[论文解读] A unified approach to truthful scheduling on related machines

Leah Epstein, Asaf Levin+1 more|arXiv (Cornell University)|Jan 6, 2013

Auction Theory and Applications参考文献 35被引用 5

一句话总结

本文提出了一种统一的、单调的多项式时间近似方案（PTAS），用于在相关机器上进行 truthful 调度，目标函数包括 makespan、最小负载和 ℓp-范数。通过避免对作业大小或速度进行舍入，并在结构化调度类上使用动态规划，该方法实现了 (1+ε)-近似，且通过一种新颖框架确保了单调性与机制设计的兼容性，从而实现 truthful 支付。

ABSTRACT

We present a unified framework for designing deterministic monotone polynomial time approximation schemes (PTAS's) for a wide class of scheduling problems on uniformly related machines. This class includes (among others) minimizing the makespan, maximizing the minimum load, and minimizing the ep norm of the machine loads vector. Previously, this kind of result was only known for the makespan objective. Monotone algorithms have the property that an increase in the speed of a machine cannot decrease the amount of work assigned to it. The key idea of our novel method is to show that for goal functions that are sufficiently well-behaved functions of the machine loads, it is possible to compute in polynomial time a highly structured nearly optimal schedule.An interesting aspect of our approach is that, in contrast to all known approximation schemes, we avoid rounding any job sizes or speeds throughout. We can therefore find the exact best structured schedule using dynamic programming. The state space encodes a sufficient amount of information such that no postprocessing is needed, allowing an elegant and relatively simple analysis without any special cases. The monotonicity is a consequence of the fact that we find the best schedule in a specific collection of schedules. Monotone approximation schemes have an important role in the emerging area of algorithmic mechanism design. In the game-theoretical setting of these scheduling problems there is a social goal, which is one of the objective functions that we study. Each machine is controlled by a selfish single-parameter agent, where its private information is its cost of processing a unit sized job, which is also the inverse of the speed of its machine. Each agent wishes to maximize its own profit, defined as the payment it receives from the mechanism minus its cost for processing all jobs assigned to it, and places a bid which corresponds to its private information. For each one of the problems, we show that we can calculate payments that guarantee truthfulness in an efficient manner. Thus, there exists a dominant strategy where agents report their true speeds, and we show the existence of a truthful mechanism which can be implemented in polynomial time, where the social goal is approximated within a factor of 1 + e for every e > 0.

研究动机与目标

• 开发一种统一的框架，用于在均匀相关机器上设计单调的 PTAS。
• 解决在算法机制设计中，针对具有自私代理的调度问题，设计 truthful 机制的挑战。
• 将已知的单调 PTAS 结果从 makespan 扩展到更广泛的目标准则函数，包括最小负载和 ℓp-范数。
• 消除近似方案中对作业大小或速度进行舍入的需要，从而实现对结构化调度的精确计算。
• 确保支付可高效计算，以保证在主导策略下实现 truthfulness。

提出的方法

• 该方法构建了一个精心设计的调度类，其编码了足够信息以避免后续处理。
• 使用动态规划在该结构化类中精确计算最优调度，无需对作业大小或机器速度进行舍入。
• 该框架依赖于目标函数的良好性质（例如，凸性、连续性）以确保多项式时间可解。
• 单调性通过算法从一个固定且明确定义的集合中选择最优调度而自然实现。
• 通过利用调度结构和代理的出价，高效计算支付，以确保真实申报是主导策略。
• 该方法避免了特殊情况，实现了对多种目标函数的简洁、统一分析。

实验结果

研究问题

• RQ1 能否使用单一框架为相关机器上的多个调度目标设计单调的 PTAS？
• RQ2 如何为机器由自私代理控制且具有私有速度的调度问题构建 truthful 机制？
• RQ3 是否可能在近似方案中不进行作业大小或速度的舍入，仍实现 (1+ε)-近似？
• RQ4 目标函数的哪些结构性质使得动态规划和单调性在调度中能够高效实现？
• RQ5 如何高效计算支付以确保在主导策略下实现 truthfulness？

主要发现

• 该框架在相关机器上对 makespan、最小负载和 ℓp-范数目标均实现了 (1+ε)-近似。
• 该算法具有单调性，即提高某台机器的速度不会减少其分配的工作负载。
• 该方法避免了对作业大小或速度的舍入，从而可通过动态规划精确计算最优结构化调度。
• 支付可在多项式时间内计算，以保证 truthfulness，使机制可高效实现。
• 该方法为多个调度目标提供了统一解决方案，扩展了以往仅限于 makespan 的结果。
• 该框架使得设计一种 truthful 机制成为可能，且代理以真实速度申报是其主导策略。