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An Improved Approximation for Packing Big Two-Bar Charts

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We consider the two-bar charts packing problem which is a generalization of the strongly NP-hard bin packing problem and 2-D vector packing problem. We propose an O(n2.5)-time 16/11-approximation algorithm for packing two-bar charts when at least one bar of each two-bar chart has height more than 1/2 and an O(n2.5)-time 5/4-approximation algorithm for packing nonincreasing or nondecreasing two-bar charts when each two-bar chart has at least one bar higher than 1/2.

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References

  1. A. Erzin et al., “Optimal investment in the development of oil and gas field,” In: Mathematical Optimization Theory and Operations Research. 19th International Conference, MOTOR 2020, Novosibirsk, Russia, July 6–10m 2020, pp. 336–349, Springer, Charm (2020).

  2. A. Erzin, G. Melidi, S. Nazarenko, and R. Plotnikov, “Two-bar charts packing problem 2D,” Optim. Lett. 15, No. 6, 1955–1971 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  3. B. S. Baker, “A new proof for the first-fit decreasing bin-packing algorithm,” J. Algorithms 6, 49–70 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Dósa, “The tight bound of first fit decreasing bin-packing algorithm is FFD(I) ⩽ 11/9 OPT(I) + 6/9,” Lect. Notes Comput. Sci. 4614, 1–11 (2007).

  5. D. S. Johnson and M. R. Garey, “A 71/60 theorem for bin packing,” J. Complexity 1, No. 1, 65–106 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Li and M. Yue, “The proof of FFD(L) ⩽ 11/9 OPT(L)+7/9,” Chin. Sci. Bull. 42, No. 15, 1262–1265 (1997).

  7. M. Yue, “A simple proof of the inequality FFD(L) ⩽ 11/9 OPT(L)+1,L, for the FFD bin-packing algorithm,” Acta Math. Appl. Sin., Engl. Ser. 7, No. 4, 321–331 (1991).

  8. M. Yue and L. Zhang, “A simple proof of the inequality MFFD(L) ⩽ 71/60 OPT(L) +1 ∀L, for the MFFD bin-packing algorithm,” Acta Math. Appl. Sin., Engl. Ser. 11, No. 3, 318–330 (1995).

  9. B. S. Baker, E. G. Coffman Jr., and R. L. Rivest, “Orthogonal packing in two dimensions,” SIAM J. Comput. 9, No. 4, 846–855 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  10. E. G. Coffman Jr., M. R. Garey, D. S. Johnson, and R. E. Tarjan, “Performance bounds for level-oriented two-dimensional packing algorithms,” SIAM J. Comput. 9, No. 4, 808–826 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Harren and R. van Stee, “Improved absolute approximation ratios for two-dimensional packing problems,” Lect. Notes Comput. Sci. 5687, 177–189 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Harren, K. Jansen, L. Prädel, and R. van Stee, “A (5/3 + epsilon)-approximation for strip packing,” Comput. Geom. 47, No. 2, 248–267 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Schiermeyer, “Reverse-fit: A 2-optimal algorithm for packing rectangles,” Lect. Notes Comput. Sci. 855, 290–299 (1994).

    Article  MathSciNet  Google Scholar 

  14. A. Steinberg, “A strip-packing algorithm with absolute performance bound 2,” SIAM J. Comput. 26, No. 2, 401–409 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Bansal, M. Eliáš, and A. Khan, “Improved approximation for vector bin packing,” In: Proceedings of the 27th annual ACM-SIAM symposium on discrete algorithms, SODA 2016, Arlington, VA, USA, January 10–12, 2016, Philadelphia,, PA, pp. 1561–1579, SIAM, New York, NY (2016).

  16. H. I. Christensen, A. Khanb, S. Pokutta, and P. Tetali, “Approximation and online algorithms for multidimensional bin packing: A survey,” Comput. Sci. Rev. 24, 63–79 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Kellerer and V. Kotov, “An approximation algorithm with absolute worst-case performance ratio 2 for two-dimensional vector packing,” Oper. Res. Lett. 31, No. 1, 35–41 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Wei, M. Lai, A. Lim, and Q. Hu, “A branch-and-price algorithm for the two-dimensional vector packing problem,” Eur. J. Oper. Res. 281, No. 1, 25–35 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. D. D. Sleator, “A 2.5 times optimal algorithm for packing in two dimensions,” Inf. Process. Lett. 10, No. 1, 37–40 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Erzin, G. Melidi, S. Nazarenko, and R. Plotnikov, “A 3/2-approximation for big two-bar charts packing,” J. Comb. Optim. 42, No. 1, 71–84 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  21. K. Paluch, “Maximum ATSP with weights zero and one via half-edges,” Theory Comput. Syst. 62, No. 2, 319–336 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  22. Y. Xie, “An O(n2.5) algorithm: For maximum matchings in general graphs,” J. Appl. Math. Phys. 6, No. 9, 1773-1782 (2018).

    Article  Google Scholar 

  23. V. V. Vazirani, Approximation Algorithms, Springer, Berlin (2001)

    MATH  Google Scholar 

  24. M. Bläser, “A 3/4-approximation algorithm for maximum ATSP with weights zero and one,” Lect. Notes Comput. Sci. 3122, 61–71 (2004).

    Article  MATH  Google Scholar 

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Correspondence to A. I. Erzin.

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JMS Source Journal International Mathematical Schools. Vol. 1. Advances in Pure and Applied Mathematics

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Erzin, A.I., Shenmaier, V. An Improved Approximation for Packing Big Two-Bar Charts. J Math Sci 267, 465–473 (2022). https://doi.org/10.1007/s10958-022-06151-w

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  • DOI: https://doi.org/10.1007/s10958-022-06151-w

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