New Efficiency Resuts for Makespan Cost Sharing Yvonne Beischwitz a, Forian Schoppmann a, a University of Paderborn, Department of Computer Science Fürstenaee, 3302 Paderborn, Germany Abstract In the context of scheduing, we study socia cost efficiency for a cost-sharing probem in which the service provider s cost is determined by the makespan of the served agents jobs. For identica machines, we give surprisingy simpe cross-monotonic cost-sharing methods that achieve the essentiay best efficiency Mouin mechanisms can guarantee. Sti, our methods match the budget-baance of previous (yet rather intricate) resuts. Subsequenty, we give a generaization for arbitrary jobs. Finay, we return to identica jobs in order to perform a fine-grained anaysis. We show that the universa worst-case efficiency bounds from [8] are overy pessimistic. Key words: Scheduing, Mechanism Design, Cost-Sharing, Cross-Monotonicity Introduction and Mode For n N, we use the abbreviations [n] := {,..., n} and H n := n i= i. Cost-Sharing. Consider n agents interested in some common service offered by a service provider. Each agent i [n] submits a bid b i R 0 indicating the amount of money she is wiing to pay for the service. Ony on the basis of these bids b := (b,..., b n ), the provider determines a set Q(b) [n] of served agents and their payments x(b) R n 0, referred to as cost shares. This is accompished via a commony-known cost-sharing mechanism (Q x) : R n 0 2 [n] R n 0. The agents bids can not be verified to indeed refect their true vauations v i R 0, since agents are assumed to act sefishy such to ony maximize their quasi-inear utiities defined as u i (b) := v i x i (b) if i Q(b) and 0 otherwise. Thus, a major chaenge is to create incentives for truthfu bidding, even if agents may coude. A mechanism (Q x) is group-strategyproof (GSP) This work was partiay supported by the IST Program of the European Union under contract number IST-5964 (AEOLUS). Internationa Graduate Schoo of Dynamic Inteigent Systems Artice pubished in Information Processing Letters 07 (2008)
if for every true vauation vector v R n 0 and any coaition K [n] there is no bid vector b R n 0, with b i = v i for i / K, such that u i (b) u i (v) for a i K and u i (b) > u i (v) for at east one i K. The ony genera technique to design GSP cost-sharing mechanisms is due to Mouin [7]. The key ingredient of a Mouin mechanism is a cross-monotonic cost-sharing method ξ : 2 [n] R n 0 such that for a A, B [n] and i A: ξ i (A) ξ i (A B). Given a bid vector b R n 0, mouin ξ := (Q x) ξ can be computed by a very simpe agorithm: Initiay, et Q(b) := [n]. Then repeatedy eiminate agents from Q(b) whose bids are beow their current cost shares (b i < ξ i (Q(b))) unti a remaining agents can afford their cost-shares. Let x(b) := ξ(q(b)). If ξ is cross-monotonic then mouin ξ is GSP [7]. We now focus on the service cost C : 2 [n] R 0, mapping each subset of agents to the cost of serving them. Typicay, costs stem from soutions to a combinatoria minimization probem and are defined ony impicitey. Computing an optima soution with cost C may be computationay intractabe. Thus, the service provider resorts to approximate soutions with cost C. At east a β-fraction of the cost incurred by serving the seected agents shoud be recovered, whie at the same time offering competitive prices. Formay, a mechanism is β-budget-baanced (β-bb, β ) if for a b R n 0: β C (Q(b)) i Q(b) x i (b) C(Q(b)). An economic objective is to design efficient mechanisms that trade off the incurred cost and the vauations of the rejected agents as good as possibe: The socia cost of a set A [n] with respect to C and C is SC v(a) := C (A) + i [n]\a v i and SC v (A) := C(A) + i [n]\a v i, respectivey. A mechanism is γ-efficient (γ-eff, γ ) if for any vauation vector v R n 0 and a A [n] it hods that SC v(q(v)) γ SC v (A). The Scheduing Probem. We consider as underying combinatoria minimization probem to schedue a subset of n jobs on m reated parae machines. The service provider administrates the machines and each agent owns exacty one job with processing time p i N. We et d := {p,..., p n }. The speed of machine j [m] is denoted as s j N, where S := j [m] s j. We say that jobs (machines) are identica if p i = for a i [n] (s j = for a j [m]). For a given assignment, et δ j be the sum of the processing times of the jobs assigned to machine j. Then, the competion time of a job assigned to j is δ j /s j and the makespan is max j [m] δ j /s j. We denote the makespan of an optima assignment for A [n] by C(A). For identica jobs, the optima makespan ony depends on the number of jobs. We therefore write c( A ) := C(A). Certainy, our mode may not very we refect rea-word appications. Agents might not ony be interested in their job being processed, but aso in its competion time. Furthermore, the makespan as the competion time of the whoe system may not refect the provider s cost propery. However, it has been shown that for some other more natura cost functions ike weighted competion time, cross-monotonic cost-sharing methods are impracticabe [2]. We thus consider our work to be a first basic step for cost-sharing scheduing scenarios. 2
Previous Work. Efficiency of cost-sharing mechanisms is traditionay defined as the maximization of the socia wefare, which is the sum of the receivers vauations minus the provider s cost. However, resuts by economists in the 970 s (e.g., [6]) impy that it is essentiay impossibe for a GSP mechanism to be both -BB and socia wefare efficient. Furthermore, Feigenbaum et a. [3] showed that even a constant factor approximation of both objectives is impossibe. Abeit this negative resut, Roughgarden and Sundararajan [8] proved that for the minimization of the socia cost, a homogeneous formuation of efficiency, a simutaneous approximation is possibe and can be investigated by the summabiity of a cost-sharing method. A cost-sharing method ξ is α- summabe (α-sum ) if for a A [n] and every order a,..., a A of A with A i := {a,..., a i } it hods that A i= ξ ai (A i ) α C(A). Theorem ([8]) Let ξ be a cross-monotonic cost-sharing method. Let α 0 be the smaest and β be the argest numbers such that ξ is α-sum and β-bb. Then mouin ξ is (α + β )-EFF and no better than max{α, β }-EFF. Brenner and Schäfer [2] introduce a genera ower bound on the summabiity: Theorem 2 ([2]) Let ξ : 2 [n] R n 0 be a β-bb cost-sharing method. If there are constants p, q and a set A [n] with A n C(A) such that C(B) p q for a B A, then ξ is not α-sum for any α < H n/p β. q Cross-monotonic cost-sharing methods for scheduing on parae machines under makespan minimization were first presented by Beischwitz and Monien []. Their methods aways guarantee a coverage of of the makespan cost of the 2d LPT agorithm [5], and even a coverage of m+ if either jobs or machines are 2m identica. On the other hand, they show that cross-monotonic methods cannot be better than m+ -BB in genera and not better than -BB for identica jobs d 2m or machines. Brenner and Schäfer [2] modified the cost-sharing method of [] for identica machines such that its summabiity improved from n to H 2 n. Our Contribution. In Section 3, we compement the work of [2] by ooking at reated machines. We give poynomiay computabe cross-monotonic cost-sharing methods for makespan scheduing that are m+ -BB for identica 2m jobs and -BB for arbitrary jobs, matching the bounds from []. The strength 2d of our approach ies in the simpicity of the new methods, aowing for first summabiity resuts. For identica jobs, our method is H n -SUM. We concude from Theorem 2 that this is the best that generay can be achieved. For arbitrary jobs, we show a tight approximate summabiity for our method. Athough in genera, a Mouin mechanism mouin ξ ir for our cost-sharing method ξ ir for identica jobs (as defined in Section 3) is no better than max{ 2m, H m+ n}-eff [8], we show better tight approximations of the socia cost for many cases in Section 4. Roughy speaking, we show that mouin ξ ir is no worse than 2-EFF if there is sufficienty arge demand. 3
2 LPT and Properties of Minimum Makespan Soutions Since computing the minimum makespan is NP-hard in genera, we assume that the service provider empoys Graham s LPT agorithm [5] to compute an aocation. LPT processes the jobs by decreasing processing times and assigns each job to a machine on which it has the smaest competion time (taking into account the jobs that have been assigned aready). Ties are resoved in a deterministic way. For a set A [n] we use LPT(A) to denote the makespan resuting from LPT. If a jobs have the same processing time, LPT computes an optima soution. For arbitrary jobs, LPT achieves an approximation ratio of 5/3 [4]. Its running time is O(n og n + nm) in genera and O(n og n + n og m) for identica jobs using a priority queue for job pacement. Lemma 3 The optimum makespan costs C and c satisfy the properties beow: (P) For a A, B [n]: C(A B) C(A) + C(B) (P2) For a k, [n]: c(k + ) + c(), especiay c(2k) 2k (P3) For a k [n]: k S (P4) For a k [n]: c(k S) = k (P5) For a k [n] with k m + : k 2m S m+ (P6) For a a [n]: min k [a] k S (P7) For a a [n]: If [a] is maximum with c() c(a) 2c(). k = min k [a], then k Proof: (P) and (P2) hod by subadditivity of optima makespan costs, (P3) and (P4) are trivia observations. To prove (P5), et q be the number of machines on which the makespan occurs (makespan machines). We change this assignment by moving one job from a makespan machine if it can achieve a stricty smaer competion time on another machine. This can be done at most q times; otherwise, the initia assignment was not optima. For a makespan machine j, it is δ j = s j and δ j + s j j [m]\{j}. Summation k yieds k +m S. Consequenty, k +(m ) S yieds the m+ bound. To prove (6), et min k [a] = c() for an [a]. By (P3), c() =. k S We continue to prove (P7). By (P2), c(2) c(). If now a 2, obviousy 2 cannot be maximum. Thus, a < 2. By (P2), c(a) c(2) 2c(). 3 Cost-Sharing Methods ξ IR for Identica Jobs on Reated Machines and ξ AR for Arbitrary Jobs on Reated Machines Definition 4 For each set A [n], define the method ξ ir : 2 [n] R n 0 by ξi ir (A) := min k [ A ] if i A and ξ ir k i (A) := 0 otherwise. 4
Theorem 5 ξ ir is cross-monotonic, m+-bb, and H 2m n-sum. For any A [n], the cost shares {ξi ir (A)} i A can be computed in time O(n og n + n og m). Proof: Cross-monotonicity is obvious. Fix A [n]. We start proving m+-bb. 2m Directy from the definition, it foows that i A ξi ir (A) c( A ). From now on, we assume strict inequaity; otherwise, we have -BB. First assume A m. Let < A be maximum with c() = min k [ A ]. k (We do not consider = A since we assume strict inequaity above, i.e., i A ξi ir (A) < c( A ).) By (P7) and m +, we get the foowing bound: i A ξi ir (A) = A c() + c( A ) m m+ c( A ) c( A ). 2 2m 2 2m Now assume that A m +. By (P5), c( A ) A 2m. With (P6), we get S m+ i A ξi ir (A) A m+ c( A ). S 2m For summabiity, et a,..., a A be an arbitrary order of A, and et A i denote the set of the first i eements. Then, A i= ξ ir a i (A i ) A i= c(i) i H n c( A ). Since LPT on A identica jobs simutaneousy computes c(),..., c( A ), the time to compute min k [ A ] = ξ ir k i (A) for a i A is O(n og n + n og m). The instance with n identica jobs and n identica machines meets the conditions of Theorem 2 with p = q =. Hence, by Theorem, O()-BB Mouin mechanisms can in genera be no better than Θ(og n)-eff. In this sense, H n -SUM (and H n -EFF) is the best achievabe. We now appy our idea for identica jobs to arbitrary jobs. As a consequence, budget-baance impairs. However, we achieve -BB whie the best that can 2d in genera be achieved with cross-monotonic methods is -BB []. d For A [n], et P(A) := {p i i A} be the set of different processing times of the jobs in A, and et A(y) := {i A p i = y} denote the set of a agents with processing time y. Definition 6 For each set A [n] define the method ξ ar : 2 [n] R n 0 by ξi ar (A) := p i min P(A) k [ A(p i ) ] if i A and ξ ar k i (A) := 0 otherwise. Let P(A) = {y,..., y P(A) }. For k [ P(A) ], et τ k be the cardinaity of the k-th argest set in {A(y )} [ P(A) ]. Define H(A) := P(A) k= H τk k. Theorem 7 ξ ar is cross-monotonic, -BB, and H([n])-summabe. For 2 P([n]) any A [n], {ξi ar (A)} i A can be computed in time O(n og n + n og m). Proof: Cross-monotonicity is obvious. Fix A [n]. To show -BB, we 2 P([n]) first bound the sum of the cost shares from above by C(A). We have that i A ξ ar i (A) = y A(y) min P(A) k [ A(y) ] k P(A) y c( A(y) ) C(A), 5
where the ast inequaity foows from y c( A(y) ) = C(A(y)) C(A). Next we give the ower bound with respect to LPT. Appying (P7), we get i A ξ ar i (A) = = P(A) 2 P(A) A(y) y C(A(y)) = min k [A(y)] k 2 P(A) 2 P(A) y c( A(y) ) LPT(A(y)) LPT(A) 2 P(A). The ast inequaity is due to the fact that adding a LPT costs for the sets A(y) can never be smaer than the LPT cost of the whoe set A. For summabiity, et a,..., a A be an arbitrary order of A, and et A i denote the set of the first i eements. Furthermore, et {y,..., y P(A) } = P(A), in the order in which the processing times first occur in a,..., a A. Then, A i= ξ ar a i (A i ) = A i= p ai P(A i ) min k [ A i (p ai ) ] k P(A) = y A(y ) i= min k [i] k This estimate hods, since the midde term is maximized when jobs of the same processing time are contiguous in the order. This is due to the term P(A i ), which in that way is aways the smaest possibe number. The remaining computation is straightforward: A i= P(A) ξa ar i (A i ) = A(y ) i= y c(i) i P(A) = H A(y ) C(A) H(A) C(A), where we utiize that for a [ P(A) ] and a i [ A(y ) ] it hods that y c(i) y c( A(y ) ) = C(A(y )) C(A).. The time to compute {ξi ar (A)} i A is determined by the computation time of c(),..., c( A ) (for identica jobs). We have aready seen for ξ ir that this can be accompished in time O(n og n + n og m). If P([n]) {, n}, then we achieve H n -summabiity. In the foowing, we show that the summabiity bound H([n]) is tight for ξ ar. The foowing instance meets the bound from Theorem 7: Let d N >0 be the number of different processing times. For k [d] and q N >0, et there be n k jobs with processing time y k = q + k. Let n = d k= n k. It hods that P([n]) = d. Let there be m = n machines of speed q. We assume that processing times are increasing, i.e., p p n. Then, [i] corresponds to the set of the first i eements of [n]. As q, we get C([n]) = and n i= ξ ar i ([i]) = n q +... + j= j q d n d j= q + d = j q d k= H nk k C([n]). 6
4 Efficiency Guarantees of mouin ξ ir Athough in genera, mouin ξ ir is not better than max{ 2m, H m+ n}-eff [8], we show better approximations for many cases with identica jobs. Theorem 8 states that ony if mouin ξ ir gives the service to ess than S agents, the worst case performance of + H min{n,s} may occur. Theorem 8 Let v R n 0, µ = Q(v) be the number of agents seected by mouin ξ ir =: (Q x) ξ ir and σ be the argest cardinaity of a sets that minimize the socia cost SC v. Then for a A [n] it hods that SC v (Q(v)) γ SC v (A), where the vaues of γ are given for specific conditions in the tabe beow: case 2 3 4 5 condition σ < µ and µ m + σ < µ and µ < m + σ > µ and µ S σ > µ and µ < S σ = µ γ 2m m+ 2m 2 m 2 µ/s µ/s + + H min{n,s} To discuss these cases, we first give exampes which aso show that the resuts from Theorem 8 are tight. For simpicity, we write (a) n := (a,..., a) R n. Case : Consider m identica machines and n = m + jobs. It is x = (,,,...,, ). Let v := ( 2 3 m m m )n. It hods that µ = m + and σ = m. The socia costs are SC v ([µ]) = 2 and SC v ([σ]) = +. Whereas we have -BB m for to m agents (c[i] = i [m]), the cost increase of when moving from m to m + agents cannot be recovered by the cross-monotonic cost-shares. Case 2 : Consider m 2 machines, where m machines have speed 2 and machine has speed. Let there be n = m jobs and et v := ( 2(m ) )n. It is x = (,,,...,, ). It hods that µ = m and σ = m. The 2 4 6 2(m ) 2(m ) socia costs are SC v ([µ]) = and SC v ([σ]) = +. As in case, the 2 2(m ) ack of budget baance in favor of cross-monotonicity causes inefficiency. Case 3 : Consider m identica machines and n = 2m jobs. Furthermore, et v := () (m+) ( m ε)(m ). It is x = (,,,..., )( 2 3 m m )m. It hods that µ = m +, σ = 2m, SC v ([µ]) = 2 + (m )( ε), and SC m v([σ]) = 2. Despite -BB for 2m agents, the fact that some vauations are just beow the cost-shares causes efficiency oss. For m and ε 0 it is γ 3. 2 Case 4 : Consider m identica machines and n = m jobs, x = (,,,..., ). 2 3 m Let v := ( ε, ε, ε,..., ε). it hods that µ = 0 and σ = m. For 2 3 m ε 0, the socia costs are SC v ([µ]) = H n and SC v ([σ]) =. We have the same source of inefficiency as in case 3. Proof of Theorem 8: The main idea of this proof is to order the agents from to n such that v... v n. Then, Q(v) = [µ], and [σ] is the maximum set of agents that minimizes SC v. In the foowing, we determine the ratio between 7
SC v ([µ]) and SC v ([σ]). mouin ξ ir is obviousy -EFF for µ = σ. We define x R n c() 0 by x k := min [k] ; x k denotes ξ ir ([k]). We frequenty use that v i x µ for a i [µ] (because mouin ξ ir has decided to serve these agents). Case : With (P5), SC v ([µ]) µ 2m + n S m+ i=µ+ v i. Appying (P6), we get SC v ([σ]) = c(σ) + µ i=σ+ v i + n i=µ+ v i σ + µ σ + n S S i=µ+ v i. Case 2: Let [µ] be maximum with x µ = c(). It is σ > 0, since σ = 0 together with SC v ([]) = c() + n i=+ v i = x µ + n i=+ v i n i= v i = SC v ([σ]) contradicts the assumption that σ is maximum. Furthermore, m = impies µ = and σ = 0. From now on we thus assume σ > 0 and m >. We start showing that σ. Assume σ <. Using c() = x x σ c(σ), σ we get SC v ([σ]) = c(σ)+ n i=σ+ v i σ σ c()+ c()+ n + v i = SC v ([]), a contradiction to σ being maximum. We now show 2σ > µ. Otherwise, µ 2 and c(2) c() (see (P2)) 2 contradicts the maximaity of. Specificay, by (P2), 2c(σ) c(2σ) > c(µ). We aso observe that µx µ = µ c() µ + c(µ) c(µ) m c(µ), 2 2 2m 2 which foows from (P7), µ > σ, and m µ +. Now, with µ i=σ+ v i (µ σ) x µ µ σ m µ σ c(µ) c(µ), it is µ 2m 2 2m 2 SC v ([µ]) SC v ([σ]) c(µ) + ni=µ+ v i µ σ c(µ) + c(µ) + n 2 2m 2 µ+ v i + µ σ 2m 2 m. 2 2m 2 Case 3: By definition, x S c(s) =. Together with (P6) we get that for a S S i S, it hods that x i =. Furthermore, for a i > µ we have that v S i < S (since they did not receive service). Together with σ > µ and (P3) we get SC v ([µ]) SC v ([σ]) c(σ) + σµ+ v i + n σ+ v i c(σ) + n σ+ v i < + (σ µ) S S σ = 2 µ σ. Let p N, q {0} [S ] such that µ = p S + q. By (P3) and (P4), SC v ([(p+) S]) < p++ σ (p+) S + n S i=σ+ v i c(σ)+ n i=σ+ v i = SC v ([σ]). Consequenty, σ (p+) S, since σ > (p+) S contradicts the maximaity of σ. Now, we have that SCv([µ]) 2 µ S SC v([σ]) µ +. S Case 4: For i > µ we have that v i < x i c(i) and therefore it foows that i σi=µ+ v i < σ c(i) i=µ+ ( ) σi= i i c(σ) = Hσ c(σ). Now, SC v ([µ]) SC v ([σ]) < c(σ) + H σ c(σ) + ni=σ+ v i c(σ) + n + H σ. i=σ+ v i Furthermore, it hods that σ S. Otherwise, c(s) + n i=s+ v i < + σ S S n n + v i c(σ) + v i, i=σ+ i=σ+ contradicting [σ] minimizing socia cost. Thus, SCv([µ]) SC v([σ]) + H min{n,s}. 8
5 Concusion and Open Probems In our opinion, our fine-grained anaysis of efficiency oss pursues an interesting direction. Whereas for identica machines we identified cases with a sma efficiency oss, we eft it for future work to investigate how ikey these cases may actuay arise. Furthermore, we consider it interesting to find out, if such an anaysis is promising for the probem of scheduing arbitrary jobs on parae machines and/or for other cost-sharing probems. Another probem is to improve both the approximate efficiency and budgetbaance for the probem of scheduing arbitrary jobs on parae machines or to show that this is impossibe. Furthermore, aternatives to the makespan scheduing cost function as we as other scheduing modes shoud be considered. There are many open issues, e.g., agents might as we be interested in the competion times of their jobs, and the provider may be interested in the number of resources expended. Finay, one coud go beyond cross-monotonicity and deveop aternatives to Mouin mechanisms with better approximate budget-baance and/or efficiency. Acknowedgements We woud ike to thank the two anonymous referees for their hepfu comments. References [] Y. Beischwitz, B. Monien, Fair cost-sharing methods for scheduing jobs on parae machines, in: Proceedings of the 6th Itaian Conference on Agorithms and Compexity, vo. 3998 of LNCS, 2006. [2] J. Brenner, G. Schäfer, Cost sharing methods for makespan and competion time scheduing, in: Proceedings of the 24th Internationa Symposium on Theoretica Aspects of Computer Science, vo. 4393 of LNCS, 2007. [3] J. Feigenbaum, A. Krishnamurthy, R. Sami, S. Shenker, Hardness resuts for muticast cost sharing, Theoretica Computer Science 304 (-3) (2003) 25 236. [4] D. K. Friesen, Tighter bounds for LPT scheduing on uniform processors, SIAM Journa on Computing 6 (3) (987) 554 560. [5] R. Graham, Bounds on mutiprocessing timing anomaies, SIAM Journa of Appied Mathematics 7 (2) (969) 46 429. [6] J. Green, J.-J. Laffont, Characterizations of satisfactory mechanisms for the reveation of preferences for pubic goods, Econometrica 45 (2) (977) 427 438. URL http://www.jstor.org/stabe/929 [7] H. Mouin, Incrementa cost sharing: Characterization by coaition strategyproofness, Socia Choice and Wefare 6 (2) (999) 279 320. 9
[8] T. Roughgarden, M. Sundararajan, New trade-offs in cost-sharing mechanisms, in: Proceedings of the 38th ACM Symposium on Theory of Computing, 2006. 0