Abstract

We consider the following single-machine scheduling problem, which is often denoted $1||\sum f_{j}$: we are given $n$ jobs to be scheduled on a single machine, where each job $j$ has an integral processing time $p_j$, and there is a nondecreasing, nonnegative cost function $f_j(C_{j})$ that specifies the cost of finishing $j$ at time $C_{j}$; the objective is to minimize $\sum_{j=1}^n f_j(C_j)$. Bansal and Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the linear program is less than 4. Finally, we show how the technique can be adapted to yield, for any $\epsilon >0$, a polynomial time $(4+\epsilon )$-approximation algorithm for this problem.

MSC codes

  1. min-sum scheduling
  2. approximation algorithm
  3. primal-dual schema
  4. knapsack-cover inequalities

MSC codes

  1. 90C27
  2. 90C05
  3. 90B35

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 825 - 838
ISSN (online): 1095-7146

History

Submitted: 27 July 2016
Accepted: 19 December 2016
Published online: 2 May 2017

MSC codes

  1. min-sum scheduling
  2. approximation algorithm
  3. primal-dual schema
  4. knapsack-cover inequalities

MSC codes

  1. 90C27
  2. 90C05
  3. 90B35

Authors

Affiliations

Funding Information

Nucleo Milenio Informacion y Coordinacion en Redes : ICM/FIC RC130003

Funding Information

Fondo Nacional de Desarrollo Científico y Tecnológico : 11140579

Funding Information

Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038 : PGS-358528

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : CCF-0832782, CCF-1017688, CCF-1526067, CCF-1522054

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