Document Open Access Logo

A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems

Authors Andrés Fielbaum, Ignacio Morales, José Verschae

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2020.46.pdf
  • Filesize: 490 kB
  • 15 pages

Document Identifiers

Author Details

Andrés Fielbaum
  • Department of Cognitive Robotics, Faculty of Mechanical, Maritime and Materials Engineering, TU Delft, The Netherlands
Ignacio Morales
  • Departamento de Ingeniería Industrial, Escuela de Ingeniería, Pontificia Universidad Católica, Santiago, Chile
José Verschae
  • Instituto de Ingeniería Matemática y Computacional, Facultad de Matemáticas y Escuela de Ingeniería, Pontificia Universidad Católica, Santiago, Chile

Funding

This article was partially funded by Fondecyt Project Nr. 1181527.

Cite As Get BibTex

Andrés Fielbaum, Ignacio Morales, and José Verschae. A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.46

Abstract

Obtaining strong linear relaxations of capacitated covering problems constitute a significant technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities has an integrality gap of 2. These inequalities are exploited in more general problems, many of which admit primal-dual approximation algorithms.
Inspired by problems from power and transport systems, we introduce a general setting in which items can be taken fractionally to cover a given demand. The cost incurred by an item is given by an arbitrary non-decreasing function of the chosen fraction. We generalize the knapsack-cover inequalities to this setting an use them to obtain a (2+ε)-approximate primal-dual algorithm. Our procedure has a natural interpretation as a bucket-filling algorithm which effectively overcomes the difficulties implied by having different slopes in the cost functions. More precisely, when some superior segment of an item presents a low slope, it helps to increase the priority of inferior segments. We also present a rounding algorithm with an approximation guarantee of 2.
We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for the setting of fractional items with non-linear costs. For this problem we obtain a (4+ε)-approximation algorithm in polynomial time, almost matching the 4-approximation algorithm known for the classical setting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Mathematics of computing → Discrete optimization
  • Mathematics of computing → Linear programming
Keywords
  • Knapsack-Cover Inequalities
  • Non-Linear Knapsack-Cover
  • Primal-Dual
  • Water-Filling Algorithm

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. H.-C. An, M. Singh, and O. Svensson. LP-based algorithms for capacitated facility location. SIAM Journal on Computing, 46(1):272-306, 2017. Google Scholar
  2. N. Bansal and J. Batra. Geometry of scheduling on multiple machines. arXiv:1907.05473 [cs], 2019. URL: http://arxiv.org/abs/1907.05473.
  3. N. Bansal and K. Pruhs. The geometry of scheduling. SIAM Journal on Computing, 43(5):1684-1698, 2014. Google Scholar
  4. A. Bar-Noy, R. Bar-Yehuda, A. Freund, J. Naor, and B. Schieber. A unified approach to approximating resource allocation and scheduling. Journal of the ACM, 48(5):1069-1090, 2001. Google Scholar
  5. R. Bar-Yehuda and S. Even. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2(2):198-203, 1981. Google Scholar
  6. R. Bar-Yehuda and D. Rawitz. On the equivalence between the primal-dual schema and the local ratio technique. SIAM Journal on Discrete Mathematics, 19(3):762-797, 2005. Google Scholar
  7. A. Bazzi, S. Fiorini, S. Huang, and O. Svensson. Small extended formulation for knapsack cover inequalities from monotone circuits. Theory of Computing, 14(1):1-29, 2018. Google Scholar
  8. Tim Carnes and David B. Shmoys. Primal-dual schema for capacitated covering problems. Mathematical Programming, 153(2):289-308, 2015. Google Scholar
  9. R. D. Carr, L. K. Fleischer, V. J. Leung, and C. A. Phillips. Strengthening integrality gaps for capacitated network design and covering problems. In Proceedings of the 11th annual ACM-SIAM symposium on Discrete algorithms (SODA 2000), pages 106-115, 2000. Google Scholar
  10. M. Cheung, J. Mestre, D. B. Shmoys, and J. Verschae. A primal-dual approximation algorithm for min-sum single-machine scheduling problems. SIAM Journal on Computing, 31(2):825-838, 2017. Google Scholar
  11. V. Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979. Google Scholar
  12. N. Coulombel and N. Monchambert. Diseconomies of scale and subsidies in urban public transportation. HAL:02373768, 2019. Google Scholar
  13. A. Fielbaum, I. Morales, and J. Verschae. A water-filling primal-dual algorithm for approximating non-linear covering problems. arXiv:1912.12151 [cs.DS], 2019. URL: http://arxiv.org/abs/1912.12151.
  14. Andrés Fielbaum, Sergio Jara-Diaz, and Antonio Gschwender. Beyond the Mohring effect: Scale economies induced by transit lines structures design. Economics of Transportation, 22:100-163, 2020. Google Scholar
  15. M. Goemans and J. Kleinberg. An improved approximation ratio for the minimum latency problem. Mathematical Programming, 82(1):111-124, 1998. Google Scholar
  16. W. Höhn, J. Mestre, and A. Wiese. How unsplittable-flow-covering helps scheduling with job-dependent cost functions. In Automata, Languages, and Programming (ICALP 2014). Springer, 2014. Google Scholar
  17. M.-Y. Kao, J. H. Reif, and S. R. Tate. Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem. Information and Computation, 131(1):63-79, 1996. Google Scholar
  18. Retsef Levi, Andrea Lodi, and Maxim Sviridenko. Approximation algorithms for the capacitated multi-item lot-sizing problem via flow-cover inequalities. Mathematics of Operations Research, 33(2):461-474, 2008. Google Scholar
  19. S. Li. Constant approximation algorithm for non-uniform capacitated multi-item lot-sizing via strong covering inequalities. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pages 2311-2325, 2017. Google Scholar
  20. S. Thomas McCormick, Britta Peis, J. Verschae, and A. Wierz. Primal-dual algorithms for precedence constrained covering problems. Algorithmica, 78(3):771-787, 2017. Google Scholar
  21. Herbert Mohring. Optimisation and scale economies in urban bus transport. American Economic Review, 62(4):591-604, 1972. Google Scholar
  22. B. Moseley. Scheduling to approximate minimization objectives on identical machines. In Automata, Languages, and Programming (ICALP 2019), pages 86:1-86:14, 2019. Google Scholar
  23. D. P. Williamson and D. B. Shmoys. The design of approximation algorithms. Cambridge University Press, 1 edition, 2011. Google Scholar
  24. J. Zhu. Optimization of power system operation, volume 47. John Wiley & Sons, 2015. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact