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 latency. 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 most 6.83, which reduces to 4.57 when the linear latency functions are homogeneous. We also show the existence of a (54,1)-approximate Nash equilibrium. Additionally we address some algorithmic 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 Applications (DIMAP), University of Warwick, Coventry, CV4 7AL, United Kingdom,
[email protected] ‡ Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China,
[email protected] § Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China,
[email protected] †
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=1 Pj with P P ∈Pi fP = 1 for every i = 1, 2, . . . , k. Each link e ∈ E bears a load with respect to f defined P as fe = P ∈P:e∈E(P ) fP , the sum of flow along e; and each e ∈ E is associated with a loaddependent 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 P 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 ) ≡ max Mi (f ) = 1≤i≤k
max
P ∈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 singlecommodity 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 e0j ) from uj to vj , j = 1, 2, . . . , h, is lej (fej ) = hfej (resp. le0j (fe0j ) = fe0j ). 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, suj vj t, going through the bottom link e0j , 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 fvj uj+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 fe0j < h(h − 1) and (ii) fsuj 0 ≥ h2 + 2 for some j 0 ∈ {1, 2, . . . , h} − {j}. From (ii) we conclude that, between s and t, either at least through ej 0 , or at least
h 2 h+1 (h + 2)
1 2 h+1 (h
+ 2) > h − 1 units of flow is routed
> h(h − 1) units of flow is routed through e0j 0 , 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 suj vj t, which goes through ej if fej < h − 1 and through e0j if fe0j < 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 0 , either fej 0 < h − 1 or fe0 0 < h(h − 1), since j
otherwise some player would benefit by switching his own flow from e
j0
to e0j 0 ) or vise versa.
Therefore, fej ≥ h − 1 and fe0j ≥ 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 , . . . , vn } and E = {ei = vi vi+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.
v n+1 = v 1
l e ( x )= x
v2
l e ( x )= x 1
1
e1
e1
s1
t1
s1
e1
s2
e2
en vn
x l e ( x )= 4
v3 e n-1
e3 v4
v n-1
e4
4
e2
t2
x l e ( x )= 4 2
7
4
t2
s2
e3
e4
x l e ( x )= 5
e3
l e ( x )= 2 2
t1
l e ( x )= x
3 x l e 3( x )= 2
( b ) PoA > 2 , PoS = 1
( C ) PoA = PoS = 8 / 7
3
( a ) R =( V , E )
e2
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 communication 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 P¯i in R, satisfying Pi ∪ P¯i = R. For convenience, let P¯ = P , i = 1, 2, . . . , k. Given feasible (unsplittable) flow f ∈ {0, 1}P with i
i
P = ∪ki=1 {Pi , P¯i }, in view of the correspondence between f ∈ {0, 1}P 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 , P¯i }, 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 ) = ae fe + 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 lPa (f ) =
X e∈E(P )
ae fe ,
||P ||a =
X
ae ,
||P ||b =
e∈E(P )
X
be ,
||P || = ||P ||a + ||P ||b .
e∈E(P )
We write Pi = {Pi , P¯i } such that ||Pi || ≤ ||P¯i || for every i = 1, 2, . . . , k. Note that ||Pi || + ||P¯i || = ||R||, for every i = 1, 2, . . . , k. 6
(2.1)
¯ i ||a and the long path strategy We say that user i adopts the short path strategy if ||Qi ||a ≤ ||Q 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 hf i 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} X
lQi (f ) ≤
le (fe + 1).
(2.2)
e∈E(Q¯i )
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
X
le (fe + 1).
(2.3)
e∈E(P¯ )
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 ∈F
M (f ) M (f ) and ρS (R, (si , ti )ki=1 , l) = min . f ∈F OPT 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 )ki=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) =
x 4,
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 , Q2 } ¯ 1, Q ¯ 2 } are Nash flows, and f ∗ is an optimum flow for the SRR instance. Hence and f = {Q 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 ∗ = {s1 s2 t1 , s2 t1 t2 } has maximum latency M (f ∗ ) = 3.5 while its unique Nash flow f = {s1 s2 t1 , s2 s1 t2 } 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 i0 ∈ {1, 2, . . . , k} can benefit from unilaterally changing his strategy provided strategies of other users remain the same, which implies that the SRR has a feasible flow f 0 = ¯ i0 , Qi0 +1 , . . . , Qk } satisfying l ¯ (f ∗ ) + ||Q ¯ i0 ||a = l ¯ (f 0 ) < lQ 0 (f ∗ ) ≤ M (f ∗ ), {Q1 , . . . , Qi0 −1 , Q Qi0 Qi0 i where the equality is directly based on the definition of new flow f 0 . Therefore, ¯ i0 ||a . lR (f ∗ ) = lQi0 (f ∗ ) + lQ¯ i0 (f ∗ ) < 2M (f ∗ ) − ||Q
(3.1)
¯ i0 || and lQ 0 (f ∗ ) ≥ ||Qi0 ||, it follows from (2.1) that Since lQ¯ i0 (f 0 ) ≥ ||Q i ¯ i || = ||R|| = ||R||a + ||R||b = ||Qi0 || + ||Q ¯ i0 || < 2M (f ∗ ) for i = 1, 2, . . . , k. ||Qi || + ||Q
(3.2)
Let f be an arbitrary Nash flow. Suppose, without loss of generality, that users 1, 2, . . . , k are ¯ 1, Q ¯ 2, . . . , Q ¯ j , Qj+1 , . . . , Qk }. Notice that ordered such that for some 1 ≤ j ≤ k, f = {Q lQ¯ i (f ) + lQi (f ) = lR (f ) for i = 1, 2, . . . , k. By (2.2), we have X lQ¯ i (f ) ≤ le (fe + 1) = e∈E(Qi )
X
(3.3)
(ae fe + ae + be ) = lQi (f ) + ||Qi ||a for i = 1, 2, . . . , j. (3.4)
e∈E(Qi )
8
Similarly, ¯ i ||a for i = j + 1, j + 2, . . . , k. lQi (f ) ≤ lQ¯ i (f ) + ||Q
(3.5)
It is instant from (3.3)–(3.5) and (1.1) that M (f ) ≤ max Q∈f
¯ a lQ (f ) + lQ¯ (f ) + ||Q|| lR (f ) + ||R||a ≤ , 2 2
(3.6)
and from which, along with (3.2), we obtain M (f ) ≤ lQi (f ) +
||Qi ||a ||R||a + , for i = 1, 2, . . . , j. 2 2
(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+ 17 2
≤ 4.57.
Proof. For homogeneous linear latency functions, we have ||·|| = ||·||a . Without loss of generality assume that γ =
¯ 1 || ||Q ||Q1 ||
= maxji=1
¯ i || ||Q ||Qi ||
≥ 1. (Note that if γ < 1, then M (f ) ≤ lR (f ) < lR (f ∗ )
j, we can similarly obtain ¯ i0 || + (γ − 1) lR (f ) ≤ 2M (f ∗ ) − ||Q
j X
||Qi ||
i=1 j X ≤ 2M (f ) − (γ − 1)||Qi0 || + (γ − 1)( ||Qi || + ||Qi0 ||) ∗
i=1 ∗
≤ 2M (f ) − (γ − 1)||Qi0 || + (γ − 1)lR (f ∗ ) ¯ i0 ||) ≤ 2M (f ∗ ) − (γ − 1)||Qi0 || + (γ − 1)(2M (f ∗ ) − ||Q = 2γM (f ∗ ) − (γ − 1)||R||. Therefore, from the above analysis and (3.6), we derive M (f ) ≤ γM (f ∗ ) −
γ−2 ||R||. 2
(3.9)
To prove the lemma, we can assume γ ≥ γ0 thanks to (3.9). Recall the assumption that ||Pi || ≤ ||P¯i || 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 X
||Pi ||.
(3.10)
i=1
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 P sum up to cl . In other words, lQ1 (f ) = cs +cl . Observe from (3.10) that cs ≤ ki=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
¯ 1 || ≥ ||P || − ||Q1 || > ||R|| − ||Q1 || = γ + 1 ||Q1 || − ||Q1 || = γ − 1 ||Q1 || ≥ γ − 1 ||P ∩ Q1 ||. ||P ∩ Q 2 2 2 2 (3.12) Therefore, the contribution of this user to lQ¯ 1 (f ) is at least i.e., to cl . So using (3.11), we get lQ¯ 1 (f ) > lQ1 (f ) + ||Q1 || >
γ−1 2 (lQ1 (f )
lQ1 (f ) +
γ−1 2 cl
>
γ−1 2
times his contribution to lQ1 (f ),
γ−1 2 (lQ1 (f )
− 2M (f ∗ )). Now (3.4) gives
− 2M (f ∗ )), which, along with γ ≥ γ0 > 4, yields
2(γ − 1) γ+1 ||R|| ||Q1 || ||R|| + < M (f ∗ ) + ||Q1 || + . 2 2 γ−3 2(γ − 3) 2 10
Therefore, according to (3.7) and (3.8), we have M (f ) ≤
2(γ − 1) γ−2 M (f ∗ ) + ||R||, γ−3 2(γ − 3)
which, together with (3.9), leads to M (f ) ≤
3γ − 2 M (f ∗ ) ≤ γ0 M (f ∗ ), γ−2
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. a (·) and || · || by || · || in our above discussions for the case of Proof. If we replace lR (·) by lR a a (f ) ≤ 2γM (f ∗ ) − (γ − 1)||R|| , which, together homogeneous latency functions, then we have lR a
with (3.6) and (3.2), gives an analogue to (3.9): M (f ) ≤
a (f ) + ||R|| + ||R|| lR γ−2 1 γ−2 a b ≤ γM (f ∗ ) − ||R||a + ||R||b ≤ (γ + 1)M (f ∗ ) − ||R||a , 2 2 2 2 (3.13)
where the first “≤” is obtained by noticing (3.2) and that a b a lR (f ) = lR (f ) + lR (f ) ≤ lR (f ) + ||R||b .
(3.14)
With the same replacement, inequality (3.12) still holds with the adjusted definition of γ. Hence a (f ) > we get lQ¯ 1 (f ) ≥ lQ ¯ 1
γ−1 2 cl
≥
γ−1 a ∗ 2 (lQ1 (f ) − 2M (f )),
which, together with (3.4) and (3.14),
leads to lQ1 (f ) + ||Q1 ||a >
γ−1 γ−1 a (lQ1 (f ) − 2M (f ∗ )) ≥ (lQ1 (f ) − 2M (f ∗ ) − ||R||b ). 2 2
Therefore, according to (3.2), we have lQ1 (f ) + ||Q1 ||a >
γ−1 (lQ1 (f ) − 4M (f ∗ ) + ||R||a ), 2
which, together with (3.7) and (3.8) with || · || replaced by || · ||a , gives us the following: M (f ) ≤
4(γ − 1) γ M (f ∗ ) − ||R||a , γ−3 2(γ − 3)
11
which, combined with (3.13), yields ½ ¾ √ 4(γ − 1) M (f ) ≤ max γ + 1, M (f ∗ ) ≤ (4 + 2 2)M (f ∗ ), γ−3 proving Lemma 2.
2
By the definitions of PoA and PoS, the combination of the above two lemmas implies Theorem 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 = {Q01 , Q02 } be an optimal flow and an arbitrary Nash flow, respectively. We have ¯ 0i ) + ||Q ¯ 0i || ≤ 2||Q ¯ 0i ||, for i = 1, 2. lf (Q0i ) ≤ 2||Q0i || and lf (Q0i ) ≤ lfa (Q
(3.15)
By symmetry, it suffices to distinguish among three cases. Case 1: Q1 = Q01 and Q2 = Q02 . Clearly, M (f )/M (f ∗ ) = 1. ¯ 0 and Q2 = Q0 . It follows from (3.15) that lf ∗ (Q1 ) = lf ∗ (Q ¯ 0 ) = la (Q ¯0 ) + Case 2: Q1 = Q 1 2 1 1 f ¯ 0 || ≥ lf (Q0 ) and lf ∗ (Q2 ) = lf ∗ (Q0 ) ≥ ||Q0 || ≥ 1 lf (Q0 ). Hence lf (Q0 ) ≤ 2M (f ∗ ) holds for ||Q 1 1 2 2 2 i 2 i = 1, 2. ¯ 0 and Q2 = Q ¯ 0 . Therefore lf ∗ (Qi ) ≥ ||Q ¯ 0 || for i = 1, 2, which, together with Case 3: Q1 = Q 1 2 i (3.15), implies lf (Q0i ) ≤ 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 hf ∗ i = {i : i ∈ {1, 2, . . . , k} and lQi (f ∗ ) = M (f ∗ )} contains as few elements as possible; (Min2) subject to (Min1), the index τ (f ∗ ) =
P e∈E
fe∗ · le (fe∗ ) 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 i0 with 1 ≤ i0 ≤ k such that lQi0 (f ∗ ) > 54
X
(ae (fe∗ + 1) + be ).
¯ 0) e∈E(Q i
So for γ ≥ 18, by (2.2) we have P a (f ∗ ) + ||Q ¯ i0 || < Claim 1 lQ¯ i0 (f ∗ ) ≤ e∈E(Q¯ 0 ) (ae (fe∗ + 1) + be ) = lQ ¯0 i
i
1 ∗ 3γ lQi0 (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 ¯ i and Qj with Q ¯ j , we obtain from f ∗ a new flow f such that fe ≤ fe∗ for all e ∈ E and at with Q 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. ¯ i0 ⊇ Q1 and hence la¯ (f ∗ ) + ||Q ¯ i0 || ≥ lQ (f ∗ ) = Observe that E(Q1 ) ∩ E(Qi0 ) 6= ∅, as otherwise Q 1 Q0 i
M (f ∗ ), which contradicts Claim 1. Without loss of generality, suppose that si0 ∈ R[s1 , t1 ]. ¯ 1 , Q2 , . . . , Qk }, which is obtained from f ∗ by changing user 1’s strategy Consider flow f1 = {Q (see Figures 3(a) and 3(b) for an illustration). We claim a (f ∗ ) + ||Q ¯ 1 || < 1 M (f ∗ ). Claim 3 lQ¯ 1 (f ∗ ) ≤ lQ¯ 1 (f1 ) = lQ ¯ γ 1
The first inequality follows directly from the definition of f1 . We only need to justify the validity of the second inequality. Recall that si0 ∈ R[s1 , t1 ], we have ¯1 ⊆ Q ¯ i0 ∪ R[t ˜ 1 , ti0 ], Q
(4.1)
˜ 1 , ti0 ] = R[t1 , ti0 ] if ti0 ∈ R[s1 , t1 ] and = ∅ otherwise. Since where R[t lQ1 ∩Qi0 (f ∗ ) = lQ1 (f ∗ ) − lQ1 ∩Q¯ i0 (f ∗ ) ≥ lQ1 (f ∗ ) − lQ¯ i0 (f ∗ ), we have ||R[t1 , ti0 ]|| ≤ lR[t1 ,ti0 ] (f ∗ ) = lQi0 ∩Q¯ 1 (f ∗ ) = lQi0 (f ∗ ) − lQ1 ∩Qi0 (f ∗ ) ≤ M (f ∗ ) − lQ1 (f ∗ ) + lQ¯ i0 (f ∗ ) = lQ¯ i0 (f ∗ ), where the last equality is due to (Max1). Therefore, according to (4.1) and Claim 1, 1 ∗ a ∗ a ∗ ∗ ¯ ¯ lQ ¯ 1 (f ) + ||Q1 || ≤ lQ ¯ 0 (f ) + ||Qi0 || + 2lR[t1 ,ti0 ] (f ) < M (f ), i γ as desired in the claim.
2
14
Q1
t1
s2
s1
t1
s2
Q2 s1
t2
Q2
Q2
s1
t2
Q1
t1
s2
t2
Q1 ( a)
f * ={ Q 1, Q 2,..., Q k}
( b)
f 1 ={ Q 1 , Q 2 ,..., Q k }
Q2
Q1
s1
-
( d)
Q2
ti t1
s2 t2
s1
-
Qi
Qj
- 2,..., Q k} - 1, Q f 2={ Q
Q1
s2 sj ti t1
si
( c)
tj
- i ,..., Q- j ,...} f 3={ Q 1 , Q 2 ,..., Q
-
Q2
s2 sj t2
s1
Q
t2
-
i
si -
( e ) f 4 ={ Q 1 , Q 2 ,..., Q i ,...}
Q1
tj
-j Q -
t1
-
-
( f ) f 5 ={ Q 1 , Q 2 ,..., Q j ,...}
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. ¯ 1 ⊇ Q2 and l ¯ (f1 ) ≥ lQ (f1 ) ≥ M (f ∗ ), It follows that E(Q1 ) ∩ E(Q2 ) 6= ∅ (otherwise Q 2 Q1 contradicting Claim 3), which, together with Claim 2, implies that Q2 intersects with both Q1 ¯ 1 . We may assume, without loss of generality, that s2 ∈ R(s1 , t1 ) and t2 ∈ R(t1 , s1 ) (see and Q Figures 3(a) and 3(b) for an illustration). Then µ ¶ 1 lR[s2 ,t1 ] (f1 ) = lQ2 (f1 ) − lR[t1 ,t2 ] (f1 ) ≥ M (f ) − lQ¯ 1 (f1 ) > 1 − M (f ∗ ), γ ∗
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 ) < γ1 M (f ∗ ).2 Claim 5 ||R[t2 , s1 ]|| ≤ ||R[t1 , s2 ]|| ≤ lQ¯ 1 (f1 ) + ||R[s1 , s2 ]|| < γ2 M (f ∗ ). Claims 3 and 4 directly imply Claim 6 lR[t1 ,s2 ] (f ∗ ) = lQ¯ 1 (f ∗ ) + lR[s1 ,s2 ] (f ∗ ) < γ2 M (f ∗ ). 15
2
¯ 1, Q ¯ 2 , Q3 , . . . , Qk } from f1 by changing user 2’s strategy (see Figure Now obtain flow f2 = {Q 3(c)). It follows that lR[s1 ,s2 ] (f2 ) = lR[s1 ,s2 ] (f ∗ ), and by Claims 3 and 4, we get Claim 7 lQ¯ 1 ∪Q¯ 2 (f2 ) = lR[t1 ,s2 ] (f2 ) ≤ 2lQ¯ 1 (f1 ) + lR[s1 ,s2 ] (f ∗ ) < γ3 M (f ∗ ).
2
Define Qs as the set of paths Qh in {Q3 , Q4 , . . . , Qk } satisfying µ ¶ 4 lQh (f ) > 1 − M (f ∗ ) and R[s1 , s2 ] ⊆ R(sh , th ). γ ∗
Similarly, define Qt as the set of paths Qh in {Q3 , Q4 , . . . , Qk } satisfying µ ¶ 4 lQh (f ) > 1 − M (f ∗ ) and R[t1 , t2 ] ⊆ R(sh , th ). γ ∗
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 − γ6 )M (f ∗ ).
2
Since max{lQ¯ 1 (f2 ), lQ¯ 2 (f2 )} ≤ lQ¯ 1 ∪Q¯ 2 (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 ¯1 ∪ Q ¯ 2 or (ii) Q` ⊆ R[s1 , t2 ]. However, not true, then we would have either (i) Q` ⊆ R[t1 , s2 ] = Q 16
(i) together with Claim 7 leads to a contradiction lQ` (f2 ) ≤ lQ¯ 1 ∪Q¯ 2 (f2 ) < implies the optimality of f2 due to M (f2 ) = lQ` (f2 ) ≤ lQ` choice of
f∗
in (Min1): hf2 i ⊆
hf ∗ i
− {1}
(f ∗ )
≤
M (f ∗ ),
3 ∗ γ M (f ).
And (ii)
which contradicts our
hf ∗ i.
On the other hand, from Figure 3(c) we see with Claim 5 that lQ` (f ∗ ) +
4 M (f ∗ ) > lQ` (f ∗ ) + 2||R[t2 , s1 ]|| ≥ lQ` (f2 ) ≥ M (f ∗ ), γ
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 ∗ ) µ ¶ 6 > 2 1− M (f ∗ ) > M (f ∗ ) = lQ1 (f ∗ ) ≥ lR[s2 ,t1 ] (f ∗ ), γ 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 µ ¶ 6 lR[sj ,ti ] (f ∗ ) > 2 1 − M (f ∗ ) − lR[sj ,ti ] (f ∗ ). γ Now we conclude from R[sj , ti ] ⊆ Q1 and (Max1) that ¶ ¶ µ µ 6 6 12 ∗ ∗ ∗ M (f ) − lQ1 (f ) = 1 − M (f ∗ ) ≥ M (f ∗ ), lR[sj ,ti ] (f ) > 2 1 − γ γ γ
17
(4.2)
¯ i, . . . , Q ¯ j , . . .} where the last inequality uses the assumption γ ≥ 18. Let flow f3 = {Q1 , Q2 , . . . , Q be obtained from f ∗ by changing the strategies of users i and j (see Figure 3(d)). It is routine to check that
∗ for e ∈ E(R[si , sj ]) ∪ E(R[ti , tj ]), fe ∗ f − 2 for e ∈ E(R[sj , ti ]), (f3 )e = e∗ fe + 2 for e ∈ E(R[tj , si ]).
(4.3)
max{lQ¯ i (f3 ), lQ¯ j (f3 )} < M (f ∗ ).
(4.4)
We show that
In fact, it is easy to see from the definition of f3 (see Figure 3(d)) that R[t1 , si ] ⊆ R[t1 , s1 ] = ¯ 1 and hence lR[t ,s ] (f3 ) ≤ 2l ¯ (f1 ). Then, noticing that R[ti , t1 ] ⊆ R[s2 , t2 ]\R[s2 , ti ] and Q Q1 1 i R[s2 , ti ] = Qi ∩ R[s2 , t1 ], we obtain lQ¯ i (f3 ) = lR[ti ,t1 ] (f ∗ ) + lR[t1 ,si ] (f3 ) ≤ lR[s2 ,t2 ] (f ∗ ) − lQi ∩R[s2 ,t1 ] (f ∗ ) + 2lQ¯ 1 (f1 ) ¶ µ 2 8 6 M (f ∗ ) + M (f ∗ ) = M (f ∗ ), ≤ M (f ∗ ) − 1 − γ γ γ where the last inequality is based on Claims 9 and 3. Similarly, since R[tj , s2 ] ⊆ R[t2 , s2 ] = ¯ 2 and hence lR[t ,s ] (f3 ) ≤ 2l ¯ (f2 ). Then, noticing that R[s2 , sj ] ⊆ R[s2 , t2 ]\R[sj , t1 ] and Q Q2 j 2 R[sj , t1 ] = Qj ∩ R[s2 , t1 ], we obtain lQ¯ j (f3 ) = lR[s2 ,sj ] (f ∗ ) + lR[tj ,s2 ] (f3 ) ≤ lR[s2 ,t2 ] (f ∗ ) − lQj ∩R[s2 ,t1 ] (f ∗ ) + 2lQ¯ 2 (f2 ) µ ¶ 6 6 12 ∗ ≤ M (f ) − 1 − M (f ∗ ) + M (f ∗ ) = M (f ∗ ), γ γ γ 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∈ ¯ i, Q ¯ j } by (4.4). From Claim 5 and (4.3), we have {Q lQ (f ∗ ) +
4 M (f ∗ ) > lQ (f ∗ ) + 2||R[t2 , s1 ]|| γ ≥ 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 ]||
0
X
X
2(2ae + be ) +
e∈E(R[tj ,si ])
e∈E(R[tj ,si ])
fe∗ =0
fe∗ >0
X
X
2(2ae + be ) −
e∈E(R[tj ,si ])
e∈E(R[tj ,si ])
fe∗ =0
fe∗ >0
X
X
4(ae + be ) −
e∈E(R[tj ,si ])
[fe∗ · le (fe∗ ) − (fe∗ + 2)le (fe∗ + 2)]
[fe∗ · le (fe∗ ) − (fe∗ + 2)(le (fe∗ ) + 2ae )]
[2fe∗ ae + 2le (fe∗ ) + 4ae ]
4le (fe∗ )
e∈E(R[tj ,si ]),fe∗ >0
≥ 2lR[sj ,ti ] (f ∗ ) − 4||R[tj , si ]|| − 4lR[tj ,si ] (f ∗ ) ≥ 2lR[sj ,ti ] (f ∗ ) − 4||R[t2 , s1 ]|| − 4lQ¯ 1 (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
¯ 2, . . . , Q ¯ i , . . .} from f ∗ by Case 2: Qs 6= ∅ = Qt . In this case, we derive flow f4 = {Q1 , Q changing the strategies of users 2 and ∗ fe f∗ − 2 f4 e = e∗ fe + 2
i (See Figure 3(e)). Similar to (4.3), we get for e ∈ E(R[si , s2 ]) ∪ E(R[ti , t2 ]), for e ∈ E(R[s2 , ti ]), for e ∈ E(R[t2 , si ]).
From Claims 7 and 9, it follows that lQ¯ 2 (f4 ) ≤ lQ¯ 2 ∪Q¯ 1 (f2 ) < lQ¯ i (f4 ) = lR[ti ,t1 ] (f ∗ ) + lR[t1 ,si ] (f4 ) ≤
3 M (f ∗ ), γ
6 9 M (f ∗ ) + lQ¯ 1 (f2 ) ≤ M (f ∗ ), γ γ
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 ∗ ), hf4 i ⊆ hf ∗ i and τ (f ∗ ) − τ (f4 ) ≥ 2lR[s2 ,ti ] (f ∗ ) − 4||R[t2 , si ]|| − 4lQ¯ 1 (f ∗ ). ³ ´ By Claims 9, 5, and 3, we obtain τ (f ∗ ) − τ (f4 ) > 2 1 − γ6 M (f ∗ ) − γ8 M (f ∗ ) − γ4 M (f ∗ ) ≥ 0 giving τ (f ∗ ) > τ (f4 ) a contradiction to (Min2). ¯ 1 , Q2 , . . . , Q ¯ j , . . .} from f ∗ by Case 3: Qs = ∅ = 6 Qt . In this case, we derive flow f5 = {Q changing strategies of users 1 and j (See Figure 3(f)). Then f5 e = fe∗ for all e ∈ E(R[t1 , tj ]) ∪ E(R[s1 , sj ]), f5 e = fe∗ − 2 for all e ∈ E(R[sj , t1 ]), and f5 e = fe∗ + 2 for all e ∈ E(R[tj , s1 ]). Furthermore, lQ¯ 1 (f5 ) ≤ lQ¯ 1 (f2 )
0. To summarize, the contradictions in the above three cases show that our contradiction assumption is false. Hence f ∗ is a (54, 1)-approximate Nash flow, giving the conclusion of Theorem 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 splittable counterpart, the splittable selfish ring routing with linear latency (SSRR), by relaxing the unsplittable constraint f ∈ {0, 1}P to splittable one: f ∈ [0, 1]P , and f (Pi ) + f (P¯i ) = 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 X e∈E(P )
³
X
ae
X
xj +
1≤j≤k:E(Pj )3e
´ (1 − xj ) + be ≤ y
1≤j≤k:E(P¯j )3e
for P ∈ {Pi , P¯i }, 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, 1}P 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 reduce the latency he experiences in the current solution and, as easily verified with the potential function technique [17, 15], we finally reach a Nash flow f with M (f ) ≤ M (f˜) in time O(k 3 n2 maxni=1 {aei +bei }) and in time O(k 3 n2 ) 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 pseudopolynomial 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 continuous 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.
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