## Truthful mechanism design for multi-dimensional scheduling via cycle monotonicity (2007)

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Venue: | In Proceedings 8th ACM Conference on Electronic Commerce (EC |

Citations: | 36 - 11 self |

### BibTeX

@INPROCEEDINGS{Lavi07truthfulmechanism,

author = {Ron Lavi},

title = {Truthful mechanism design for multi-dimensional scheduling via cycle monotonicity},

booktitle = {In Proceedings 8th ACM Conference on Electronic Commerce (EC},

year = {2007},

pages = {252--261}

}

### OpenURL

### Abstract

We consider the problem of makespan minimization on m unrelated machines in the context of algorithmic mechanism design, where the machines are the strategic players. This is a multidimensional scheduling domain, and the only known positive results for makespan minimization in such a domain are O(m)-approximation truthful mechanisms [22, 20]. We study a well-motivated special case of this problem, where the processing time of a job on each machine may either be “low ” or “high”, and the low and high values are public and job-dependent. This preserves the multidimensionality of the domain, and generalizes the restricted-machines (i.e., {pj, ∞}) setting in scheduling. We give a general technique to convert any c-approximation algorithm to a 3capproximation truthful-in-expectation mechanism. This is one of the few known results that shows how to export approximation

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Citation Context ...) ∑ = i g(i, t)2 . We seek an integer max-flow in this flow network whose cost is minimum over all integer max-flows. It is known that such a flow can be computed in time poly ( log(T), ∑ ) j log Lj (=-=Ahuja et al., 1993-=-; Goldberg and Tarjan, 1990); since T is at most ∑ j Hj, this is polynomial in the number of bits required to encode the scheduling instance. Let g be a min-cost integer max-flow in the above flow net... |

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Citation Context ... itscorresponds to maximizing the minimum welfare and the notion of max-min fairness, and appears to be a much harder problem from the viewpoint of mechanism design. In particular, the celebrated VCG =-=[26, 9, 10]-=- family of mechanisms does not apply here, and we need to devise new techniques. The possibility of constructing a truthful mechanism for makespan minimization is strongly related to assumptions on th... |

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Citation Context ...he only private parameter of machine i. This assumption makes the domain of players’ types single-dimensional. Truthfulness in such domains is equivalent to a convenient value-monotonicity condition (=-=Myerson, 1981-=-; Archer and Tardos, 2001). This provides ample flexibility in mechanism design and appears to make it significantly easier to design truthful mechanisms in such domains. Archer and Tardos (2001) firs... |

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Citation Context ... itscorresponds to maximizing the minimum welfare and the notion of max-min fairness, and appears to be a much harder problem from the viewpoint of mechanism design. In particular, the celebrated VCG =-=[26, 9, 10]-=- family of mechanisms does not apply here, and we need to devise new techniques. The possibility of constructing a truthful mechanism for makespan minimization is strongly related to assumptions on th... |

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Citation Context ... itscorresponds to maximizing the minimum welfare and the notion of max-min fairness, and appears to be a much harder problem from the viewpoint of mechanism design. In particular, the celebrated VCG =-=[26, 9, 10]-=- family of mechanisms does not apply here, and we need to devise new techniques. The possibility of constructing a truthful mechanism for makespan minimization is strongly related to assumptions on th... |

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Citation Context ...ines are the strategic players. This is a multidimensional scheduling domain, and the only known positive results for makespan minimization in such a domain are O(m)-approximation truthful mechanisms =-=[22, 20]-=-. We study a well-motivated special case of this problem, where the processing time of a job on each machine may either be “low” or “high”, and the low and high values are public and job-dependent. Th... |

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Citation Context ...gorithmic problem (i.e., without requiring truthfulness) of makespan-minimization on unrelated machines is well understood and various 2-approximation algorithms are known. Lenstra, Shmoys and Tardos =-=[18]-=- gave the first such algorithm. Shmoys and Tardos [25] later gave a 2approximation algorithm for the generalized assignment problem, a generalization where there is a cost cij for assigning a job j to... |

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Citation Context ...the only private parameter of machine i. This assumption makes the domain of players’ types single-dimensional. Truthfulness in such domains is equivalent to a convenient value-monotonicity condition =-=[21, 3]-=-, which appears to make it significantly easier to design truthful mechanisms in such domains. Archer and Tardos [3] first considered the related machines setting and gave a randomized 3-approximation... |

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Citation Context ...s) of makespan-minimization on unrelated machines is well understood and various 2-approximation algorithms are known. Lenstra, Shmoys and Tardos [18] gave the first such algorithm. Shmoys and Tardos =-=[25]-=- later gave a 2approximation algorithm for the generalized assignment problem, a generalization where there is a cost cij for assigning ajobjto a machine i, and the goal is to minimize the cost subjec... |

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Citation Context ...nal domains [3, 7, 4, 1, 13]. For convex domains (i.e., each players’ set of private values is convex), it is known that cycle monotonicity is implied by a simpler condition, called weak monotonicity =-=[15, 6, 24]-=-. But even this simpler condition has not found much application in truthful mechanism design for multidimensional problems. Objectives other than social-welfare maximization and revenue maximization ... |

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Citation Context ...ained by first constructing a fractional truthful mechanism for a fractional relaxation of the problem, and then converting it into a truthfulin-expectation mechanism. This builds upon a technique of =-=[16]-=-, and shows the usefulness of fractional mechanisms in truthful mechanism design. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without... |

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Citation Context ...terms of payments, it turns out that truthfulness actually boils down to a certain algorithmic condition of monotonicity. This seems to have been first observed for multidimensional domains by Rochet =-=[23]-=- in 1987, and has been used successfully in algorithmic mechanism design several times, but for singledimensional domains. However for multidimensional domains, the monotonicity condition is more invo... |

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Citation Context ...e seek an integer max-flow in this flow network whose cost is minimum over all integer max-flows. It is known that such a flow can be computed in time poly ( log(T), ∑ ) j log Lj (Ahuja et al., 1993; =-=Goldberg and Tarjan, 1990-=-); since T is at most ∑ j Hj, this is polynomial in the number of bits required to encode the scheduling instance. Let g be a min-cost integer max-flow in the above flow network. We show that g is a p... |

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Citation Context ...sign of truthful mechanisms for singledimensional domains, where value-monotonicity yields a convenient characterization enabling one to concentrate on the algorithmic side of the problem (see, e.g., =-=[3, 7, 4, 1, 13]-=-). But for multidimensional domains, almost all positive results have relied on explicit price specifications in order to prove truthfulness (an exception is the work on unknown single-minded players ... |

57 | On approximately fair allocations of indivisible goods
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Citation Context ...lem, and Nisan [15] showed that the former objective cannot be implemented truthfully; Bezakova and Dani [5] gave a 0.5-approximation mechanism for two players with additive valuations. Lipton et al. =-=[19]-=- showed that the latter objective cannot be implemented truthfully. These lower bounds were strengthened in [20]. 2. PRELIMINARIES 2.1 The scheduling domain In our scheduling problem, we are given n j... |

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Citation Context ...nal domains [3, 7, 4, 1, 13]. For convex domains (i.e., each players’ set of private values is convex), it is known that cycle monotonicity is implied by a simpler condition, called weak monotonicity =-=[15, 6, 24]-=-. But even this simpler condition has not found much application in truthful mechanism design for multidimensional problems. Objectives other than social-welfare maximization and revenue maximization ... |

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Citation Context ...nal domains [3, 7, 4, 1, 13]. For convex domains (i.e., each players’ set of private values is convex), it is known that cycle monotonicity is implied by a simpler condition, called weak monotonicity =-=[15, 6, 24]-=-. But even this simpler condition has not found much application in truthful mechanism design for multidimensional problems. Objectives other than social-welfare maximization and revenue maximization ... |

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Citation Context ... approximation ratio better than 2, and strengthened this lower bound to m for two specific classes of deterministic mechanisms. Recently, [20] extended this lower bound to randomized mechanisms, and =-=[8]-=- improved the deterministic lower bound. In stark contrast with the above state of affairs, much stronger (and many more) positive results are known for a special case of the unrelated machines proble... |

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Citation Context ...n optimal schedule. (Recall that in the multidimensional unrelated machines setting, it is impossible to obtain a truthful mechanism with approximation ratio better than 2.) Various follow-up results =-=[2, 4, 1, 13]-=- have strengthened the notion of truthfulness and/or improved the approximation ratio. Such difficulties in moving from the single-dimensional to the multidimensional setting also arise in other mecha... |

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Citation Context ...onotonicity conditions for truthful mechanism design in arbitrary domains. The monotonicity condition we use, which is sometimes called cycle monotonicity, was first proposed by Rochet [23] (see also =-=[11]-=-). It is a generalization of value-monotonicity and completely characterizes truthfulness in every domain. Our methods and analyses demonstrate the potential benefitssof this characterization, and sho... |

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Citation Context ...kespan minimization is a common objective in scheduling environments, and has been well studied algorithmically in both the Computer Science and Operations Research communities (see, e.g., the survey =-=[12]-=-). Following the work of Nisan and Ronen [22], we consider each machine to be a strategic player or agent who privately knows its own processing time for each job, and may misrepresent these values in... |

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Citation Context ...n optimal schedule. (Recall that in the multidimensional unrelated machines setting, it is impossible to obtain a truthful mechanism with approximation ratio better than 2.) Various follow-up results =-=[2, 4, 1, 13]-=- have strengthened the notion of truthfulness and/or improved the approximation ratio. Such difficulties in moving from the single-dimensional to the multidimensional setting also arise in other mecha... |

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Citation Context ... optimal schedule. (Recall that in the multidimensional unrelated machines setting, it is impossible to obtain a truthful mechanism with approximation ratio better than 2.) Various follow-up results (=-=Archer, 2004-=-; Auletta et al., 2004; Andelman et al., 2007; Kovács, 2005) have given polyonomial time mechanisms that have strengthened the notion of truthfulness and/or improved the approximation ratio. Such diff... |

26 | Setting lower bounds on truthfulness
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(Show Context)
Citation Context ...ines are the strategic players. This is a multidimensional scheduling domain, and the only known positive results for makespan minimization in such a domain are O(m)-approximation truthful mechanisms =-=[22, 20]-=-. We study a well-motivated special case of this problem, where the processing time of a job on each machine may either be “low” or “high”, and the low and high values are public and job-dependent. Th... |

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Citation Context ...ule. (Recall that in the multidimensional unrelated machines setting, it is impossible to obtain a truthful mechanism with approximation ratio better than 2.) Various follow-up results (Archer, 2004; =-=Auletta et al., 2004-=-; Andelman et al., 2007; Kovács, 2005) have given polyonomial time mechanisms that have strengthened the notion of truthfulness and/or improved the approximation ratio. Such difficulties in moving fro... |

14 | Allocating indivisible goods
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Citation Context ... by a player, and computing an envy-minimizing allocation have been studied briefly. Lavi, Mu’alem, and Nisan [15] showed that the former objective cannot be implemented truthfully; Bezakova and Dani =-=[5]-=- gave a 0.5-approximation mechanism for two players with additive valuations. Lipton et al. [19] showed that the latter objective cannot be implemented truthfully. These lower bounds were strengthened... |

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Approximation algorithms for scheduling unrelated parallel machines
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Citation Context ...gorithmic problem (i.e., without requiring truthfulness) of makespan-minimization on unrelated machines is well understood and various 2-approximation algorithms are known. Lenstra, Shmoys and Tardos =-=[18]-=- gave the first such algorithm. Shmoys and Tardos [25] later gave a 2approximation algorithm for the generalized assignment problem, a generalization where there is a cost cij for assigning ajobjto a ... |

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Citation Context ...mation ratio. We then use a suitable randomized rounding procedure to convert the fractional assignment into a random integral assignment. For this, we use a recent rounding procedure of Kumar et al. =-=[14]-=- that is tailored for unrelated-machine scheduling. This preserves truthfulness, but we lose another additive factor equal to the approximation ratio. Our construction uses and extends some observatio... |

1 | A 1 + OE lower bound for truthful scheduling - Koutsoupias, Vidali - 2007 |

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1 |
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Citation Context ...lower bounds have been improved and extended in a series of recent results (Mu’alem and Schapira, 2007; Christodoulou, Koutsoupias, and Vidali , 2007; Christodoulou, Koutsoupias, and Kovács , 2007; 3=-=Koutsoupias and Vidali, 2007-=-). In stark contrast with the above state of affairs, much stronger (and many more) positive results are known for a special case of the unrelated machines problem, namely, the setting of related mach... |

1 | Proofs from Section 4 Proof of Lemma 4.1 : The proof is borrowed from Lavi and Swamy (2005), where a somewhat more general statement was proved. We are given that X(p) is the random assignment R(x(p)) that satisfies E [ ] X(p)ij = x(p)ij for all i, j. (Re - unknown authors |

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