We study the scheduling problem on unrelated machines in the mechanism design setting. This problem was proposed and studied in the seminal paper (Nisan and Ronen 1999), where they gave a 1.75-approximation randomized truthful mechanism for the case of two machines. We improve this result by a 1.6737-approximation randomized truthful mechanism. We also generalize our result to a @math -approximation mechanism for task scheduling with @math machines, which improve the previous best upper bound of $0.875m(Mu'alem and Schapira 2007).
@cite_4: 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 3c-approximation truthful-in-expectation mechanism. This is one of the few known results that shows how to export approximation algorithms for a multidimensional problem into truthful mechanisms in a black-box fashion. When the low and high values are the same for all jobs, we devise a deterministic 2-approximation truthful mechanism. These are the first truthful mechanisms with non-trivial performance guarantees for a multidimensional scheduling domain. Our constructions are novel in two respects. First, we do not utilize or rely on explicit price definitions to prove truthfulness; instead we design algorithms that satisfy cycle monotonicity. Cycle monotonicity [23] is a necessary and sufficient condition for truthfulness, is a generalization of value monotonicity for multidimensional domains. However, whereas value monotonicity has been used extensively and successfully to design truthful mechanisms in single-dimensional domains, ours is the first work that leverages cycle monotonicity in the multidimensional setting. Second, our randomized mechanisms are obtained by first constructing a fractional truthful mechanism for a fractional relaxation of the problem, and then converting it into a truthful-in-expectation mechanism. This builds upon a technique of [16], and shows the usefulness of fractional mechanisms in truthful mechanism design.
Lavi and Swarmy considered a restricted variant, where each task @math only has two values of running time , and gave a @math -approximation randomized truthful mechanism @cite_4 . They first use the cycle monotonicity in designing mechanisms.