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Between a rock and a hard place: the two-to-one assignment problem

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Abstract

We describe the two-to-one assignment problem, a problem in between the axial three-index assignment problem and the three-dimensional matching problem, having applications in various domains. For the (relevant) case of decomposable costs satisfying the triangle inequality we provide, on the positive side, two constant factor approximation algorithms. These algorithms involve solving minimum weight matching problems and transportation problems, leading to a 2-approximation, and a \({\frac{3}{2}}\) -approximation. Moreover, we further show that the best of these two solutions is a \({\frac{4}{3}}\) -approximation for our problem. On the negative side, we show that the existence of a polynomial time approximation scheme for our problem would imply P  =  NP. Finally, we report on some computational experiments showing the performance of the described heuristics.

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References

  • Bandelt H-J, Crama Y, Spieksma FCR (1994) Approximation algorithms for multidimensional assignment problems with decomposable costs. Discret Appl Math 49: 25–50

    Article  MathSciNet  MATH  Google Scholar 

  • Biyani P, Wu X, Sinha A (2005) Joint classification and pairing of human chromosomes. IEEE/ACM Trans Comput Biol Bioinf 2: 102–109

    Article  Google Scholar 

  • Burkard RE, Rudolf R, Woeginger GJ (1996) Three-dimensional axial assignment problems with decomposable cost coefficients. Discret Appl Math 65: 123–140

    Article  MathSciNet  MATH  Google Scholar 

  • Crama Y, Spieksma FCR (1992) Approximation algorithms for three-dimensional assignment problems with triangle inequalities. Eur J Oper Res 60: 273–279

    Article  MATH  Google Scholar 

  • Dutta A, Tsiotras P (2008) Egalitarian peer-to-peer satellite refueling strategy. J Spacecr Rocket 45: 608–618

    Article  Google Scholar 

  • Friesen D, Langston M (1991) Analysis of a compound bin packing algorithm. SIAM J Discret Math 4: 61–79

    Article  MathSciNet  MATH  Google Scholar 

  • Goossens D, Polyakovskiy S, Spieksma FCR, Woeginger GJ (2010) The approximability of three-dimensional assignment problems with bottleneck objective. Optim Lett 4: 7–16

    Article  MathSciNet  MATH  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco

    MATH  Google Scholar 

  • Kann V (1991) Maximum bounded 3-dimensional matching is MAX SNP-complete. Inf Process Lett 37: 27–35

    Article  MathSciNet  MATH  Google Scholar 

  • Magyar G, Johnsson M, Nevalainen O (2000) An adaptive hybrid genetic algorithm for the three-matching problem. IEEE Trans Evol Comput 4: 135–146

    Article  Google Scholar 

  • Petrank E (1994) The hardness of approximation: gap location. Comput Complex 4: 133–157

    Article  MathSciNet  MATH  Google Scholar 

  • Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin

    MATH  Google Scholar 

  • Spieksma FCR, Woeginger GJ (1996) Geometric three-dimensional assignment problems. Eur J Oper Res 91: 611–618

    Article  MATH  Google Scholar 

  • Urban T, Russel R (2003) Scheduling sport competitions on multiple venues. Eur J Oper Res 148: 302–311

    Article  MATH  Google Scholar 

  • Wilson B (1963) Surf city. Liberty records

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Correspondence to Frits C. R. Spieksma.

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This research has been supported by the Netherlands Organisation for Scientific Research (NWO) under grant 639.033.403, by BSIK grant 03018 (BRICKS: Basic Research in Informatics for Creating the Knowledge Society), and by OT Grant OT/07/015. An earlier version of this work appeared in the Proceedings of the 7th Workshop on Approximation and Online Algorithms (WAOA 2009).

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Goossens, D., Polyakovskiy, S., Spieksma, F.C.R. et al. Between a rock and a hard place: the two-to-one assignment problem. Math Meth Oper Res 76, 223–237 (2012). https://doi.org/10.1007/s00186-012-0397-2

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  • DOI: https://doi.org/10.1007/s00186-012-0397-2

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