Truthful mechanism design via correlated tree rounding

One of the most powerful algorithmic techniques for truthful mechanism design are maximal-indistributional- range (MIDR) mechanisms. Unfortunately, many algorithms using this paradigm rely on heavy algorithmic machinery and require the ellipsoid method or (approximate) solution of convex programs. In this paper, we present a simple and natural correlated rounding technique for designing mechanisms that are truthful in expectation. Our technique is elementary and can be implemented quickly. The main property we rely on is that the domain offers fractional optimum solutions with a tree structure. In auctions based on the generalized assignment problem, each bidder has a publicly known knapsack constraint that captures the subsets of items that are of value to him. He has a private valuation for each item and strives to maximize the value of assigned items minus payment. For this domain we design a mechanism for social welfare maximization. Our technique gives a truthful 2-approximate MIDR mechanism without using the ellipsoid method or convex programming. In contrast to some previous work, our mechanism achieves exact truthfulness. In restricted-related scheduling with selfish machines, each job comes with a public weight, and it must be assigned to a machine from a public job-specific subset. Each machine has a private speed and strives to maximize payments minus workload of jobs assigned to it. For this domain we design a mechanism for makespan minimization. Although this is a single-parameter domain, the approximation status of the underlying optimization problem is similar to unrelated scheduling: The best known algorithm gives a (nontruthful) 2-approximation for unrelated machines, and there is 1.5-hardness. Our mechanism matches this bound and provides a truthful 2-approximation.