Optimization Letters https://doi.org/10.1007/s11590-018-1285-3 SHORT COMMUNICATION
Improved price of anarchy for machine scheduling games with coordination mechanisms Long Zhang1 · Yuzhong Zhang1 · Donglei Du2 · Qingguo Bai1 Received: 4 December 2017 / Accepted: 8 June 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract We study several machine scheduling games, each involving n jobs to be processed on m uniformly related machines. Each job, as an agent, selects a machine for processing to minimize his disutility (e.g., the completion time of the agent). We analyze the price of anarchy (PoA) for these scheduling games, where the PoA is defined as maximum ratio of the central objective value of the worst pure Nash equilibrium over the optimal central objective value among all problem instances. We improve several existing results in the literature. First, we give an improved upper bound of the PoA for the scheduling game studied by Hoeksma and Uetz (WAOA’11 proceedings of the 9th international conference on approximation and online algorithms, vol 9. Springer, Berlin, pp 261–273, 2011). Then, we present a better lower bound of the PoA for the scheduling game studied by Lee et al. (Eur J Oper Res 220:305–313, 2012). Finally, we provide improved upper bounds of the PoA in terms of the number of machines, for another scheduling game proposed by Chen and Gürel (J Sched 15:157–164, 2012). Keywords Machine scheduling games · Coordination mechanisms · Nash equilibrium · Price of anarchy
B
Yuzhong Zhang
[email protected] Long Zhang
[email protected] Donglei Du
[email protected] Qingguo Bai
[email protected]
1
Institute of Operations Research, School of Management, Qufu Normal University, Rizhao 276826, Shandong, China
2
Faculty of Business Administration, University of New Brunswick, Fredericton, NB E3B 5A3, Canada
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1 Introduction We consider several machine scheduling games where n jobs are processed on m uniformly related machines. In contrast to the classical scheduling problem where a central authority makes scheduling decisions, each job is an agent whose strategy space is the set of machines. Agents are selfish, trying to maximize their own utilities, or equivalently, minimizing their disutility, such as the completion time in the scheduling setting. A Nash equilibrium (NE) is a solution such that no agent can improve their utility by unilaterally changing their own strategy. A system normally has a central objective (also called social cost) that could be different from the agents’ objectives. In a scheduling game, due to the selfish behaviour of its agents, optimal social cost may not be achievable. In other words, Nash equilibria do not necessarily lead to globally optimal solutions for the central objective. We quantify this inefficiency via the concept of price of anarchy (PoA) first proposed by Koutsoupias and Papadimitriou [1]: max f (σ )
PoA(G) = sup PoA(g) = sup g∈G
g∈G
σ ∈ψ(g)
Opt(g)
,
where G is a family of games, ψ(g) is the set of Nash equilibria of the game g ∈ G, f (σ ) is the social cost of σ ∈ ψ(g), and Opt(g) is the optimal objective value for the game g ∈ G. A machine scheduling game throughout this paper is characterized by a set of machines M = {1, 2, . . . , m}, and a set of jobs N = {1, 2, . . . , n}, where a job j ∈ N has a length (i.e., processing time) p j , and a machine i ∈ M has a speed si . The set of possible strategy profiles of jobs is denoted by Ω. A strategy profile of jobs is denoted by σ = (σ1 , σ2 , . . . , σn ) ∈ Ω, where σ j ∈ M denotes the machine that job j selects. The strategy profile is also referred to as a schedule. For a given schedule σ , the completion time of job j is denoted by C j (σ ). For simplicity, σ will be omitted unless it is required. The central objective function is the total completion time of all jobs, namely nj=1 C j . In a machine scheduling game, each job tries to maximize its own utility or, equivalently, minimize its own disutility (e.g., the utility of job j is the opposite of the completion time of the job defined as −C j ). Jobs assigned to a given machine compete for an earliest completion, which may cause chaos. To resolve this conflict, each machine will announce, in advance, its sequencing rule to schedule all jobs assigned to it. The sequencing rule used by each machine is usually called the local policy of the machine. In this work, we assume that the Shortest Processing Time first (SPT) rule is used by each machine, i.e., jobs assigned to any machine are sequenced in a non-decreasing order of their processing times. For alternate sequencing rules in scheduling games, we refer the readers to Christodoulou et al. [2], Ye and Zhang [3], Nong et al. [4], and Chen et al. [5]. We utilize the following three-field notation α|β|γ to classify different scheduling games, as proposed by Lee et al. [6]. The α field contains two entries that specify the machine environment and the local policy of each machine, the β field describes job
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specifications and the utility function of the job in the form of ut = utility function, and the γ field refers to the central objective function. For example, P2 (SPT)|ut = −C j |Cmax denotes a machine scheduling game where there are two identical machines, each of whose local policy is SPT. Each job tries to minimize its own completion time, and the central objective function is the makespan, Cmax = max1≤ j≤n C j . There has been extensive interest in machine scheduling games with the SPT rule in the last decade. These results of the scheduling games considered in this paper are summarized in Table 1. There are several results for the load-balancing game Pm (·)|ut = −L σ j | L σ j , where (·) means that the local policy of the machine is immaterial, and L σ j = k:σk =σ j pk is the load of the machine to which job j assigned. These results are summarized in Table 2. For more recent work on scheduling games, we refer the readers to Feldman and Tamir [7], Caragiannis [8], Lin et al. [9], Liu et al. [10], Xie et al. [11], Xie et al. [12]. The rest of this paper is organized as follows. In Sect. 2, we characterize the optimal C j . In Sect. 3, we give an improved upper bound solutions for Qm (SPT)|ut = −C j | on the PoA for Qm (SPT)|ut = −C j | C j . In Sect. 4, we provide an improved lower bound on the PoA for Q2 (SPT)|ut = −C j | C j . In Sect. 5, we offer an improved upper bounds on the PoA for Pm (·)|ut = −L σ j | L σ j . Section 6 offers concluding remarks.
Table 1 The existing PoA for some scheduling games with SPT rule Problem
PoA
References
Pm (SPT)|ut = −C j |Cmax
2 − m1
Immorlica et al. [13]
Θ(log m)
Aspnes et al. [15]
1
Conway et al. [16]
√ 1+ 5 2
Q2 (SPT)|ut = −C j |Cmax Qm (SPT)|ut = −C j |Cmax Pm (SPT)|ut = −C j | C j Q2 (SPT)|ut = −C j | C j Qm (SPT)|ut = −C j | C j
Table 2 The existing PoAs for the load-balancing game Pm (·)|ut = −L σ j | L σ j
√ √ 3+ 3 , 1+ 5 4 2
≤ 2
Cho and Sahni [14]
Lee et al. [6] Hoeksma and Uetz [17]
PoA
References
2 ≤ 2 − m+1 ≤ 4 pmax
Vöcking [18]
n) ≤ Θ( m
Lee et al. [6]
≤ ρ1 + 1
Chen and Gürel [20]
Berenbrink et al. [19]a
a PoA ≤ 4 p
max can be extended to m uniform machines, where pmax is the maximum job processing time. ρ1 is the ratio of average job processing time to minimum job processing time
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2 Characterization of optimal solutions In this section we present some basic properties of the structure of an optimal schedule for Qm (SPT)|ut = −C j | C j . It is not difficult to see that SPT is the optimal local policy of the machine for this social cost. W.l.o.g., we always assume that the job set N is indexed in an non-decreasing processing time order, and the machine set M is indexed in an non-decreasing speed order. For a given schedule σ , let Ji (σ ) be the set of jobs scheduled on machine i (i = 1, . . . , m), namely Ji (σ ) = { j|σ j = i}. Let n i (σ ) = |Ji (σ )| be the number of jobs on machine i in schedule σ . Let h iσ ( j) = |{k > j|σk = i}| be the number of jobs that have a higher index than j on machine i in schedule σ . Let C j (σ j , ν− j ) = p j /sσ j + k∈Jσ j (ν),k< j pk /sσ j be the completion time of job j when it unilaterally deviates from machine ν j to machine σ j in the schedule ν and is processed with the SPT local policy. We need the following two known lemmas. Lemma 1 (Lemma 1 of [17]) A schedule σ is optimal for Qm (SPT)| ut = −C j | C j if and only if h lσ ( j)/sl ≤ (h iσ ( j) + 1)/si for all machines i and l. Lemma 2 (Algorithm MFT of [21]) There exists an optimal schedule for Q m (SPT)|ut = −C j | C j , such that job n is processed on the fastest machine, assumed to be machine m, w.l.o.g. Theorem 1 For an optimal schedule σ of Q m (SPT)|ut = −C j | C j such that n i (σ ) > 0 for all machines i, we have sm n m (σ ) + 1 . ≤ si n i (σ ) Proof Suppose there is a machine i such that n i (σ ) > 0 and optimal schedule σ . Then n i (σ ) n m (σ ) + 1 > si sm
sm si
>
n m (σ )+1 n i (σ )
in an (1)
for machine i. Therefore, there are at least n m (σ ) + 1 jobs on machine m in the optimal schedule σ from (1), a contradiction since there n m (σ ) jobs on machine m in the optimal schedule σ . Theorem 2 For any optimal schedule σ of Q m (SPT)|ut = −C j | C j , we have n
C j (σ ) ≤
j=1
(n + m)(n + 1) pn . 2 sm
Proof According to the definitions of notations h iσ ( j) and n i (σ ), the sum of the completion times of jobs in Ji (σ ) is as follows. j∈Ji (σ )
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C j (σ ) =
j∈Ji (σ )
(h σσ j ( j) + 1)
pj sσ j
Improved price of anarchy for machine…
≤
j∈Ji (σ )
=
n i (σ )
k
k=1
(h σσ j ( j) + 1)
pn sσ j
pn si
Summing over all machines from 1 to m, the sum of the completion times of jobs N in the optimal schedule σ is as follows. n
C j (σ ) =
m
C j (σ )
i=1 j∈Ji (σ )
j=1
≤
m n i (σ ) k pn si i=1 k=1
=
m n i (σ ) sm p n k si sm i=1 k=1
Together with Theorem 1, we have m n m n i (σ ) i (σ ) sm p n n m (σ ) + 1 pn k ≤ k si sm n i (σ ) sm i=1 k=1
i=1 k=1
= (n m (σ ) + 1) = (n m (σ ) + 1)
m pn n i (σ )(n i (σ ) + 1) sm 2n i (σ )
pn sm
i=1 m i=1
n i (σ ) + 1 2
pn n + m . = (n m (σ ) + 1) sm 2 For n m (σ ) ≤ n, we have (n m (σ ) + 1)
pn n + m pn n + m ≤ (n + 1) sm 2 sm 2 (n + m)(n + 1) pn . = 2 sm
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3 Upper bound on the PoA for Qm (SPT)|ut = −Cj |
Cj
In thissection we give an improved upper bound on the PoA for Qm (SPT)|ut = −C j | C j over PoA ≤ 2 by Hoeksma and Uetz [17]. Theorem 3 For the game Q m (SPT)|ut = −C j | C j , we have PoA ≤ 2 −
2 . (n + m)(n + 1)
Proof Let schedule σ and ν be an optimal schedule and a Nash equilibrium, respectively. The sum of the completion times of all jobs that are on machine i in σ is given by ⎛
C j (σ j , ν− j ) =
j∈Ji (σ )
⎞
⎜ pj pk ⎟ ⎜ ⎟ ⎝ si + si ⎠
j∈Ji (σ )
=
j∈Ji (σ )
=
j∈Ji (σ )
k∈Ji (ν) k< j
pj + si
pk si
j∈Ji (σ ) k∈Ji (ν) k< j
pj pk + h iσ (k) . si si k∈Ji (ν)
Summing over all machines from 1 to m, we obtain that m
C j (σ j , ν− j ) =
i=1 j∈Ji (σ )
m i=1 j∈Ji (σ )
m pj σ pk + h i (k) si si i=1 k∈Ji (ν)
n n pj pj = + h σν j ( j) . sσ j sν j j=1
(2)
j=1
Note that h σσ j ( j) ≥ 0. So n n n pj pj ≤ (h σσ j ( j) + 1) = C j (σ ). sσ j sσ j j=1
j=1
j=1
Note that h σνn (n) = 0, so n j=1
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h σν j ( j)
pj pj = h σν j ( j) . sν j sν j n−1 j=1
(3)
Improved price of anarchy for machine…
Lemmas 1, 2 imply that n−1
h σν j ( j)
j=1
n−1 pj pj ≤ (h σσ j ( j) + 1) sν j sσ j
= =
j=1 n
(h σσ j ( j) + 1)
j=1 n
C j (σ ) −
j=1
pj pn − sσ j sσn
pn . sm
(4)
Theorem 2, (2), (3) and (4) together imply that m
C j (σ j , ν− j ) ≤ 2
i=1 j∈Ji (σ )
n j=1
n pn 2 C j (σ ) − ≤ 2− C j (σ ). sm (n + m)(n + 1) j=1
Since ν is a Nash equilibrium, we have n
C j (ν) ≤
j=1
=
n
C j (σ j , ν− j )
j=1 m
C j (σ j , ν− j )
i=1 j∈Ji (σ )
≤ 2−
n 2 C j (σ ). (n + m)(n + 1) j=1
Therefore an upper bound on the PoA for Qm (SPT)|ut = −C j | 2 . (n + m)(n + 1)
4 Lower bound on the PoA for Q2 (SPT)|ut = −Cj |
C j is 2 −
Cj
In thissection we show an improved lower bound on the PoA for Q2 (SPT)|ut = √ −C j | C j over PoA ≥ (3 + 3)/4 in Lee et al. [6]. We show that the following instance g gives us the improved lower bound. Instance g The number of jobs n = 7, with processing times p1 = s and p j = s ) j−1 for j = 2, 3, . . . , 7, and the number of machines m = 2, with speeds ( s−1 s1 = 1 and s2 = s ∈ (3/2, 5/3). Theorem 4 For the game Q 2 (SPT)|ut = −C j | C j , we have PoA ≥ 1.1875.
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Proof We consider the schedule σ where jobs 1, 3, and 6 are processed on machine 1 and jobs 2, 4, 5, and 7 are processed on machine 2 with SPT local policy. Therefore, we have
2
5
3 n s s s 1 s +3 C j (σ ) = 3s + 2 + + 4 s−1 s−1 s s−1 s−1 j=1
4
6 s s (5) +2 + s−1 s−1 3s 7 − 15s 6 + 46s 5 − 81s 4 + 88s 3 − 59s 2 + 23s − 4 . (s − 1)6
=
On the other hand, it is easy to verify that the schedule ν where all jobs are processed on the machine 2 with SPT local policy is a Nash equilibrium. Therefore, we have
2
3 s 1 s s +5 C j (ν) = +4 7s + 6 s s−1 s−1 s−1 j=1
4
5
6 s s s +3 +2 + . s−1 s−1 s−1
n
Computing
s s−1
(6)
∗ (6) − (6), we have
2
3 s s s−1 s s −6 + C j (ν) = + 7 s s−1 s−1 s−1 s−1 j=1
4
5
6
7 s s s s + + + + s−1 s−1 s−1 s−1
8 s−1 s −s = (s − 1) s s−1
7 s = (s − 1) − (s − 1). s−1
n
(7)
Combining (5) and (7), we have n
j=1 C j (ν)
PoA ≥ n
j=1 C j (σ )
=
3s 7
− 15s 6
+ 46s 5
s 7 − (s − 1)7 = f (s). − 81s 4 + 88s 3 − 59s 2 + 23s − 4
An approximation of the maximum value of function f (s) is 1.1875 when s equals to 1.5737 which is obtained via MATLAB.
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5 Upper bound on the PoA for Pm (·)|ut = −Lj |
L j
For this game, we provide an improved upper bound on PoA ≤ ρ1 + 1 − 1/m over PoA ≤ ρ1 + 1 in Chen and Gürel [20]. Let P = nj=1 p j be the total processing time for all jobs, let pmin be the minimum job processing time, and let ρ1 = P/(npmin ) be the ratio of average job processing time to minimum job processing time. Lemma 3 (Theorem n 5 of [6]) For any Nash equilibrium ν of Pm (·)|ut = −L σ j | L σ j , the social cost j=1 L ν j satisfies the following inequality n
Lνj ≤
j=1
m−1 n P+ P. m m
Theorem 5 For any optimal schedule σ of Pm (·)|ut = −L σ j | n j=1 L σ j satisfies the following inequalities n
L σ j ≥ P , and
j=1
n
Lσj ≥
j=1
L σ j , the social cost
n2 pmin . m
Proof Let n i (σ ) denote the number of jobs on machine i in schedule σ , and let L i (σ ) denote the load of the machine i in schedule σ . We have n
Lσj =
j=1
m
m
n i (σ )L i (σ ) ≥
i=1
L i (σ ) =
i=1
n
p j = P.
j=1
Note that for all the machine i, we have L i (σ ) ≥ n i (σ ) pmin . So, n
Lσj =
j=1
With Cauchy inequality
m i=1
m
(n i (σ ))2 ≥
i=1
n, we have
n i (σ )L i (σ ) ≥
m
m
(n i (σ ))2 pmin .
i=1
1 m
m
2 n i (σ )
and noticing that
i=1
(n i (σ ))2 pmin ≥
i=1
m
n i (σ ) =
i=1
n2 pmin . m
Theorem 6 For the game Pm (·)|ut = −L σ j |
L σ j , we have
PoA ≤ ρ1 + 1 − 1/m.
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Proof Lemma 3 and Theorem 5 imply that n j=1
PoA ≤ n
j=1
Lνj Lσj
≤
n m n2 m
P
pmin
+
m−1 m P
P
≤ ρ1 + 1 −
1 . m
6 Conclusions In this paper, we considered three machine scheduling games. For the first game Qm (SPT)|ut = −C j | C j , we give an improved upper bound on the PoA ≤ 2 − 2/((n + m)(n + 1)) over PoA ≤ 2. For the second game Q2 (SPT)| ut = −C j |√ C j , we show an improved lower bound on the PoA ≥ 1.1875 over PoA ≥ (3 + 3)/4. For the last game Pm (·)|ut = −L σ j | L σ j , we provide an improved upper bound on PoA ≤ ρ1 + 1 − 1/m over PoA ≤ ρ1 + 1. However, we leave open the question to find the tight PoA’s for these games. Acknowledgements The work of Yuzhong Zhang, Long Zhang and Qingguo Bai was supported in part by the National Natural Science Foundation of China under grants 11771251 and 71771138, the Natural Science Foundation of Shandong Province, China under grant ZR2017MG009, and the Key Project of Shandong Provincial Natural Science Foundation of China under grant ZR2015GZ009. The work of Donglei Du was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 283106 and National Science Foundation of China (NNSF) Grants (11771386 and 11728104).
References 1. Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Conference on Theoretical Aspects of Computer Science, pp. 404–413. Springer (1999) 2. Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. Theor. Comput. Sci. 410, 3327–3336 (2009) 3. Ye, D., Zhang, G.: Coordination mechanisms for selfish parallel jobs scheduling. In: Theory and Applications of Models of Computation, pp. 225–236. Springer, Berlin (2012) 4. Nong, Q.Q., Fan, G.Q., Fang, Q.Z.: A coordination mechanism for a scheduling game with parallelbatching machines. J. Comb. Optim. 33(2), 1–13 (2016) 5. Chen, Q., Lin, L., Tan, Z., Yan, Y.: Coordination mechanisms for scheduling games with proportional deterioration. Eur. J. Oper. Res. 263(2), 380–389 (2017) 6. Lee, K., Leung, Y.T., Pinedo, M.L.: Coordination mechanisms for parallel machine scheduling. Eur. J. Oper. Res. 220(2), 305–313 (2012) 7. Feldman, M., Tamir, T.: Conflicting congestion effects in resource allocation games. Oper. Res. 60(3), 529–540 (2012) 8. Caragiannis, I.: Efficient coordination mechanisms for unrelated machine scheduling. Algorithmica 66(3), 512–540 (2013) 9. Lin, L., Xian, X., Yan, Y., He, X., Tan, Z.: Inefficiency of equilibria for scheduling game with machine activation costs. Theor. Comput. Sci. 607(P2), 193–207 (2015) 10. Liu, L., Tang, G., Fan, B., Wang, X.: Two-person cooperative games on scheduling problems in outpatient pharmacy dispensing process. J. Comb. Optim. 30(4), 938–948 (2015) 11. Xie, F., Xu, Z., Zhang, Y., Bai, Q.: Scheduling games on uniform machines with activation cost. Theor. Comput. Sci. 580, 28–35 (2015) 12. Xie, F., Zhang, Y., Bai, Q., Xu, Z.: Inefficiency analysis of the scheduling game on limited identical machines with activation costs. Inf. Process. Lett. 116(4), 316–320 (2016)
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Improved price of anarchy for machine… 13. Immorlica, N., Li, L., Mirrokni, V.S., Schulz, A.: Coordination mechanisms for selfish scheduling. Theor. Comput. Sci. 410(17), 1589–1598 (2009) 14. Cho, Y., Sahni, S.: Bounds for list schedules on uniform processors. SIAM J. Comput. 9(1), 91–103 (1980) 15. Aspnes, J., Azar, Y., Fiat, A., Plotkin, S., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM 44(3), 486–504 (1997) 16. Conway, R.W., Maxwell, W.L., Miller, L.W.: Theory of Scheduling. Addison-Wesley Publishing Company, Boston (1967) 17. Hoeksma, R., Uetz, M.: The price of anarchy for minsum related machine scheduling. In: WAOA’11 Proceedings of the 9th International Conference on Approximation and Online Algorithms, vol. 9, pp. 261–273. Springer, Berlin (2011) 18. Vöcking, B.: Selfish load balancing. Algorithmic Game Theory 20, 517–542 (2007) 19. Berenbrink, P., Goldberg, L.A., Goldberg, P.W., Martin, R.: Utilitarian resource assignment. J. Discrete Algorithms 4(4), 567–587 (2006) 20. Chen, B., Gürel, S.: Efficiency analysis of load balancing games with and without activation costs. J. Sched. 15(2), 157–164 (2012) 21. Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. J. ACM 23(2), 317–327 (1976)
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