Bayesian Truthful Mechanisms for Job Scheduling from Bi-criterion Approximation Algorithms

We provide polynomial-time approximately optimal Bayesian mechanisms for makespan min-imization on unrelated machines as well as for max-min fair allocations of indivisible goods, withapproximation factors of 2 and min{m− k + 1, Õ(√k)} respectively, matching the approxima-tion ratios of best known polynomial-time algorithms (for max-min fairness, the latter claimis true for certain ratios of the number of goods m to people k). Our mechanisms are ob-tained by establishing a polynomial-time approximation-sensitive reduction from the problemof designing approximately optimal mechanisms for some arbitrary objective O to that of de-signing bi-criterion approximation algorithms for the same objective O plus a linear allocationcost term. Our reduction is itself enabled by extending the celebrated “equivalence of sepa-ration and optimization” [26, 31] to also accommodate bi-criterion approximations. Moreover,to apply the reduction to the specific problems of makespan and max-min fairness we developpolynomial-time bi-criterion approximation algorithms for makespan minimization with costsand max-min fairness with costs, adapting the algorithms of [44], [9] and [3] to the type ofbi-criterion approximation that is required by the reduction. Supported by a Sloan Foundation Fellowship, a Microsoft Research Faculty Fellowship and NSF Award CCF-0953960 (CAREER) and CCF-1101491.Supported by a Microsoft Research Graduate Fellowship, NSF Graduate Research Fellowship and NSF awardCCF-1101491.