Truthful mechanism design for multidimensional scheduling via cycle monotonicity

We consider the makespan-minimization problem on unrelated machines in the context of algorithmic mechanism design. No truthful mechanisms with non-trivial approximation guarantees are known for this multidimensional domain. We study a well-motivated special case (also a multidimensional domain), where the processing time of a job on each machine is either “low” or “high.” We give a general technique to convert any c-approximation algorithm (in a black-box fashion) to a 3c-approximation truthful-in-expectation mechanism. Our construction uses fractional truthful mechanisms as a building block, and builds upon a technique of Lavi and Swamy Lavi, R., Swamy, C., 2005. Truthful and near-optimal mechanism design via linear programming. In: Proc. 46th FOCS, pp. 595–604]. When all jobs have identical low and high values, we devise a deterministic 2-approximation truthful mechanism. The chief novelty of our results is that we do not utilize explicit price definitions to prove truthfulness. Instead we design algorithms that satisfy cycle monotonicity Rochet, J., 1987. A necessary and sufficient condition for rationalizability in a quasilinear context. J. Math. Econ. 16, 191–200], a necessary and sufficient condition for truthfulness in multidimensional settings; this is the first work that leverages this characterization.