8 research outputs found
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
Smooth Inequalities and Equilibrium Inefficiency in Scheduling Games
We study coordination mechanisms for Scheduling Games (with unrelated
machines). In these games, each job represents a player, who needs to choose a
machine for its execution, and intends to complete earliest possible. Our goal
is to design scheduling policies that always admit a pure Nash equilibrium and
guarantee a small price of anarchy for the l_k-norm social cost --- the
objective balances overall quality of service and fairness. We consider
policies with different amount of knowledge about jobs: non-clairvoyant,
strongly-local and local. The analysis relies on the smooth argument together
with adequate inequalities, called smooth inequalities. With this unified
framework, we are able to prove the following results.
First, we study the inefficiency in l_k-norm social costs of a strongly-local
policy SPT and a non-clairvoyant policy EQUI. We show that the price of anarchy
of policy SPT is O(k). We also prove a lower bound of Omega(k/log k) for all
deterministic, non-preemptive, strongly-local and non-waiting policies
(non-waiting policies produce schedules without idle times). These results
ensure that SPT is close to optimal with respect to the class of l_k-norm
social costs. Moreover, we prove that the non-clairvoyant policy EQUI has price
of anarchy O(2^k).
Second, we consider the makespan (l_infty-norm) social cost by making
connection within the l_k-norm functions. We revisit some local policies and
provide simpler, unified proofs from the framework's point of view. With the
highlight of the approach, we derive a local policy Balance. This policy
guarantees a price of anarchy of O(log m), which makes it the currently best
known policy among the anonymous local policies that always admit a pure Nash
equilibrium.Comment: 25 pages, 1 figur
Scheduling Games with Machine-Dependent Priority Lists
We consider a scheduling game in which jobs try to minimize their completion
time by choosing a machine to be processed on. Each machine uses an individual
priority list to decide on the order according to which the jobs on the machine
are processed. We characterize four classes of instances in which a pure Nash
equilibrium (NE) is guaranteed to exist, and show, by means of an example, that
none of these characterizations can be relaxed. We then bound the performance
of Nash equilibria for each of these classes with respect to the makespan of
the schedule and the sum of completion times. We also analyze the computational
complexity of several problems arising in this model. For instance, we prove
that it is NP-hard to decide whether a NE exists, and that even for instances
with identical machines, for which a NE is guaranteed to exist, it is NP-hard
to approximate the best NE within a factor of for all
. In addition, we study a generalized model in which players'
strategies are subsets of resources, each having its own priority list over the
players. We show that in this general model, even unweighted symmetric games
may not have a pure NE, and we bound the price of anarchy with respect to the
total players' costs.Comment: 19 pages, 2 figure
Coordinating selfish players in scheduling games
We investigate coordination mechanisms that schedule n jobs on m unrelated machines. The objective is to minimize the makespan. It was raised as an open question whether it is possible to design a coordination mechanism that has constant price of anarchy using preemption. We give a negative answer. Next we introduce multi-job players that control a set of jobs, with the aim of minimizing the sum of the completion times of theirs jobs. In this setting, previous mechanisms designed for players with single jobs are inadequate, e.g., having large price of anarchy, or not guaranteeing pure Nash equilibria. To meet this challenge, we design three mechanisms that induce pure Nash equilibria while guaranteeing relatively small price of anarchy.
Then we consider multi-job players where each player\u27s objective is to minimize the weighted sum of completion time of her jobs, while the social cost is the sum of players\u27 costs. We first prove that if machines order jobs according to Smith-rule, then the coordination ratio is at most 4, moreover this is best possible among non-preemptive policies. Then we design a preemptive policy, em externality that has coordination ratio 2.618, and complement this result by proving that this ratio is best possible even if we allow for randomization or full information. An interesting consequence of our results is that an local optima of for the jump neighborhood can be found in polynomial time and is within a factor of 2.618 of the optimal solution.Wir betrachten Koordinationsmechanismen um n Jobs auf m Maschinen mit individuellen Bearbeitungszeiten zu verteilen. Ziel dabei ist es den Makespan zu minimieren. Es war eine offene Frage, ob es möglich ist einen preämptiven Koordinationsmechanismus zu entwickeln, der einen konstanten Price of Anarchy hat. Wir beantworten diese Frage im negativen Sinne. Als nächstes führen wir Multi-Job-Spieler ein, die eine Menge von Jobs kontrollieren können, mit dem Ziel die Summe der Fertigstellungszeiten ihrer Jobs zu minimieren. In diesem Szenario sind bekannte Mechanismen, die für Ein-Job-Spieler entworfen worden sind, nicht gut genug, und haben beispielsweise einen hohen Price of Anarchy oder können kein reines Nash Gleichgewicht garantieren. Wir entwickeln drei Mechanismen die jeweils ein reines Nash Gleichgewicht besitzen, und einen relativ kleinen Price of Anarchy haben.
Zusätzlich betrachten wir Multi-Job-Spieler, mit dem Ziel jeweils die gewichtete Summe der Fertigstellungszeiten ihrer Jobs zu minimieren, während die Gesamtkosten die Summe der Kosten der Spieler sind. Wir zeigen zuerst, dass das Koordinationsverhältnis höchstens ist, wenn die Maschinen die Jobs nach der Smith-Regel sortieren, was bei nicht-preämptiven Verfahren optimal ist. Danach entwickeln wir ein preämptives Verfahren, Externality, welches ein Koordinationsverhältnis von 2.618 hat, und ergänzen dieses Ergebniss indem wir beweisen, dass dieses Verhältnis optimal ist, auch für den Fall, dass wir Randomisierung oder volle Information erlauben. Eine interessante Folge unserer Ergebnisse ist, dass ein -lokales Optimum von für die Jump-Neighborhood in Polynomialzeit gefunden werden kann, und innerhalb eines Faktors von 2.618 von der optimalen Lösung ist