We propose an optimal mechanism for a sequencing problem where the jobs' processing times and waiting costs are private. Given public priors for jobs' private data, we seek to find a scheduling rule and incentive compatible payments that minimize the total expected payments to the jobs. Here, incentive compatible refers to a Bayes-Nash equilibrium. While the problem can be efficiently solved when jobs have single dimensional private data, we here address the problem with two dimensional private data. We show that the problem can be solved in polynomial time by linear programming techniques, answering an open problem in [13]. Our implementation is randomized and truthful in expectation. The main steps are a compactification of an exponential size linear program, and a combinatorial algorithm to decompose feasible interim schedules. In addition, in computational experiments with random instances, we generate some more insights.
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