Balanced Optimization with Vector Costs

  • Annette M. C. Ficker
  • Frits C. R. Spieksma
  • Gerhard J. Woeginger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10138)


An instance of a balanced optimization problem with vector costs consists of a ground set X, a vector cost for every element of X, and a system of feasible subsets over X. The goal is to find a feasible subset that minimizes the spread (or imbalance) of values in every coordinate of the underlying vector costs.

We investigate the complexity and approximability of balanced optimization problems in a fairly general setting. We identify a large family of problems that admit a 2-approximation in polynomial time, and we show that for many problems in this family this approximation factor 2 is best-possible (unless P = NP). Special attention is paid to the balanced assignment problem with vector costs, which is shown to be NP-hard even in the highly restricted case of sum costs.


Balanced optimization Assignment problem Computational complexity Approximation 



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 the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.


  1. 1.
    Ahuja, R.: The balanced linear programming problem. Eur. J. Oper. Res. 101(1), 29–38 (1997)CrossRefMATHGoogle Scholar
  2. 2.
    Camerini, P., Maffioli, F., Martello, S., Toth, P.: Most and least uniform spanning trees. Discrete Appl. Math. 15(23), 181–197 (1986)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cappanera, P., Scutellà, M.G.: Balanced paths in acyclic networks: tractable cases and related approaches. Networks 45(2), 104–111 (2005)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Dokka, T., Crama, Y., Spieksma, F.C.R.: Multi-dimensional vector assignment problems. Discrete Optim. 14, 111–125 (2014)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Galil, Z., Schieber, B.: On finding most uniform spanning trees. Discrete Appl. Math. 20(2), 173–175 (1988)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Kamura, Y., Nakamori, M.: Modified balanced assignment problem in vector case: system construction problem. In: 2014 International Conference on Computational Science and Computational Intelligence (CSCI), vol. 2, pp. 52–56. IEEE (2014)Google Scholar
  7. 7.
    Katoh, N., Iwano, K.: Efficient algorithms for minimum range cut problems. Networks 24(7), 395–407 (1994)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Kinable, J., Smeulders, B., Delcour, E., Spieksma, F.C.R.: Exact algorithms for the Equitable Traveling Salesman Problem. Research report, KU Leuven (2016)Google Scholar
  9. 9.
    Larusic, J., Punnen, A.: The balanced traveling salesman problem. Comput. Oper. Res. 38(5), 868–875 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Martello, S., Pulleyblank, W., Toth, P., De Werra, D.: Balanced optimization problems. Oper. Res. Lett. 3(5), 275–278 (1984)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Turner, L.: Variants of shortest path problems. Algorithmic Oper. Res. 6(2), 91–104 (2012)MATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Annette M. C. Ficker
    • 1
  • Frits C. R. Spieksma
    • 1
  • Gerhard J. Woeginger
    • 2
  1. 1.Operations Research GroupKU LeuvenLeuvenBelgium
  2. 2.Department of MathematicsTU EindhovenEindhovenNetherlands

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