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Truthful job scheduling is a
mechanism design Mechanism design (sometimes implementation theory or institution design) is a branch of economics and game theory. It studies how to construct rules—called Game form, mechanisms or institutions—that produce good outcomes according to Social ...
variant of the
job shop scheduling Job-shop scheduling, the job-shop problem (JSP) or job-shop scheduling problem (JSSP) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. In a general job scheduling problem, we are gi ...
problem from
operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and ...
. We have a project composed of several "jobs" (tasks). There are several workers. Each worker can do any job, but for each worker it takes a different amount of time to complete each job. Our goal is to allocate jobs to workers such that the total
makespan In operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods t ...
of the project is minimized. In the standard job shop scheduling problem, the timings of all workers are known, so we have a standard optimization problem. In contrast, in the truthful job scheduling problem, the timings of the workers are not known. We ask each worker how much time he needs to do each job, but, the workers might lie to us. Therefore, we have to give the workers an incentive to tell us their true timings by paying them a certain amount of money. The challenge is to design a payment mechanism which is
incentive compatible In game theory and economics, a mechanism is called incentive-compatible (IC) if every participant can achieve their own best outcome by reporting their true preferences. For example, there is incentive compatibility if high-risk clients are bette ...
. The truthful job scheduling problem was introduced by Nisan and Ronen in their 1999 paper on
algorithmic mechanism design Algorithmic mechanism design (AMD) lies at the intersection of economic game theory, optimization, and computer science. The prototypical problem in mechanism design is to design a system for multiple self-interested participants, such that the part ...
.


Definitions

There are n jobs and m workers ("m" stands for "machine", since the problem comes from scheduling jobs to computers). Worker i can do job j in time T_. If worker i is assigned a set of jobs J_i, then he can execute them in time: :T_i(J_i) = \sum_ t_ Given an allocation J_1,\dots,J_m of jobs to workers, The
makespan In operations research Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods t ...
of a project is: :MakeSpan(J_1,\dots,J_n) = \max_ An ''optimal allocation'' is an allocation of jobs to workers in which the makespan is minimized. The minimum makespan is denoted by MinMakeSpan. A ''mechanism'' is a function that takes as input the matrix T (the time each worker needs to do each job) and returns as output: * An allocation of jobs to workers, J_1,\dots,J_n; * A payment to each worker, p_1,\dots,p_n. The utility of worker i, under such mechanism, is: :u_i = p_i - T_i(J_i) I.e, the agent gains the payment, but loses the time that it spends in executing the tasks. Note that payment and time are measured in the same units (e.g., we can assume that the payments are in dollars and that each time-unit costs the worker one dollar). A mechanism is called truthful (or
incentive compatible In game theory and economics, a mechanism is called incentive-compatible (IC) if every participant can achieve their own best outcome by reporting their true preferences. For example, there is incentive compatibility if high-risk clients are bette ...
) if every worker can attain a maximum utility by reporting his true timing vector (i.e., no worker has an incentive to lie about his timings). The ''approximation factor'' of a mechanism is the largest ratio between Makespan and MinMakespan (smaller is better; an approximation factor of 1 means that the mechanism is optimal). The research on truthful job scheduling aims to find upper (positive) and lower (negative) bounds on approximation factors of truthful mechanisms.


Positive bound – m – VCG mechanism

The first solution that comes to mind is VCG mechanism, which is a generic truthful mechanism. A VCG mechanism can be used to minimize the sum of costs. Here, we can use VCG to find an allocation which minimizes the "make-total", defined as: :MakeTotal(J_1,\dots,J_n) = \sum_ Here, minimizing the sum can be done by simply allocating each job to the worker who needs the shortest time for that job. To keep the mechanism truthful, each worker that accepts a job is paid the second-shortest time for that job (like in a
Vickrey auction A Vickrey auction or sealed-bid second-price auction (SBSPA) is a type of sealed-bid auction. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest ...
). Let OPT be an allocation which minimizes the makespan. Then: :MakeSpan CG\leq MakeTotal CG\leq MakeTotal PT\leq m\cdot MakeSpan PT/math> (where the last inequality follows from the
pigeonhole principle In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, of three gloves, at least two must be right-handed or at least two must be l ...
). Hence, the approximation factor of the VCG solution is at most m – the number of workers. The following example shows that the approximation factor of the VCG solution can indeed be exactly m. Suppose there are n=m jobs and the timings of the workers are as follows: * Worker 1 can do every job in time 1. * The other workers can do every job in time 1+\epsilon, where \epsilon>0 is a small constant. Then, the VCG mechanism allocates all tasks to worker 1. Both the "make-total" and the makespan are n=m. But, when each job is assigned to a different worker, the makespan is 1+\epsilon. An approximation factor of m is not very good, and many researchers have tried to improve it over the following years. On the other hand, there are some impossibility results that prove that the approximation factor cannot be too small.


Negative bound – 2

The approximation factor of every truthful deterministic mechanism is at least 2. The proof is typical of lower bounds in mechanism design. We check specific scenarios (in our case, specific timings of the workers). By truthfulness, when a single worker changes his declaration, he must not be able to gain from it. This induces some constraints on the allocations returned by the mechanism in the different scenarios. In the following proof sketch, for simplicity we assume that there are 2 workers and that the number of jobs is even, n = 2k. We consider the following scenarios: # The timings of both workers to all jobs are 1. Since the mechanism is deterministic, it must return a unique allocation J_1,J_2. Suppose, without loss of generality, that , J_1, \leq , J_2, (worker 1 is assigned at most as many jobs as worker 2). # The timings of worker 1 to the jobs in J_1 are \epsilon (a very small positive constant); the timings of worker 1 to the jobs in J_2 are 1+\epsilon; and the timings of worker 2 to all jobs is still 1. Worker 1 knows that, if he lies and says that his timings to all jobs are 1, the (deterministic) mechanism will allocate him the jobs in J_1 and his cost will be very near 0. In order to remain truthful, the mechanism must do the same here, so that worker 1 does not gain from lying. However, the makespan can be made half as large by dividing the jobs in J_2 equally between the agents. Hence, the approximation factor of the mechanism must be at least 2.


Monotonicity and Truthfulness

Consider the special case of
Uniform-machines scheduling Uniform machine scheduling (also called uniformly-related machine scheduling or related machine scheduling) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. We are given ''n'' jobs ...
, in which the workers are single-parametric: for each worker there is a speed, and the time it takes the worker to do a job is the job length divided by the speed. The speed is the worker's private information, and we want to incentivize machines to reveal their true speeds. Archer and Tardos prove that a scheduling algorithm is truthful if and only if it is monotone. This means that, if a machine reports a higher speed, and all other inputs remain the same, then the total processing time allocated to the machine weakly increases. For this problem: * Auletta, De Prisco, Penna and Persiano presented a 4-approximation monotone algorithm, which runs in polytime when the number of machines is fixed. * Ambrosio and Auletta proved that the Longest Processing Time algorithm is monotone whenever the machine speeds are powers of some c ≥ 2, but not when c ≤ 1.78. In contrast,
List scheduling List scheduling is a greedy algorithm for Identical-machines scheduling. The input to this algorithm is a list of jobs that should be executed on a set of ''m'' machines. The list is ordered in a fixed order, which can be determined e.g. by the pr ...
is not monotone for ''c'' > 2. * Andelman, Azar and Sorani presented a 5-approximation monotone algorithm, which runs in polytime even when the number of machines is variable. * Kovacz presented a 3-approximation monotone algorithm.


References

{{Scheduling problems Mechanism design Optimal scheduling