# Using dual approximation algorithms for scheduling problems: Theoretical and practical results

@article{Hochbaum1985UsingDA, title={Using dual approximation algorithms for scheduling problems: Theoretical and practical results}, author={Dorit S. Hochbaum and David B. Shmoys}, journal={26th Annual Symposium on Foundations of Computer Science (sfcs 1985)}, year={1985}, pages={79-89} }

The problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well-studied problem in the theory of approximation algorithms for NP-hard optimization problems. In this paper we present the strongest possible type of result for this problem, a polynomial approximation scheme. More precisely, for each ε, we give an algorithm that runs in time O((n/ε)1/ε2) and has relative error at most ε. For algorithms that are polynomial in n and m…

## 701 Citations

### Approximability of scheduling with fixed jobs

- Computer ScienceSODA '99
- 1999

This paper focuses on simple algorithms for the scheduling problem of minimizing the makespan which have a reasonable performance guarantee and are easy to implement in practical settings, and shows that there exists no FPTAS in the constant machine case, unless P = NP.

### Approximation algorithms for scheduling unrelated parallel machines

- Business28th Annual Symposium on Foundations of Computer Science (sfcs 1987)
- 1987

It is proved that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unless P = NP, and a complexity classification for all special cases with a fixed number of processing times is obtained.

### A Polynomial Approximation Scheme for Scheduling on Uniform Processors: Using the Dual Approximation Approach

- Computer ScienceSIAM J. Comput.
- 1988

A family of polynomial-time algorithms are given such that the last job to finish is completed as quickly as possible and the algorithm delivers a solution that is within a relative error of the optimum.

### Near-Linear Approximation Algorithms for Scheduling Problems with Batch Setup Times

- Computer ScienceSPAA
- 2019

This work gives near-linear approximation algorithms for the scheduling of n jobs divided into c classes on m identical parallel machines and presents the first algorithm improving the previously best approximation ratio of 2 to a better ratio of 3/2 in the preemptive case.

### Approximation algorithms for scheduling parallel machines with capacity constraints ∗

- Computer Science
- 2013

The key idea is to establish a non-standard ILP (Integer Linear Programming) formulation for the scheduling problem, where a set of crucial constraints (called proportional constraints) is introduced to derive an integer solution from a relaxed fractional one without violating constraints.

### Approximation Algorithms for Scheduling with Reservations

- Computer ScienceHiPC
- 2007

This work uses an approach based on algorithms for multiple subset sum problems to derive a polynomial time approximation scheme (PTAS) which is best possible in the sense that the problem does not admit an FPTAS unless P = NP.

### Approximation schemes for the Min-Max Starting Time Problem

- Mathematics, Computer ScienceActa Informatica
- 2004

This work presents techniques that are designed to handle order constraints imposed by the sequence of jobs and shows that the makespan problem in the linear hierarchical model may be reduced to the min-max starting time problem, thus concluding that the former problem also admits a PTAS.

### Minimizing machine assignment costs over Δ-approximate solutions of the scheduling problem P||Cmax

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### Improved approximation algorithms for scheduling with fixed jobs

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- 2009

The main feature of the algorithms is a new technique based on an interesting cyclic shifting argument in combination with a network flow model for the assignment of jobs to large gaps via flexible rounding that is essentially tight via suitable inapproximability results.

### On the extension complexity of scheduling

- Computer Science, MathematicsArXiv
- 2019

This work studies the minimum makespan problem on identical machines, and proves that the canonical formulation for this problem has extension complexity $2^{\Omega(n/\log n)}$, even if each job has size 1 or 2 and the optimal makespan is 2.

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