Truthful mechanism design via correlated tree rounding

Abstract

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.

New Bounds for Truthful Scheduling on Two Unrelated Selfish Machines

“…Our approach is to formulate a mathematical optimization problem for R n . This approach was not common until recently when several successful truthful or truthful in expectation mechanisms have been constructed using linear or nonlinear programs [2,3,7,15]. This paper continues the trend to combine optimization with mechanism design and has the following contributions: (13) in Corollary 3.…”

“…The minimum makespan problem is one of many optimization problems considered in algorithmic mechanism design. These include, among others, combinatorial auctions (see, e.g., [2], [6] and references therein) and graph theoretic problems, such as the shortest paths tree [11] and the maximum matching problem [27].…”

“…The generalization considers a broader class of algorithms and provides stronger upper bounds for n ≤ 4. Our approach is fundamentally different from the methods in [2,7,15]. To begin with, our method is suitable for the minimum makespan problem on unrelated machines, while the methods in [2,7,15] are not guaranteed to work for this problem.…”

“…Next, we compare our approach to the existing methods for upper bounds in [2,3,7,15] that use optimization. Our method for upper bounds generalizes the approach by Chen et al [3].…”

“…Monotone in expectation task allocation algorithms can be used in truthful in expectation mechanisms. For certain classes of problems (e.g., for combinatorial auctions), one can convert LP relaxations with rounding into truthful in expectation mechanisms, see Azar et al [2], Elbassioni et al [7] or Lavi and Swamy [15]. Truthful in expectation mechanisms could perform better in expectation than the universally truthful ones.…”