122 research outputs found
New bounds for truthful scheduling on two unrelated selfish machines
We consider the minimum makespan problem for tasks and two unrelated
parallel selfish machines. Let 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 formulation for , as well as upper and lower bounds on
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 for small
. In particular, we obtain almost tight bounds for showing that
.Comment: 28 pages, 3 tables, 1 figure. Theory Comput Syst (2019
A New Lower Bound for Deterministic Truthful Scheduling
We study the problem of truthfully scheduling tasks to selfish
unrelated machines, under the objective of makespan minimization, as was
introduced in the seminal work of Nisan and Ronen [STOC'99]. Closing the
current gap of on the approximation ratio of deterministic truthful
mechanisms is a notorious open problem in the field of algorithmic mechanism
design. We provide the first such improvement in more than a decade, since the
lower bounds of (for ) and (for ) by
Christodoulou et al. [SODA'07] and Koutsoupias and Vidali [MFCS'07],
respectively. More specifically, we show that the currently best lower bound of
can be achieved even for just machines; for we already get
the first improvement, namely ; and allowing the number of machines to
grow arbitrarily large we can get a lower bound of .Comment: 15 page
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
-approximation mechanism for task scheduling with machines, which
improve the previous best upper bound of $0.875m(Mu'alem and Schapira 2007)
Non-clairvoyant Scheduling Games
In a scheduling game, each player owns a job and chooses a machine to execute
it. While the social cost is the maximal load over all machines (makespan), the
cost (disutility) of each player is the completion time of its own job. In the
game, players may follow selfish strategies to optimize their cost and
therefore their behaviors do not necessarily lead the game to an equilibrium.
Even in the case there is an equilibrium, its makespan might be much larger
than the social optimum, and this inefficiency is measured by the price of
anarchy -- the worst ratio between the makespan of an equilibrium and the
optimum. Coordination mechanisms aim to reduce the price of anarchy by
designing scheduling policies that specify how jobs assigned to a same machine
are to be scheduled. Typically these policies define the schedule according to
the processing times as announced by the jobs. One could wonder if there are
policies that do not require this knowledge, and still provide a good price of
anarchy. This would make the processing times be private information and avoid
the problem of truthfulness. In this paper we study these so-called
non-clairvoyant policies. In particular, we study the RANDOM policy that
schedules the jobs in a random order without preemption, and the EQUI policy
that schedules the jobs in parallel using time-multiplexing, assigning each job
an equal fraction of CPU time
A deterministic truthful PTAS for scheduling related machines
Scheduling on related machines () 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
The Anarchy of Scheduling Without Money
We consider the scheduling problem on n strategic unrelated machines when no payments are allowed, under the objective of minimizing the makespan. We adopt the model introduced in [Koutsoupias 2014] where a machine is bound by her declarations in the sense that if she is assigned a particular job then she will have to execute it for an amount of time at least equal to the one she reported, even if her private, true processing capabilities are actually faster. We provide a (non-truthful) randomized algorithm whose pure Price of Anarchy is arbitrarily close to 1 for the case of a single task and close to n if it is applied independently to schedule many tasks, which is asymptotically optimal for the natural class of anonymous, task-independent algorithms. Previous work considers the constraint of truthfulness and proves a tight approximation ratio of (n+1)/2 for one task which generalizes to n(n+1)/2 for many tasks. Furthermore, we revisit the truthfulness case and reduce the latter approximation ratio for many tasks down to n, asymptotically matching the best known lower bound. This is done via a detour to the relaxed, fractional version of the problem, for which we are also able to provide an optimal approximation ratio of 1. Finally, we mention that all our algorithms achieve optimal ratios of 1 for the social welfare objective
A characterization of 2-player mechanisms for scheduling
We study the mechanism design problem of scheduling unrelated machines and we
completely characterize the decisive truthful mechanisms for two players when
the domain contains both positive and negative values. We show that the class
of truthful mechanisms is very limited: A decisive truthful mechanism
partitions the tasks into groups so that the tasks in each group are allocated
independently of the other groups. Tasks in a group of size at least two are
allocated by an affine minimizer and tasks in singleton groups by a
task-independent mechanism. This characterization is about all truthful
mechanisms, including those with unbounded approximation ratio.
A direct consequence of this approach is that the approximation ratio of
mechanisms for two players is 2, even for two tasks. In fact, it follows that
for two players, VCG is the unique algorithm with optimal approximation 2.
This characterization provides some support that any decisive truthful
mechanism (for 3 or more players) partitions the tasks into groups some of
which are allocated by affine minimizers, while the rest are allocated by a
threshold mechanism (in which a task is allocated to a player when it is below
a threshold value which depends only on the values of the other players). We
also show here that the class of threshold mechanisms is identical to the class
of additive mechanisms.Comment: 20 pages, 4 figures, ESA'0
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