### Mechanism design for fractional scheduling on unrelated machines

George Christodoulou, Elias Koutsoupias, Annamária Kovács
2010 ACM Transactions on Algorithms
Scheduling on unrelated machines is one of the most general and classical variants of the task scheduling problem. Fractional scheduling is the LP-relaxation of the problem, which is polynomially solvable in the non-strategic setting, and is a useful tool to design deterministic and randomized approximation algorithms. The mechanism design version of the scheduling problem was introduced by Nisan and Ronen. In this paper, we consider the mechanism design version of the fractional variant of
more » ... problem. We give lower bounds for any fractional truthful mechanism. Our lower bounds also hold for any (randomized) mechanism for the integral case. In the positive direction, we propose a truthful mechanism that achieves approximation 3/2 for 2 machines, matching the lower bound. This is the first new tight bound on the approximation ratio of this problem, after the tight bound of 2, for 2 machines, obtained by Nisan and Ronen. For n machines, our mechanism achieves an approximation ratio of n+1 2 . Motivated by the fact that all the known deterministic and randomized mechanisms for the problem, assign each task independently from the others, we focus on an interesting subclass of allocation algorithms, the task-independent algorithms. We give a lower bound of n+1 2 , that holds for every (not only monotone) allocation algorithm that takes independent decisions. Under this consideration, our truthful independent mechanism is the best that we can hope from this family of algorithms. * A preliminary version of this work appeared in [10] . The second author was partially supported by IST programs IST-2005-15964 (AEOLUS) and IST-2008-215270 (FRONTS). 1 In game-theoretic settings n is used to denote the number of the players, while in scheduling literature, usually m is used to denote the cardinality of the machines set. In our case, the aforementioned sets coincide. We prefer to use the former notation, in order to be compatible with the original paper [25] .