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).
Scheduling on unrelated machines is one of the most fundamental scheduling Problems. For this NP-hard optimization problem, there is a polynomial time algorithm with approximation ratio of @math @cite_11 . Especially if the number of machines is bounded by some constant, Angel, Bampis and Kononov gave an FPTAS @cite_6 . However there is no corresponding payment strategy to make either of the above allocation algorithms truthful.
@cite_6 @cite_11
2046825771 2102201348
We consider the following scheduling problem. There are m parallel machines and n independent jobs. Each job is to be assigned to one of the machines. The processing of job j on machine i requires time pij. The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3 2 unless P = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times. We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep ij . The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3 2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.