A Quasi-Polynomial Approximation for the Restricted Assignment Problem

Jansen, Klaus; Rohwedder, Lars

doi:10.1007/978-3-319-59250-3_25

Cited by 14 publications

(18 citation statements)

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“…While the additive integrality gap of the assignment-LP can be as large as p max , this gap is only ǫ • p max for the slot-MILP. We show this using a rounding procedure inspired by a local search method for the restricted assignment problem [28,18,19]. The local search algorithm repeatedly moves jobs between machines, eventually converging to a good solution.…”

Section: Introductionmentioning

confidence: 99%

Additive Approximation Schemes for Load Balancing Problems

Buchem¹,

Rohwedder²,

Vredeveld³

et al. 2020

Preprint

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In this paper we introduce the concept of additive approximation schemes and apply it to load balancing problems. Additive approximation schemes aim to find a solution with an absolute error in the objective of at most ǫh for some suitable parameter h. In the case that the parameter h provides a lower bound an additive approximation scheme implies a standard multiplicative approximation scheme and can be much stronger when h ≪ OPT. On the other hand, when no PTAS exists (or is unlikely to exist), additive approximation schemes can provide a different notion for approximation.We consider the problem of assigning jobs to identical machines with given lower and upper bounds for the loads of the machines. This setting generalizes problems like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For the last problem, in which the objective is to minimize the difference between the maximum and minimum load, the optimal objective value may be zero and hence it is NP-hard to obtain any multiplicative approximation guarantee. For this class of problems we present additive approximation schemes for h = p max , the maximum processing time of the jobs.Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. We identify structural properties of (near-)optimal solutions of the slot-MILP, which allow us to solve it efficiently in polynomial time, assuming that there are O(1) different lower and upper bounds on the machine loads (which is the relevant setting for the three problems mentioned above). The second technical contribution is a local-search based algorithm which rounds a solution to the slot-MILP introducing an additive error on the target load intervals of at most ǫ • p max .

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Section: Introductionmentioning

confidence: 99%

Additive Approximation Schemes for Load Balancing Problems

Buchem¹,

Rohwedder²,

Vredeveld³

et al. 2020

Preprint

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“…Again, our FPTAS has a better dependence on m and ∆ 1 , than an exact algorithm from [3]. Additionally, it gives a better dependence on ∆ 1 , than an exact algorithm from [9].…”

Section: Description Of the Results And Related Workmentioning

confidence: 81%

“…Additionally, we need to note that better complexity bound for searching of an exact solution can be achieved for the unbounded version of the mdimensional knapsack problem. More precisely, for this case the paper [9] gives the complexity bound…”

Section: Description Of the Results And Related Workmentioning

confidence: 99%

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An FPTAS for the $Δ$-modular multidimensional knapsack problem

Gribanov

2021

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It is known that there is no EPTAS for the m-dimensional knapsack problem unless W [1] = F P T . It is true already for the case, when m = 2. But an FPTAS still can exist for some other particular cases of the problem.In this note we show that the m-dimensional knapsack problem with a ∆-modular constraints matrix admits an FPTAS with the arithmetical complexity, where T LP is the linear programming complexity bound. In particular, for fixed m the arithmetical complexity bound becomesOur algorithm is actually a generalisation of the classical FPTAS for the 1dimensional case.The goal of the paper is only to prove the existence of an FPTAS, and more accurate analysis can give better constants in exponents. Moreover, we are not worry to much about memory usage, it can be roughly estimated as O(n • (1/ε) m+3 • (2m) 2m+6 • ∆).

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“…In a breakthrough result, Svensson [36] showed that a certain integer linear program modeling the problem has an integrality gap of at most 33/17, which implies an algorithm approximating the optimal objective value with rate 33/17+ε for any ε > 0 without producing a corresponding schedule. This has been improved by Jansen and Rohwedder [18] to a rate of 11/6, and in [19] the same authors provide a quasi-polynomial approximation algorithm with rate 11/6 + ε that also outputs a corresponding schedule. For the special case of restricted assignment with only two distinct processing times (not counting ∞) an approximation algorithm due to Chakrabarty, Khanna and Li [4] with a rate slightly below 2 is known.…”

Section: Introductionmentioning

confidence: 99%

Inapproximability Results for Scheduling with Interval and Resource Restrictions

Maack¹,

Jansen²

2019

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In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions.For the case of interval restrictions-where the machines can be totally ordered such that jobs are eligible on consecutive machines-we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known.Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 ≈ 1.02 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives.

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A Quasi-Polynomial Approximation for the Restricted Assignment Problem

Cited by 14 publications

References 14 publications

Additive Approximation Schemes for Load Balancing Problems

Additive Approximation Schemes for Load Balancing Problems

An FPTAS for the $Δ$-modular multidimensional knapsack problem

Inapproximability Results for Scheduling with Interval and Resource Restrictions

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