Mechanism Design for Scheduling

We consider mechanism design issues for scheduling problems and we survey some recent developments on this important problem in Algorithmic Game Theory. We treat both the related and the unrelated version of the problem. 1 The scheduling problem The problem of scheduling unrelated machines [21, 14] is one of the most fundamental algorithmic problems: There are n machines and m tasks∗ and machine i can execute task j in time ti j. These times can be totally unrelated (thus the name of the problem). The objective is to allocate the tasks to machines to minimize the makespan (the time needed to finish all tasks). Thus the output is simply a partition of the m tasks into n sets. A convenient way to express it is to use indicator variables xi j ∈ {0, 1}: xi j is 1 iff task j is allocated to machine i. Each task j is allocated to exactly one machine, therefore we must have ∑n i=1 xi j = 1 for every j. With this notation, the computational problem can be expressed more precisely: given n × m values ti j, find appropriate xi j ∈ {0, 1} which satisfy these constraints and minimize maxi=1 ∑ j xi jti j. From the traditional algorithmic point of view, the unrelated machines scheduling problem is one of the most important open problems. We know that the problem is NP-hard; it is even NP-hard to approximate it within 3/2 [21]; this lower bound applies also to some special cases [11]. On the positive side, there is a polynomial-time approximation algorithm with approximation ratio 2 [21]. ∗Partially supported by IST-15964 (AEOLUS) and IST-2008-215270 (FRONTS). ∗We opt for the game-theoretic notation here and we denote the number of machines and tasks with n and m respectively. In the scheduling literature, they usually use the opposite notation. Closing the gap between the lower and upper bounds on the approximation ratio remains a long-standing major algorithmic problem. There are many interesting variants of the problem. When, for example, the times ti j are inversely proportional to the speed of the machine, that is, when there are speeds si and times p j such that ti j = p j/si, we have the special case of the problem called the related machines scheduling problem. Also, when we allow a task to be split across the machines, which is to say that xi j are nonnegative reals instead of integers, we call this the fractional scheduling problem. The computational complexity of these problems is completely settled: There is a polynomial time approximation scheme (PTAS) [13] for the related machines problem and a fully-PTAS (FPTAS) [15] when the number of machines is fixed; the general case is strongly NP-complete, so we don’t expect to find an FPTAS unless P=NP. For the fractional version of the problem, there is a polynomial-time algorithm (because it can be expressed as a linear program). Nisan and Ronen in their seminal work [27, 28] which started the area of Algorithmic Mechanism Design considered the unrelated machines problem from a game-theoretic point of view: suppose that each machine i is a rational agent who is the only one knowing the values of row ti. Suppose further that the machines want to minimize their execution time. Without any incentive, the machines will lie in order to avoid getting any task. To coerce the machines to cooperate, we pay them to execute the tasks. The payments do not have to be proportional to the execution times, but can be arbitrary functions. The combination of the algorithmic problem of allocating the tasks to machines together with the incentives in the form of payments is called a mechanism. In this article, we survey recent developments in this area of mechanisms for the scheduling problem. We consider direct revelation mechanisms with dominant truthful strategies. Direct revelation means that the players—who know the mechanism in advance— declare their hidden values to the mechanism which collects the values and computes an allocation of tasks and appropriate payments to the players. In such a mechanism, a player may have an incentive to lie and declare values other than his true values. If the mechanism is such that, independently of the values of the other players, a player has no incentive to lie, we say that the mechanism is truthful (with dominant truthful strategies). These mechanisms are very desirable and easy to be implemented since there is no reason for machines to strategize. There other weaker notions of truthfulness but we don’t consider them in this note.