An Improved Randomized Truthful Mechanism for Scheduling Unrelated Machines

We study the scheduling problem on unrelated machines in the mechanism design setting. This problem was proposed and studied in the seminal paper (Nisan and Ronen 1999), where they gave a 1.75-approximation randomized truthful mechanism for the case of two machines. We improve this result by a 1.6737-approximation randomized truthful mechanism. We also generalize our result to a $0.8368m$-approximation mechanism for task scheduling with $m$ machines, which improve the previous best upper bound of $0.875m(Mu'alem and Schapira 2007)

New bounds for truthful scheduling on two unrelated selfish machines

We consider the minimum makespan problem for $n$ tasks and two unrelated parallel selfish machines. Let $R_n$ be the best approximation ratio of randomized monotone scale-free algorithms. This class contains the most efficient algorithms known for truthful scheduling on two machines. We propose a new $Min-Max$ formulation for $R_n$, as well as upper and lower bounds on $R_n$ based on this formulation. For the lower bound, we exploit pointwise approximations of cumulative distribution functions (CDFs). For the upper bound, we construct randomized algorithms using distributions with piecewise rational CDFs. Our method improves upon the existing bounds on $R_n$ for small $n$. In particular, we obtain almost tight bounds for $n=2$ showing that $|R_2-1.505996|<10^{-6}$.Comment: 28 pages, 3 tables, 1 figure. Theory Comput Syst (2019

Average-case Approximation Ratio of Scheduling without Payments

Apart from the principles and methodologies inherited from Economics and Game Theory, the studies in Algorithmic Mechanism Design typically employ the worst-case analysis and approximation schemes of Theoretical Computer Science. For instance, the approximation ratio, which is the canonical measure of evaluating how well an incentive-compatible mechanism approximately optimizes the objective, is defined in the worst-case sense. It compares the performance of the optimal mechanism against the performance of a truthful mechanism, for all possible inputs. In this paper, we take the average-case analysis approach, and tackle one of the primary motivating problems in Algorithmic Mechanism Design -- the scheduling problem [Nisan and Ronen 1999]. One version of this problem which includes a verification component is studied by [Koutsoupias 2014]. It was shown that the problem has a tight approximation ratio bound of (n+1)/2 for the single-task setting, where n is the number of machines. We show, however, when the costs of the machines to executing the task follow any independent and identical distribution, the average-case approximation ratio of the mechanism given in [Koutsoupias 2014] is upper bounded by a constant. This positive result asymptotically separates the average-case ratio from the worst-case ratio, and indicates that the optimal mechanism for the problem actually works well on average, although in the worst-case the expected cost of the mechanism is Theta(n) times that of the optimal cost

A deterministic truthful PTAS for scheduling related machines

Scheduling on related machines ($Q||C_{\max}$) is one of the most important problems in the field of Algorithmic Mechanism Design. Each machine is controlled by a selfish agent and her valuation can be expressed via a single parameter, her {\em speed}. In contrast to other similar problems, Archer and Tardos \cite{AT01} showed that an algorithm that minimizes the makespan can be truthfully implemented, although in exponential time. On the other hand, if we leave out the game-theoretic issues, the complexity of the problem has been completely settled -- the problem is strongly NP-hard, while there exists a PTAS \cite{HS88,ES04}. This problem is the most well studied in single-parameter algorithmic mechanism design. It gives an excellent ground to explore the boundary between truthfulness and efficient computation. Since the work of Archer and Tardos, quite a lot of deterministic and randomized mechanisms have been suggested. Recently, a breakthrough result \cite{DDDR08} showed that a randomized truthful PTAS exists. On the other hand, for the deterministic case, the best known approximation factor is 2.8 \cite{Kov05,Kov07}. It has been a major open question whether there exists a deterministic truthful PTAS, or whether truthfulness has an essential, negative impact on the computational complexity of the problem. In this paper we give a definitive answer to this important question by providing a truthful {\em deterministic} PTAS

Truthful Allocation in Graphs and Hypergraphs

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L^p-norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ?, and extend the results to any p > 0