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A 3/2-approximation for big two-bar charts packing

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We consider a Two-Bar Charts Packing Problem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs) in a unit-height strip of minimum length. The problem is a generalization of the Bin Packing Problem. Earlier, we proposed an \(O(n^2)\)–time algorithm that constructs the packing of n arbitrary 2-BCs, whose length is at most \(2\cdot OPT+1\), where OPT is the minimum packing length. This paper proposes two new 3/2–approximate algorithms based on sequential matching. One has time complexity \(O(n^4)\) and is applicable when at least one bar of each 2-BC is greater than 1/2. Another has time complexity \(O(n^{3.5})\) and is applicable when, additionally, all BCs are non-increasing or non-decreasing. We prove the estimate’s tightness and conduct a simulation to compare the constructed packings with the optimal solutions or a lower bound of optimum.

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  • Baker BS (1985) A new proof for the first-fit decreasing bin-packing algorithm. J Algorithms 6:49–70

    Article  MathSciNet  Google Scholar 

  • Dósa Gy (2007) The Tight Bound of First Fit Decreasing Bin-Packing Algorithm Is \(FFD(I)\le 11/9 OPT(I)+6/9\). Lecture Notes in Computer Sciences 4614:1–11

    Article  Google Scholar 

  • Erzin A et al (2020) Optimal investment in the development of oil and gas field. Commun Comput Inf Sci 1275:336–349

    MATH  Google Scholar 

  • Erzin A et al (2020) Two-Bar Charts Packing Problem. Optim Lett (online).

    Article  Google Scholar 

  • Gabow HN (1990) Data Structures for Weighted Matching and Nearest Common Ancestors with Linking. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434–443

  • Galil Z (1986) Efficient algorithms for finding maximum matching in graphs. ACM Comput Surv 18(1):23–38

    Article  MathSciNet  Google Scholar 

  • Gimadi E, Sevastianov S, On Solvability of the Project Scheduling Problem with Accumulative Resources of an Arbitrary Sign, Selected papers in Operations Research Proceedings 2002, Berlin-Heidelberg: Springer, pp. 241–246 (2003)

  • Goncharov E (2014) A stochastic greedy algorithm for the resource-constrained project scheduling problem. Dis Anal Oper Res 21(3):11–24

    MathSciNet  MATH  Google Scholar 

  • Goncharov EN, Leonov VV (2017) Genetic algorithm for the resource-constrained project scheduling problem. Autom Remote Control 78(6):1101–1114

    Article  MathSciNet  Google Scholar 

  • Hartmann S (2002) A self-adapting genetic algorithm for project scheduling under resource constraints. Naval Res Logist 49:433–448

    Article  MathSciNet  Google Scholar 

  • Johnson D.S, (1973) Near-optimal bin packing algorithms. Massachusetts Institute of Technology. PhD thesis

  • Johnson DS, Garey MR (1985) A 71/60 theorem for bin packing. J Complex 1(1):65–106

    Article  MathSciNet  Google Scholar 

  • Kolisch R, Hartmann S (2006) Experimental investigation of heuristics for resource-constrained project scheduling: an update. Eur J Oper Res 174:23–37

    Article  Google Scholar 

  • Lewis R (2009) A general-purpose hill-climbing method for order independent minimum grouping problems: a case study in graph colouring and bin packing. Comput Oper Res 36(7):2295–2310

    Article  MathSciNet  Google Scholar 

  • Li R, Yue M (1997) The proof of \(FFD(L)\le 11/9 OPT(L)+7/9\). Chin Sci Bull 42(15):1262–1265

    Article  Google Scholar 

  • VaziraniV V (2001) Approxim Algorithms. Springer, Berlin

    Google Scholar 

  • Xie YT (2018) An \(O(n^{2.5})\) algorithm: for maximum matchings in general graphs. J Appl Math Phys 6:1773–1782

    Article  Google Scholar 

  • Yue M (1991) A simple proof of the inequality \(FFD(L)\le 11/9 OPT(L)+1, \forall L\), for the FFD bin-packing algorithm. Acta Math Appl Sinica 7(4):321–331

    Article  MathSciNet  Google Scholar 

  • Yue M, Zhang L (1995) A simple proof of the inequality \(MFFD(L)\le 71/60 OPT(L)+1, \forall L\), for the MFFD bin-packing algorithm. Acta Math Appl Sinica 11(3):318–330

    Article  MathSciNet  Google Scholar 

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Correspondence to Adil Erzin.

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The research is carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project 0314–2019–0014)

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Erzin, A., Melidi, G., Nazarenko, S. et al. A 3/2-approximation for big two-bar charts packing. J Comb Optim 42, 71–84 (2021).

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