2γM(f∗)− (γ − 1)||R|| − ||Qi′ ||≤ 2γM(f∗)− (γ − 1)||R||.

9

If i′ > j, we can similarly obtain

lR(f) ≤ 2M(f∗)− ||Qi′ ||+ (γ − 1)j∑

i=1

||Qi||

≤ 2M(f∗)− (γ − 1)||Qi′ ||+ (γ − 1)(j∑

i=1

||Qi||+ ||Qi′ ||)

≤ 2M(f∗)− (γ − 1)||Qi′ ||+ (γ − 1)lR(f∗)

≤ 2M(f∗)− (γ − 1)||Qi′ ||+ (γ − 1)(2M(f∗)− ||Qi′ ||)= 2γM(f∗)− (γ − 1)||R||.

Therefore, from the above analysis and (3.6), we derive

M(f) ≤ γM(f∗)− γ − 22

||R||. (3.9)

To prove the lemma, we can assume γ ≥ γ0 thanks to (3.9). Recall the assumption that

||Pi|| ≤ ||Pi|| for i = 1, 2, . . . , k. It is easy to see that the ring load lR(f∗) with respect to the

optimal flow f∗ has the following lower bound:

lR(f∗) ≥k∑

i=1

||Pi||. (3.10)

Among all users whose strategies contribute to lQ1(f), some adopt short path strategies and their

contributions sum up to cs, while the others adopt long path strategies and their contributions

sum up to cl. In other words, lQ1(f) = cs+cl. Observe from (3.10) that cs ≤∑k

i=1 ||Pi|| ≤ lR(f∗).

Hence by (3.1), we have

cl ≥ lQ1(f)− lR(f∗) > lQ1(f)− 2M(f∗). (3.11)

Let us consider an arbitrary user whose strategy, path P , contributes to cl. Recall that ||P || >||R||

2 . It can be deduced from (3.8) that

||P ∩ Q1|| ≥ ||P ||− ||Q1|| > ||R||2

−||Q1|| = γ + 12

||Q1||− ||Q1|| = γ − 12

||Q1|| ≥ γ − 12

||P ∩Q1||.(3.12)

Therefore, the contribution of this user to lQ1(f) is at least γ−1

2 times his contribution to lQ1(f),

i.e., to cl. So using (3.11), we get lQ1(f) > γ−1

2 cl > γ−12 (lQ1(f) − 2M(f∗)). Now (3.4) gives

lQ1(f) + ||Q1|| > γ−12 (lQ1(f)− 2M(f∗)), which, along with γ ≥ γ0 > 4, yields

lQ1(f) +||Q1||

2+||R||

2<

2(γ − 1)γ − 3

M(f∗) +γ + 1

2(γ − 3)||Q1||+ ||R||

2.

10

Therefore, according to (3.7) and (3.8), we have

M(f) ≤ 2(γ − 1)γ − 3

M(f∗) +γ − 2

2(γ − 3)||R||,

which, together with (3.9), leads to

M(f) ≤ 3γ − 2γ − 2

M(f∗) ≤ γ0M(f∗),

establishing the lemma. 2

Lemma 2 Given any SRR instance, either every optimal flow is a Nash flow, or the price of

anarchy of the instance is at most 4 + 2√

2 < 6.83.

Proof. If we replace lR(·) by laR(·) and || · || by || · ||a in our above discussions for the case of

homogeneous latency functions, then we have laR(f) ≤ 2γM(f∗)− (γ − 1)||R||a, which, together

with (3.6) and (3.2), gives an analogue to (3.9):

M(f) ≤ laR(f) + ||R||b + ||R||a2

≤ γM(f∗)− γ − 22

||R||a +12||R||b ≤ (γ +1)M(f∗)− γ − 2

2||R||a,(3.13)

where the first “≤” is obtained by noticing (3.2) and that

lR(f) = laR(f) + lbR(f) ≤ laR(f) + ||R||b. (3.14)

With the same replacement, inequality (3.12) still holds with the adjusted definition of γ. Hence

we get lQ1(f) ≥ la

Q1(f) > γ−1

2 cl ≥ γ−12 (laQ1

(f)− 2M(f∗)), which, together with (3.4) and (3.14),

leads to

lQ1(f) + ||Q1||a >γ − 1

2(laQ1

(f)− 2M(f∗)) ≥ γ − 12

(lQ1(f)− 2M(f∗)− ||R||b).

Therefore, according to (3.2), we have

lQ1(f) + ||Q1||a >γ − 1

2(lQ1(f)− 4M(f∗) + ||R||a),

which, together with (3.7) and (3.8) with || · || replaced by || · ||a, gives us the following:

M(f) ≤ 4(γ − 1)γ − 3

M(f∗)− γ

2(γ − 3)||R||a,

11

which, combined with (3.13), yields

M(f) ≤ max

γ + 1,4(γ − 1)γ − 3

M(f∗) ≤ (4 + 2

√2)M(f∗),

proving Lemma 2. 2

By the definitions of PoA and PoS, the combination of the above two lemmas implies The-

orem 1 immediately.

We conclude this section with better bounds 8/7 ≤ PoS ≤ PoA = 2 for the simplest SRR

that has only two non-cooperative users, for which we have the following result.

Theorem 2 The price of anarchy is 2 and the price of stability is at least 8/7 for the SRR

problem with k = 2 users.

Proof. Recall that the SRR instance with homogenous linear latency exhibited in Figure 2(b)

has PoA at least 2, and that the SRR instance given in Figure 2(c) has PoS equal to 8/7. By

(2.4), it suffices to show PoA ≤ 2.

Let f∗ = Q1, Q2 and f = Q′1, Q

′2 be an optimal flow and an arbitrary Nash flow,

respectively. We have

lf (Q′i) ≤ 2||Q′

i|| and lf (Q′i) ≤ laf (Q′

i) + ||Q′i|| ≤ 2||Q′

i||, for i = 1, 2. (3.15)

By symmetry, it suffices to distinguish among three cases.

Case 1: Q1 = Q′1 and Q2 = Q′

2. Clearly, M(f)/M(f∗) = 1.

Case 2: Q1 = Q′1 and Q2 = Q′

2. It follows from (3.15) that lf∗(Q1) = lf∗(Q′1) = laf (Q′

1) +

||Q′1|| ≥ lf (Q′

1) and lf∗(Q2) = lf∗(Q′2) ≥ ||Q′

2|| ≥ 12 lf (Q′

2). Hence lf (Q′i) ≤ 2M(f∗) holds for

i = 1, 2.

Case 3: Q1 = Q′1 and Q2 = Q′

2. Therefore lf∗(Qi) ≥ ||Q′i|| for i = 1, 2, which, together with

(3.15), implies lf (Q′i) ≤ 2M(f∗) for i = 1, 2.

Combining the three cases, we deduce from the arbitrariness of f that PoA ≤ 2 as desired.

2

4 Instability of the optimal

In this section we investigate how close an optimum flow could be to an equilibrium. In terms

of (α, β) approximation, we upper bound α while keeping β = 1.

Specifically, we will find an optimal flow that is a 54-approximate Nash flow. Roughly

speaking, to achieve this, we define two indices for every flow. Then beginning from an optimal

12

flow, we perform a number of iterations: in each iteration, we change strategies of at most two

users so that the resulting flow is optimal and has smaller indices. When we terminate at an

optimal flow f∗ with smallest indices, the optimal flow f∗ is proved to be a 54-approximate Nash

– if not, we should have further iteration to produce optimal flow with even smaller indices.

Let us elaborate the above high-level idea in detailed contradiction argument, and devote

the rest of the section to the proof of the following theorem.

Theorem 3 The SRR problem admits a (54, 1)-approximate Nash flow.

We may assume ae + be > 0 for all e ∈ E in view that shrink of e ∈ E with ae + be = 0 into

one node has no effect on our results. Let the optimal flow f∗ = Q1, Q2, . . . , Qk be taken such

that it has smallest indices, more precisely, the following (Min1) and (Min2) are satisfied:

(Min1) the set 〈f∗〉 = i : i ∈ 1, 2, . . . , k and lQi(f∗) = M(f∗) contains as few elements as

possible;

(Min2) subject to (Min1), the index τ(f∗) =∑

e∈E f∗e · le(f∗e ) is as small as possible.

We aim to show that f∗ is a (54, 1)-approximate Nash.

Suppose to the contrary that f∗ is not 54-approximate. We will deduce contradictions to

either (Min1) or (Min2), and therefore establish the theorem. Our proof is justification of a

series of claims that lead to our desired contradictions. First, since f∗ is not 54-approximate,

by (2.3) there exists i′ with 1 ≤ i′ ≤ k such that

lQi′ (f∗) > 54

∑

e∈E(Qi′ )

(ae(f∗e + 1) + be).

So for γ ≥ 18, by (2.2) we have

Claim 1 lQi′(f∗) ≤ ∑

e∈E(Qi′ )(ae(f∗e + 1) + be) = la

Qi′(f∗) + ||Qi′ || < 1

3γ lQi′ (f∗) ≤ 1

3γ M(f∗). 2

If E(Qi)∪E(Qj) = E and E(Qi)∩E(Qj) 6= ∅ for some 1 ≤ i ≤ j ≤ k, then by replacing Qi

with Qi and Qj with Qj , we obtain from f∗ a new flow f such that fe ≤ f∗e for all e ∈ E and at

least one of these inequalities is strict. It follows that f is also an optimal flow and additionally

τ(f) < τ(f∗), which contradicts our choice of f∗ according to (Min1) and (Min2). Therefore,

we have

Claim 2 For any 1 ≤ i, j ≤ k with E(Qi) ∩ E(Qj) 6= ∅, we have E(Qi) ∪ E(Qj) 6= E. 2

For any 1 ≤ i ≤ k, swapping si and ti if necessary, we suppose Qi = R[si, ti] is the clockwise

path in R from si to ti. Without loss of generality, let

13

(Max1) lQ1(f∗) = M(f∗) with |E(Q1)| maximized.

Observe that E(Q1)∩E(Qi′) 6= ∅, as otherwise Qi′ ⊇ Q1 and hence laQi′

(f∗)+ ||Qi′ || ≥ lQ1(f∗) =

M(f∗), which contradicts Claim 1. Without loss of generality, suppose that si′ ∈ R[s1, t1].

Consider flow f1 = Q1, Q2, . . . , Qk, which is obtained from f∗ by changing user 1’s strategy

(see Figures 3(a) and 3(b) for an illustration). We claim

Claim 3 lQ1(f∗) ≤ lQ1

(f1) = laQ1

(f∗) + ||Q1|| < 1γ M(f∗).

The first inequality follows directly from the definition of f1. We only need to justify the validity

of the second inequality. Recall that si′ ∈ R[s1, t1], we have

Q1 ⊆ Qi′ ∪ R[t1, ti′ ], (4.1)

where R[t1, ti′ ] = R[t1, ti′ ] if ti′ ∈ R[s1, t1] and = ∅ otherwise. Since

lQ1∩Qi′ (f∗) = lQ1(f

∗)− lQ1∩Qi′(f∗) ≥ lQ1(f

∗)− lQi′(f∗),

we have

||R[t1, ti′ ]|| ≤ lR[t1,ti′ ](f∗) = lQi′∩Q1

(f∗) = lQi′ (f∗)− lQ1∩Qi′ (f

∗)

≤ M(f∗)− lQ1(f∗) + lQi′

(f∗) = lQi′(f∗),

where the last equality is due to (Max1). Therefore, according to (4.1) and Claim 1,

laQ1(f∗) + ||Q1|| ≤ laQi′

(f∗) + ||Qi′ ||+ 2lR[t1,ti′ ](f∗) <

1γ

M(f∗),

as desired in the claim. 2

14

s2 Q2

s1

t1

t2

Q1

( )a = , ,..., f Q Q Q1 2 k* ( )b = , , , Q Q Qk1 2

s2 Q2

s1

t1

t2

Q1-

-( )c = , , , Q Q Q2 k1

s2Q2

s1

t1

t2

Q1-

-

-

-

s2

Q2

s1

t1

t2

Q1

s i

s j t i

t j

Qi-

Qj-

s2

Q2

s1

t1

t2

Q1

s i

t i

Qi--

( )e = , , , , Q Q1 2 Q i--

s2

Q2

s1

t1

t2

Q1

s j

t j

-Q j-

( )f = , , , , Q Q21 Q j--

f1 ... f

2 ...

... ... ... ...f4

f5( )d = , , , , , , Q Q1 2 Q Qi j

- -... ... ...f

3

Figure 3. Approximate Nash flows

By the optimality of f∗, we have M(f1) ≥ M(f∗). In view of Claim 3, we may assume

(Max2) lQ2(f1) = M(f1) ≥ M(f∗) with |E(Q2)| maximized.

It follows that E(Q1) ∩ E(Q2) 6= ∅ (otherwise Q1 ⊇ Q2 and lQ1(f1) ≥ lQ2(f1) ≥ M(f∗),

contradicting Claim 3), which, together with Claim 2, implies that Q2 intersects with both Q1

and Q1. We may assume, without loss of generality, that s2 ∈ R(s1, t1) and t2 ∈ R(t1, s1) (see

Figures 3(a) and 3(b) for an illustration). Then

lR[s2,t1](f1) = lQ2(f1)− lR[t1,t2](f1) ≥ M(f∗)− lQ1(f1) >

(1− 1

γ

)M(f∗),

where the last inequality is due to Claim 3. Therefore, we have

Claim 4 ||R[s1, s2]|| ≤ lR[s1,s2](f∗) = lQ1(f∗)− lR[s2,t1](f∗) ≤ M(f∗)− lR[s2,t1](f1) < 1

γ M(f∗).2

Claim 5 ||R[t2, s1]|| ≤ ||R[t1, s2]|| ≤ lQ1(f1) + ||R[s1, s2]|| < 2

γ M(f∗). 2

Claims 3 and 4 directly imply

Claim 6 lR[t1,s2](f∗) = lQ1(f∗) + lR[s1,s2](f∗) < 2

γ M(f∗).

15

Now obtain flow f2 = Q1, Q2, Q3, . . . , Qk from f1 by changing user 2’s strategy (see Figure

3(c)). It follows that lR[s1,s2](f2) = lR[s1,s2](f∗), and by Claims 3 and 4, we get

Claim 7 lQ1∪Q2(f2) = lR[t1,s2](f2) ≤ 2lQ1

(f1) + lR[s1,s2](f∗) < 3γ M(f∗). 2

Define Qs as the set of paths Qh in Q3, Q4, . . . , Qk satisfying

lQh(f∗) >

(1− 4

γ

)M(f∗) and R[s1, s2] ⊆ R(sh, th).

Similarly, define Qt as the set of paths Qh in Q3, Q4, . . . , Qk satisfying

lQh(f∗) >

(1− 4

γ

)M(f∗) and R[t1, t2] ⊆ R(sh, th).

We are to show that Qs ∪Qt 6= ∅. But first it is immediate from Claim 2 and the maximality

of |E(Q1)| and |E(Q2)| (stated in (Max1) and (Max2)) that

Claim 8 (i) t1, t2 6∈ Q for any Q ∈ Qs; (ii) s1, s2 6∈ Q for any Q ∈ Qt; and (iii) Qs ∩Qt = ∅.

To see (i), suppose to the contrary that some Qh ∈ Qs contains tg with g ∈ 1, 2. From

R[s1, s2] ⊆ R(sh, th), we see that neither s1 nor s2 is an end of Qh, i.e., s1, s2 ∩ sh, th = ∅,and that, depending on tg is at the top or the tail of Qh, either R[tg, s1] or R[s2, tg] is a subpath

of Qh. In the case of R[tg, s1] ⊆ Qh, we have E(Qh) ∪ E(Q2) = E, and Claim 2 enforces

E(Qh) ∩ E(Q2) = ∅, yielding sh = t2 and th = s2, a contradiction to s1, s2 ∩ sh, th = ∅. In

the case of R[s2, tg] ⊆ Qh, path Q1 chosen in (Max1) turns out to be a subpath of Qh, and

it follows from (Max1) that Qh = Q1, contradicting s1, s2 ∩ sh, th = ∅ again. Hence (i) is

established.

Statement (ii) can be proved by applying symmetric argument. Statement (iii) is instant

from Statements (i) and (ii). 2

The following is directly implied by Claim 6.

Claim 9 For any Q ∈ Qs ∪Qt, lQ∩R[s2,t1](f∗) ≥ lQ(f∗)− lR[t1,s2](f∗) > (1− 6γ )M(f∗). 2

Since maxlQ1(f2), lQ2

(f2) ≤ lQ1∪Q2(f2) < M(f∗) ≤ M(f2) as implied by Claim 7, we may

assume lQ`(f2) = M(f2) ≥ M(f∗) for some ` ∈ 3, 4, . . . , k. Note that le(f2) ≤ le(f∗) for

any link e on R[s1, t2] (see Figure 3(c)), we conclude: (a) Either s2 or t1 belongs to R(s`, t`) =

Q`\s`, t`. (b) Either s1 or t2 belongs to R(s`, t`). If at least one of the two statements were

not true, then we would have either (i) Q` ⊆ R[t1, s2] = Q1 ∪ Q2 or (ii) Q` ⊆ R[s1, t2]. However,

16

(i) together with Claim 7 leads to a contradiction lQ`(f2) ≤ lQ1∪Q2

(f2) < 3γ M(f∗). And (ii)

implies the optimality of f2 due to M(f2) = lQ`(f2) ≤ lQ`

(f∗) ≤ M(f∗), which contradicts our

choice of f∗ in (Min1): 〈f2〉 ⊆ 〈f∗〉 − 1 〈f∗〉.On the other hand, from Figure 3(c) we see with Claim 5 that

lQ`(f∗) +

4γ

M(f∗) > lQ`(f∗) + 2||R[t2, s1]|| ≥ lQ`

(f2) ≥ M(f∗),

which, together with statements (a) and (b) above, implies

Q` ∈ Qs ∪Qt ⇒ Qs ∪Qt 6= ∅.

When Qs 6= ∅, let Qi ∈ Qs be such that |E(R[si, s1])| is maximized. By definition of Qs and

Claim 8(i), si ∈ R(t2, s1] and ti ∈ R[s2, t1). When Qt 6= ∅, let Qj ∈ Qt be such that |E(R[t2, tj ])|is maximized. By definition of Qt and Claim 8(ii), sj ∈ R(s2, t1] and tj ∈ R[t2, s1). See Figure

3(d) for an illustration. Next we distinguish among three cases depending on Qs 6= ∅ 6= Qt,

Qs 6= ∅ = Qt or Qs = ∅ 6= Qt.

Case 1: Qs 6= ∅ 6= Qt. We deduce from Claim 9, γ ≥ 18, and (Max1) that

lR[s2,ti](f∗) + lR[sj ,t1](f

∗) = lQi∩R[s2,t1](f∗) + lQj∩R[s2,t1](f

∗)

> 2(

1− 6γ

)M(f∗) > M(f∗) = lQ1(f

∗) ≥ lR[s2,t1](f∗),

which implies sj ∈ R(s2, ti) and ti ∈ R(sj , t1), and hence tj ∈ R[t2, si) from Claim 2. Noticing

that Qi ∩ R[s2, t1] = R[s2, sj ] ∪ R[sj , ti] and Qj ∩ R[s2, t1] = R[sj , ti] ∪ R[ti, t1], it is clear from

Qi ∩Qj ∩R[s2, t1] = R[sj , ti] that

lR[s2,t1](f∗) + lR[sj ,ti](f

∗) = lQi∩R[s2,t1](f∗) + lQj∩R[s2,t1](f

∗),

which, together with Claim 9, gives

lR[sj ,ti](f∗) > 2

(1− 6

γ

)M(f∗)− lR[sj ,ti](f

∗).

Now we conclude from R[sj , ti] ⊆ Q1 and (Max1) that

lR[sj ,ti](f∗) > 2

(1− 6

γ

)M(f∗)− lQ1(f

∗) =(

1− 12γ

)M(f∗) ≥ 6

γM(f∗), (4.2)

17

where the last inequality uses the assumption γ ≥ 18. Let flow f3 = Q1, Q2, . . . , Qi, . . . , Qj , . . .be obtained from f∗ by changing the strategies of users i and j (see Figure 3(d)). It is routine

to check that

(f3)e =

f∗e for e ∈ E(R[si, sj ]) ∪ E(R[ti, tj ]),f∗e − 2 for e ∈ E(R[sj , ti]),f∗e + 2 for e ∈ E(R[tj , si]).

(4.3)

We show that

maxlQi(f3), lQj

(f3) < M(f∗). (4.4)

In fact, it is easy to see from the definition of f3 (see Figure 3(d)) that R[t1, si] ⊆ R[t1, s1] =

Q1 and hence lR[t1,si](f3) ≤ 2lQ1(f1). Then, noticing that R[ti, t1] ⊆ R[s2, t2]\R[s2, ti] and

R[s2, ti] = Qi ∩R[s2, t1], we obtain

lQi(f3) = lR[ti,t1](f

∗) + lR[t1,si](f3)

≤ lR[s2,t2](f∗)− lQi∩R[s2,t1](f

∗) + 2lQ1(f1)

≤ M(f∗)−(

1− 6γ

)M(f∗) +

2γ

M(f∗) =8γ

M(f∗),

where the last inequality is based on Claims 9 and 3. Similarly, since R[tj , s2] ⊆ R[t2, s2] =

Q2 and hence lR[tj ,s2](f3) ≤ 2lQ2(f2). Then, noticing that R[s2, sj ] ⊆ R[s2, t2]\R[sj , t1] and

R[sj , t1] = Qj ∩R[s2, t1], we obtain

lQj(f3) = lR[s2,sj ](f

∗) + lR[tj ,s2](f3)

≤ lR[s2,t2](f∗)− lQj∩R[s2,t1](f

∗) + 2lQ2(f2)

≤ M(f∗)−(

1− 6γ

)M(f∗) +

6γ

M(f∗) =12γ

M(f∗),

where the last inequality is based on Claims 9 and 7. Therefore, (4.4) is established. Now let

us prove the following final claim.

Claim 10 E(Q) ∩ E(R[tj , si]) = ∅ for all Q ∈ f3 with lQ(f3) = M(f3).

To see this, let us consider arbitrary Q ∈ f3 with lQ(f3) = M(f3) ≥ M(f∗). We have Q 6∈Qi, Qj by (4.4). From Claim 5 and (4.3), we have

lQ(f∗) +4γ

M(f∗) > lQ(f∗) + 2||R[t2, s1]||

≥ lQ(f∗) + 2||R[tj , si]|| ≥ lQ(f3) ≥ M(f∗).

18

Suppose to the contrary that E(Q) ∩ E(R[tj , si]) 6= ∅. From Claims 6 and 5 we obtain

lR[t1,s2](f3) ≤ lR[t1,s2](f∗) + 2||R[tj , si]||

≤ lR[t1,s2](f∗) + 2||R[t2, s1]|| < 6

γM(f∗),

which implies that Q * R[t1, s2] and hence either Q ⊇ R[s1, s2] or Q ⊇ R[t1, t2]. Therefore,

Q ∈ Qs ∪Qt. However, the facts that Q 6∈ Qi, Qj and Q ∩ E(R[tj , si]) 6= ∅ contradict one of

our two earlier choices of Qi and Qj with the maximality of |E(R[si, s1])| and the maximality

of |E(R[t2, tj ])|. Hence the above claim is justified. 2

In consequence, we deduce from (4.3) that

M(f3) ≤ M(f∗) implies that f3 is an optimal flow, and 〈f3〉 ⊆ 〈f∗〉, (4.5)

and

τ(f∗)− τ(f3)

=∑

e∈E(R[sj ,ti])

[f∗e · le(f∗e )− (f∗e − 2)le(f∗e − 2)] +∑

e∈E(R[tj ,si])

[f∗e · le(f∗e )− (f∗e + 2)le(f∗e + 2)]

≥∑

e∈E(R[sj ,ti])

2le(f∗e )−∑

e∈E(R[tj,si])

f∗e =0

2(2ae + be) +∑

e∈E(R[tj ,si])

f∗e >0

[f∗e · le(f∗e )− (f∗e + 2)le(f∗e + 2)]

= 2lR[sj ,ti](f∗)−

∑

e∈E(R[tj ,si])

f∗e =0

2(2ae + be) +∑

e∈E(R[tj ,si])

f∗e >0

[f∗e · le(f∗e )− (f∗e + 2)(le(f∗e ) + 2ae)]

= 2lR[sj ,ti](f∗)−

∑

e∈E(R[tj ,si])

f∗e =0

2(2ae + be)−∑

e∈E(R[tj ,si])

f∗e >0

[2f∗e ae + 2le(f∗e ) + 4ae]

≥ 2lR[sj ,ti](f∗)−

∑

e∈E(R[tj,si])

4(ae + be)−∑

e∈E(R[tj ,si]),f∗e >0

4le(f∗e )

≥ 2lR[sj ,ti](f∗)− 4||R[tj , si]|| − 4lR[tj ,si](f

∗)

≥ 2lR[sj ,ti](f∗)− 4||R[t2, s1]|| − 4lQ1

(f∗)

Using (4.2), Claims 5 and 3, we get τ(f∗) − τ(f3) > 0, which, in combination with (4.5), gives

a contradiction to (Min2).

19

Case 2: Qs 6= ∅ = Qt. In this case, we derive flow f4 = Q1, Q2, . . . , Qi, . . . from f∗ by

changing the strategies of users 2 and i (See Figure 3(e)). Similar to (4.3), we get

f4e =

f∗e for e ∈ E(R[si, s2]) ∪ E(R[ti, t2]),f∗e − 2 for e ∈ E(R[s2, ti]),f∗e + 2 for e ∈ E(R[t2, si]).

From Claims 7 and 9, it follows that

lQ2(f4) ≤ lQ2∪Q1

(f2) <3γ

M(f∗),

lQi(f4) = lR[ti,t1](f

∗) + lR[t1,si](f4) ≤ 6γ

M(f∗) + lQ1(f2) ≤ 9

γM(f∗),

and subsequently, the arguments in Case 1 apply with subscript 3 replaced by 4 and subscript

j replaced by 2, yielding M(f4) ≤ M(f∗), 〈f4〉 ⊆ 〈f∗〉 and

τ(f∗)− τ(f4) ≥ 2lR[s2,ti](f∗)− 4||R[t2, si]|| − 4lQ1

(f∗).

By Claims 9, 5, and 3, we obtain τ(f∗) − τ(f4) > 2(1− 6

γ

)M(f∗) − 8

γ M(f∗) − 4γ M(f∗) ≥ 0

giving τ(f∗) > τ(f4) a contradiction to (Min2).

Case 3: Qs = ∅ 6= Qt. In this case, we derive flow f5 = Q1, Q2, . . . , Qj , . . . from f∗ by

changing strategies of users 1 and j (See Figure 3(f)). Then f5e = f∗e for all e ∈ E(R[t1, tj ]) ∪E(R[s1, sj ]), f5e = f∗e − 2 for all e ∈ E(R[sj , t1]), and f5e = f∗e + 2 for all e ∈ E(R[tj , s1]).

Furthermore, lQ1(f5) ≤ lQ1

(f2) < 3γ M(f∗) by Claim 7, and the subsequent arguments go as

in Case 1 with subscript 3 replaced by 5 and subscript i replaced by 1. Finally we reach a

contradiction to (Min2) given by M(f5) ≤ M(f∗), 〈f5〉 ⊆ 〈f∗〉 and τ(f∗)− τ(f5) > 0.

To summarize, the contradictions in the above three cases show that our contradiction as-

sumption is false. Hence f∗ is a (54, 1)-approximate Nash flow, giving the conclusion of Theo-

rem 3.

5 Algorithmic consequences

Now we discuss briefly how to compute an (α, β)-approximate Nash flow for the SRR problem

such that α and β are small, in order to guarantee the stability and efficiency of the flow, and to

make the flow a satisfactory initial solution of the SRR network game as expected by network

designers.

20

Finding near optimal flow in polynomial time

In order to obtain a good social solution to the SRR problem efficiently, we resort to its split-

table counterpart, the splittable selfish ring routing with linear latency (SSRR), by relaxing the

unsplittable constraint f ∈ 0, 1P to splittable one: f ∈ [0, 1]P , and f(Pi) + f(Pi) = 1 for

i = 1, 2, . . . , k. Since the latency is linear, lP (f) can be expressed as a linear combination of

xi = f(Pi), i = 1, 2, . . . , k, finding an optimal solution to the SSRR amounts to solving the

following linear program: Minimize y subject to

∑

e∈E(P )

ae

( ∑

1≤j≤k:E(Pj)3e

xj +∑

1≤j≤k:E(Pj)3e

(1− xj))

+ be ≤ y

for P ∈ Pi, Pi, i = 1, 2, . . . , k. In polynomial time we can obtain an optimal solution (x∗1, x∗2, . . . , x

∗k, y

∗)

to the above linear program and, therefore, an optimal flow f∗ ∈ [0, 1]P to the SSRR with

f∗(Pi) = x∗i , i = 1, 2, . . . , k, and M(f∗) = y∗. We round f∗ to a feasible atomic unsplittable

flow f ∈ 0, 1P for the SRR problem in such a way that f(Pi) = 1 iff f(Pi) = x∗i ≥ 0.5,

i = 1, 2, . . . , k. It is evident that

M(f) ≤ 2y∗ ≤ 2OPT. (5.1)

Finding good Nash in pseudo-polynomial time

If f obtained above is not a Nash flow, we iteratively change the strategy of a user to re-

duce the latency he experiences in the current solution and, as easily verified with the po-

tential function technique [17, 15], we finally reach a Nash flow f with M(f) ≤ M(f) in time

O(k3n2 maxni=1aei +bei) and in time O(k3n2) when latency all equal to loads (see also Theorem

1 in [12]).

Corollary 4 The feasible flow f for the SRR problem computed as above is a (1, β)-approximate

Nash flow with β ≤ 13.66 and β ≤ 9.13 if the linear latency functions are homogenous.

Proof. If f is a Nash, then it is apparent that β = 2. Otherwise, apply verbatim the arguments

in Section 3 with f in place of f∗. It follows from (5.1) that β can be no more than twice the

PoA stated in Lemmas 1 and 2. 2

Reducing instability of near optimal flow

The proof of Theorem 3 suggests a pseudo-polynomial time approach to “stabilizing” a given

optimal flow f∗ iteratively – changing the strategy of one user or the strategies of two users

21

simultaneously in each iteration such that either fewer users suffer from the maximum latency

OPT or the resulting flow has smaller index τ (cf. (Min1) and (Min2) in the proof of Theorem 3).

This approach works on f , which is considered a substitute for f∗, and provides a 54-approximate

Nash flow f whose maximum latency M(f) equals M(f). In consequence, (5.1) asserts that f

is a (54, 2)-approximate Nash flow.

To summarize, (α, β)-approximate Nash flow in any given SRR can be constructed in pseudo-

polynomial time for (α, β) = (1, 13.66) and (α, β) = (54, 2), and for (α, β) = (1, 9.13) for

homogenous linear latency.

6 Concluding remarks

Positive results established in this paper, particularly in Lemma 1, Lemma 2, and Theorem

3, provide us with (α, β)-approximate Nash equilibria for unsplittable selfish ring routing with

linear latency. In addition to much room for improvement on bounding α and β, quantitive

relations between these two bounds deserve further research efforts. In this paper we have

focused on undirected selfish ring routing, challenging issues in its directed counterpart require

more and deeper insights into the interplay of users’ selfish behaviors and directed ring latency.

Regarding nonatomic/atomic splittable selfish ring routing, it would be tempting to extend

our methodology for the atomic unsplittable setting to derive similar results, although the con-

tinuous version is more complex than its discrete counterpart.

Roughgarden and Tardos [19] prove that, for general continuous and nondecreasing latency

functions, the “average latency” of the routes chosen by selfish network users is no more than the

“average latency” incurred by optimally routing twice as much traffic. It would be interesting to

see if some analogue (with “maximum latency” in place of “average latency”) exists for weighted

selfish ring routing to minimize maximum latency.

References

[1] A. Aggarwal, A. Bar-Noy, D. Coppersmith, R. Ramaswami, B. Schieber, and M. Sudan,

Efficient routing and scheduling algorithms for optical networks, Proceedings of the 5th

annual ACM-SIAM Symposium on Discrete Algorithms, 1994, 412–423.

[2] E. Anshelevich, A. Dasgupta, J. Kleinberg, E. Tardos, T. Wexler, and T. Roughgarden,

The price of stability for network design with fair cost allocation, Proceedings of the 45th

annual Symposium on Foundations of Computer Science, 2004, 295–304.

22

[3] B. Awerbuch, Y. Azar, and L. Epstein, The price of routing unsplittable flow, Proceedings

of the 37th annual ACM Symposium on Theory of Computing, 2005, 57–66.

[4] A. Blum, A. Kalai, and J. Kleinberg, Addmission control to minimize rejections, Lecture

Notes in Computer Science, 2152 (2001) 155–164.

[5] C. Busch and M. Magdon-Ismail, Atomic routing games on maximum congestion, Lecture

Notes in Computer Science, 4041 (2006) 79–91.

[6] G. Christodoulou and E. Koutsoupias, The price of anarchy of finite congestion games,

Proceedings of the 37th annual ACM Symposium on Theory of Computing, 2005, 67–73.

[7] R. Cominetti, J. R. Correa, and N. E. Stier-Moses, Network games with atomic players,

Lecture Notes in Computer Science, 4051 (2006) 525–536.

[8] J. R. Correa, A. S. Schulz, and N. E. Stier-Moses, Fast, fair, and efficient flows in networks,

Oprerations Research, 55 (2007) 215–225.

[9] A. Czumaj, Selfish routing on the Internet, in Handbook of Scheduling: Algorithms, Mod-

els, and Performance Analysis, Chapman & Hall/CRC Computer and Information Science

Series (J. Leung, ed.), Vol. 1, chapter 42, CRC Press, Boca Raton, FL, 2004.

[10] A. Fabrikant, C. Papadimitriou, and K. Talwar, The complexity of pure Nash equilibria,

Proceedings of the 36th annual ACM Syposimum on Theory of Computing, 2004, 604–612.

[11] R. Feldmann, M. Gairing, T. Lucking, B. Monien, and M. Rode, Nashification and the

coordination ratio for a selfish routing game, Lecture Notes in Computer Science, 2719

(2003) 514–526.

[12] D. Fotakisa, S. Kontogiannisa, and P. Spirakis, Selfish unsplittable flows, Theoretical Com-

puter Sciences, 348 (2005) 226–239.

[13] S. Kontogiannis and P. Spirakis, Atomic selfish routing in networks: a survey, Lecture Notes

in Computer Science, 3828 (2005) 989–1002.

[14] E. Koutsoupias and C. H. Papadimitriou, Worst-case equilibria, Lecture Notes in Computer

Science, 1563 (1999) 404–413.

[15] D. Monderer and L. Shapley, Potential games, Game and Economic Behavior, 14 (1996)

124–143.

[16] H. Lin, T. Roughgarden, E. Tardos and A. Walkover, Braess’s paradox, Fibonacci Numbers,

and exponential inapproximability, Lecture Notes in Computer Science, 3580 2005, 497–

512.

[17] R.W. Rosenthal, A class of games possessing pure-strategy Nash equilibira, International

Jouranl of Game Theory, 2 (1973) 65–67.

23

[18] T. Roughgarden, The price of anarchy is independent of the network topology, Jouranl of

Computer and System Sciences, 67 (2003) 342–364.

[19] T. Roughgarden and E. Tardos, How bad is selfish routing? Journal of the ACM, 49 (2002)

236–259.

[20] A. Schrijver, P. Seymour, and P. Winkler, The ring loading problem SIAM Journal on

Discrete Mathematics, 11 (1998) 1–14.

[21] D. Weitz, The price of anarchy, Unpublished manuscript, 2001.

[22] G. Wilfong and P. Winkler, Ring routing and wavelength translation, Proceedings of the

9th annual ACM-SIAM Symposium on Discrete Algorithms, 1998, 333–341.

24