Approximation of the parallel machine scheduling problem with additional unit resources

Emmanuel Hebrard, Marie-José Huguet, Nicolas Jozefowiez, Adrien Maillard, Cédric Pralet, Gérard Verfaillie

Abstract: We consider the problem of minimizing the makespan of a schedule on m<math><mi is="true">m</mi></math> parallel machines of n<math><mi is="true">n</mi></math> jobs, where each job requires exactly one of s<math><mi is="true">s</mi></math> additional unit resources. This problem collapses to P∥Cmax<math><mi is="true">P</mi><mo is="true">∥</mo><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mstyle mathvariant="normal" is="true"><mi is="true">max</mi></mstyle></mrow></msub></math> if every job requires a different resource. It is therefore NP-hard even if we fix the number of machines to 2<math><mn is="true">2</mn></math> and strongly NP-hard in general.Although very basic, its approximability is not known, and more general cases, such as scheduling with conflicts, are often not approximable. We give a (2−2m+1)<math><mrow is="true"><mo is="true">(</mo><mn is="true">2</mn><mo is="true">−</mo><mfrac is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">m</mi><mo is="true">+</mo><mn is="true">1</mn></mrow></mfrac><mo is="true">)</mo></mrow></math>-approximation algorithm for this problem, and show that when the deviation in jobs processing times is bounded by a ratio ρ<math><mi is="true">ρ</mi></math>, the same algorithm approximates the problem within a tight factor 1+ρ(m−1)n<math><mn is="true">1</mn><mo is="true">+</mo><mi is="true">ρ</mi><mfrac is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">m</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></mrow><mrow is="true"><mi is="true">n</mi></mrow></mfrac></math>.This problem appears in the design of download plans for Earth observation satellites, when scheduling the transfer of the acquired data to ground stations. Within this context, it may be required to process jobs by batches standing for the set of files related to a single observation. We show that there exists a (2−1m)<math><mrow is="true"><mo is="true">(</mo><mn is="true">2</mn><mo is="true">−</mo><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">m</mi></mrow></mfrac><mo is="true">)</mo></mrow></math>-approximation algorithm respecting such batch sequences. Moreover, provided that the ratio ρ<math><mi is="true">ρ</mi></math>, between maximum and minimum processing times, is bounded by ⌊s−1m−1⌋<math><mrow is="true"><mo is="true">⌊</mo><mfrac is="true"><mrow is="true"><mi is="true">s</mi><mo is="true">−</mo><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">m</mi><mo is="true">−</mo><mn is="true">1</mn></mrow></mfrac><mo is="true">⌋</mo></mrow></math>, we show that the proposed algorithm approximates the optimal schedule within a factor 1+s−1n<math><mn is="true">1</mn><mo is="true">+</mo><mfrac is="true"><mrow is="true"><mi is="true">s</mi><mo is="true">−</mo><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></mfrac></math>.