Multiprocessor Scheduling with Machine Allotment and Parallelism Constraints

Shachnai, Hadas; Tamir, Tami

doi:10.1007/s00453-001-0098-3

Cited by 17 publications

(19 citation statements)

References 18 publications

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“…Beier et al (2004) showed that it is NP-hard to decide whether a PSNE exists within such settings. In addition, Shachnai and Tamir (2002); Krysta et al (2003) obtained similar results for k-splittable congestion games in the job scheduling domain.…”

Section: Related Workmentioning

confidence: 57%

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

Tran-Thanh

Polukarov

Chapman

et al. 2011

Algorithmic Game Theory

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Abstract. We study the existence of pure strategy Nash equilibria (PSNE) in integer-splittable weighted congestion games (ISWCGs), where agents can strategically assign different amounts of demand to different resources, but must distribute this demand in fixed-size parts. Such scenarios arise in a wide range of application domains, including job scheduling and network routing, where agents have to allocate multiple tasks and can assign a number of tasks to a particular selected resource. Specifically, in an ISWCG, an agent has a certain total demand (aka weight) that it needs to satisfy, and can do so by requesting one or more integer units of each resource from an element of a given collection of feasible subsets.1 Each resource is associated with a unit-cost function of its level of congestion; as such, the cost to an agent for using a particular resource is the product of the resource unit-cost and the number of units the agent requests. While general ISWCGs do not admit PSNE (Rosenthal, 1973b), the restricted subclass of these games with linear unit-cost functions has been shown to possess a potential function (Meyers, 2006), and hence, PSNE. However, the linearity of costs may not be necessary for the existence of equilibria in pure strategies. Thus, in this paper we prove that PSNE always exist for a larger class of convex and monotonically increasing unit-costs. On the other hand, our result is accompanied by a limiting asumption on the structure of agents' strategy sets: specifically, each agent is associated with its set of accessible resources, and can distribute its demand across any subset of these resources. Importantly, we show that neither monotonicity nor convexity on its own guarantees this result. Moreover, we give a counterexample with monotone and semi-convex cost functions, thus distinguishing ISWCGs from the class of infinitely-splittable congestion games for which the conditions of monotonicity and semi-convexity have been shown to be sufficient for PSNE existence (Rosen, 1965). Furthermore, we demonstrate that the finite improvement path property (FIP) does not hold for convex increasing ISWCGs. Thus, in contrast to the case with linear costs, a potential function argument cannot be used to prove our result. Instead, we provide a procedure that converges to an equilibrium from an arbitrary initial strategy profile, and in doing so show that ISWCGs with convex increasing unit-cost functions are weakly acyclic.

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Section: Related Workmentioning

confidence: 57%

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

Tran-Thanh

Polukarov

Chapman

et al. 2011

Algorithmic Game Theory

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“…The authors prove that this problem does not admit a PTAS, and provide a dual PTAS and an asymptotic PTAS. In a multiprocessor scheduling context, another related problem is scheduling with allotment and parallelism constraints [26]. The goal is to schedule a certain number of tasks, where each task comes with a bound on the number of machines that can process it simultaneously and a bound on the overall number of machines that can participate in its execution.…”

Section: Related Workmentioning

confidence: 99%

“…This problem can also be seen as a splittable packing problem, but this time with a bound k i on the number of times an item can be split. In [26], an approximation algorithm of ratio…”

Section: Related Workmentioning

confidence: 99%

Heterogeneous Resource Allocation under Degree Constraints

Beaumont

Eyraud-Dubois

Thraves

et al. 2013

IEEE Trans. Parallel Distrib. Syst.

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Abstract-In this paper, we consider the problem of assigning a set of clients with demands to a set of servers with capacities and degree constraints. The goal is to find an allocation such that the number of clients assigned to a server is smaller than the server's degree and their overall demand is smaller than the server's capacity, while maximizing the overall throughput. This problem has several natural applications in the context of independent tasks scheduling or virtual machines allocation. We consider both the offline (when clients are known beforehand) and the online (when clients can join and leave the system at any time) versions of the problem. We first show that the degree constraint on the maximal number of clients that a server can handle is realistic in many contexts. Then, our main contribution is to prove that even if it makes the allocation problem more difficult (NP-Complete), a very small additive resource augmentation on the servers degree is enough to find in polynomial time a solution that achieves at least the optimal throughput. After a set of theoretical results on the complexity of the offline and online versions of the problem, we propose several other greedy heuristics to solve the online problem and we compare the performance (in terms of throughput) and the cost (in terms of disconnections and reconnections) of proposed algorithms through a set of extensive simulation results.

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“…All the proofs of upper bounds use only these bounds on OPT, and therefore the knowledge of the actual values of OPT is not required. Naturally, those bounds are not always tight as the offline problem is NP-complete already for identical machines and any constant [19]. Note that in almost all cases in this paper where we got tight bounds on the competitive ratio, the value of OPT is actually given by the maximum of the two bounds on OPT.…”

Section: Computing Optmentioning

confidence: 98%

“…Previous work: The basic model (with = s = 1) was studied in a sequence of papers, each improving either the upper bound or the lower bound on the competitive ratio [10,7,2,12,1,9,6,11]. The offline splittable jobs problem was studied by Shachnai and Tamir [19]. They showed that the problem is NP-hard (already for identical machines) and gave a PTAS for uniformly related machines.…”

Section: Introductionmentioning

confidence: 99%

Online Scheduling of Splittable Tasks in Peer-to-Peer Networks

Epstein

Stee

2004

Algorithm Theory - SWAT 2004

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a Software ENgineeringOnline Scheduling of Splittable Tasks in Peer-ToPeer Networks ABSTRACT We consider online scheduling of splittable tasks on parallel machines. In our model, each task can be split into a limited number of parts, that can then be scheduled independently. We consider both the case where the machines are identical and the case where some subset of the machines have a (fixed) higher speed than the others. We design a class of algorithms which allows us to give tight bounds for a large class of cases where tasks may be split into relatively many parts. For identical machines we also improve upon the natural greedy algorithm in other classes of cases. AbstractWe consider online scheduling of splittable tasks on parallel machines. In our model, each task can be split into a limited number of parts, that can then be scheduled independently. We consider both the case where the machines are identical and the case where some subset of the machines have a (fixed) higher speed than the others. We design a class of algorithms which allows us to give tight bounds for a large class of cases where tasks may be split into relatively many parts. For identical machines we also improve upon the natural greedy algorithm in other classes of cases.

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Multiprocessor Scheduling with Machine Allotment and Parallelism Constraints

Cited by 17 publications

References 18 publications

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

On the Existence of Pure Strategy Nash Equilibria in Integer–Splittable Weighted Congestion Games

Heterogeneous Resource Allocation under Degree Constraints

Online Scheduling of Splittable Tasks in Peer-to-Peer Networks

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