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#### Max-min Online Allocations with a Reordering Puffer

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##### Citation

Epstein, L., Levin, A., & van Stee, R. (2010). Max-min Online Allocations with
a Reordering Puffer. In S. Abramsky, C. Gavoille, C. Kirchner, F. Meyer auf der Heide, & P. G. Spirakis (*Automata, Languages and Programming* (pp. 336-347). Berlin: Springer. doi:10.1007/978-3-642-14165-2_29.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000F-1679-D

##### Abstract

We consider a scheduling problem where each job is controlled by a
selfish agent, who is only interested in minimizing its own cost,
which is defined as the total load on the machine that its job is
assigned to. We consider the objective of maximizing the minimum load
(cover) over the machines.
Unlike the regular makespan minimization problem, which was extensively
studied in a game theoretic context, this problem has not been
considered in this setting before.
We study the price of anarchy (\poa) and the price of stability (\pos).
Since these measures are unbounded already for two uniformly
related machines, we focus on identical machines. We show that
the $\pos$ is 1, and we derive tight bounds on the $\poa$ for
$m\leq6$ and nearly tight bounds for general $m$. In particular,
we show that the $\poa$ is at least 1.691 for larger $m$ and at
most 1.7. Hence, surprisingly, the $\poa$ is less than the $\poa$
for the makespan problem, which is 2. To achieve the upper bound
of 1.7, we make an unusual use of weighting functions. Finally,
in contrast we show that the mixed $\poa$ grows exponentially with
$m$ for this problem, although it is only $\Theta(\log m/\log
\log m)$ for the makespan.
In addition we consider a similar setting with a different
objective which is minimizing the maximum ratio between the loads
of any pair of machines in the schedule. We show that under this
objective for general $m$ the $\pos$ is 1, and the $\poa$ is 2.