Truthful mechanisms for two-range-values variant of unrelated scheduling

In this paper, we consider a restricted variant of the scheduling problem, where the machines are the strategic players. For this multi-parameter mechanism design problem, the only known truthful mechanisms use task independent allocation algorithms and only have approximation ratio O(m) [N. Nisan, A. Ronen. Algorithmic mechanism design (extended abstract), in: STOC'99: Proceedings of the thirty-first annual ACM symposium on Theory of computing, ACM, New York, NY, USA, 1999. pp. 129-140; A. Mu'alem, M. Schapira, Setting lower bounds on truthfulness: Extended abstract, in: SODA'07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2007, pp. 1143-1152; P. Lu, C. Yu, An improved randomized truthful mechanism for scheduling unrelated machines, in: 25th International Symposium on Theoretical Aspects of Computer Science, STACS, 2008, pp. 527-538; P. Lu, C. Yu, Randomized truthful mechanisms for scheduling unrelated machines, in: C.H. Papadimitriou, S. Zhang (Eds.), Proceedings of WINE, in: Lecture Notes in Computer Science, vol. 5385, Springer, 2008, pp. 402-413]. Lavi and Swamy first use the cycle monotone condition and design a 3-approximation truthful mechanism for a two value variant in [R. Lavi, C. Swamy, Truthful mechanism design for multi-dimensional scheduling via cycle monotonicity, in: EC'07: Proceedings of the 8th ACM conference on Electronic commerce, ACM, New York, NY, USA, 2007, pp. 252-261], where the processing time of task j on machine i, say t"i"j, can only be either a lower value L"j or a higher value H"j. We consider a generalized variant, where t"i"j lies in [L"j,L"j(1+@e)]@?[H"j,H"j(1+@e)] and @e is a parameter satisfying some condition. We consider two special cases, case A when H"j/L"j>2,@?j and case B when H"j/L"j@?2,@?j, and give randomized truthful mechanisms with approximation ratio 4(1+@e) for both cases. Based on these two cases' results, we are also able to deal with the general case of our two-range-values scheduling problem. We use a combination of two mechanisms, which is also a novel method in mechanism design for scheduling problems, and finally we give a randomized truthful mechanism with approximation ratio 7(1+@e). Although the generalization seems a little incremental, we actually use a very novel technique in the key step of proving truthfulness for case A, as well as a new mechanism scheme for case B. Besides, the results in this paper are the first truthful mechanisms with constant approximation ratios when a machine (player) can report infinitely possible values, which is quite different from the two value variant, in which only finite values are available. Furthermore, together with Lavi and Swamy's work, our results suggest that such a task-dependent approach can really do much better for the scheduling unrelated machines problem.