266 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
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
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)
Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms
We reconsider the well-studied Selfish Routing game with affine latency
functions. The Price of Anarchy for this class of games takes maximum value
4/3; this maximum is attained already for a simple network of two parallel
links, known as Pigou's network. We improve upon the value 4/3 by means of
Coordination Mechanisms.
We increase the latency functions of the edges in the network, i.e., if
is the latency function of an edge , we replace it by
with for all . Then an
adversary fixes a demand rate as input. The engineered Price of Anarchy of the
mechanism is defined as the worst-case ratio of the Nash social cost in the
modified network over the optimal social cost in the original network.
Formally, if \CM(r) denotes the cost of the worst Nash flow in the modified
network for rate and \Copt(r) denotes the cost of the optimal flow in the
original network for the same rate then [\ePoA = \max_{r \ge 0}
\frac{\CM(r)}{\Copt(r)}.]
We first exhibit a simple coordination mechanism that achieves for any
network of parallel links an engineered Price of Anarchy strictly less than
4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25.
Then, for the case of two parallel links, we describe an optimal mechanism; its
engineered Price of Anarchy lies between 1.191 and 1.192.Comment: 17 pages, 2 figures, preliminary version appeared at ESA 201
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