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Efficient Algorithms and Implementations for Optimizing the Sum of Linear Fractional Functions, with Applications

Author

Listed:
  • Danny Z. Chen

    (University of Notre Dame)

  • Ovidiu Daescu

    (University of Texas at Dallas)

  • Yang Dai

    (University of Illinois at Chicago)

  • Naoki Katoh

    (Kyoto University Katsura)

  • Xiaodong Wu

    (University of Texas-Pan American)

  • Jinhui Xu

    (State University of New York at Buffalo)

Abstract

This paper presents an improved algorithm for solving the sum of linear fractional functions (SOLF) problem in 1-D and 2-D. A key subproblem to our solution is the off-line ratio query (OLRQ) problem, which asks to find the optimal values of a sequence of m linear fractional functions (called ratios), each ratio subject to a feasible domain defined by O(n) linear constraints. Based on some geometric properties and the parametric linear programming technique, we develop an algorithm that solves the OLRQ problem in O((m+n)log (m+n)) time. The OLRQ algorithm can be used to speed up every iteration of a known iterative SOLF algorithm, from O(m(m+n)) time to O((m+n)log (m+n)), in 1-D and 2-D. Implementation results of our improved 1-D and 2-D SOLF algorithm have shown that in most cases it outperforms the commonly-used approaches for the SOLF problem. We also apply our techniques to some problems in computational geometry and other areas, improving the previous results.

Suggested Citation

  • Danny Z. Chen & Ovidiu Daescu & Yang Dai & Naoki Katoh & Xiaodong Wu & Jinhui Xu, 2005. "Efficient Algorithms and Implementations for Optimizing the Sum of Linear Fractional Functions, with Applications," Journal of Combinatorial Optimization, Springer, vol. 9(1), pages 69-90, February.
    • Handle: RePEc:spr:jcomop:v:9:y:2005:i:1:d:10.1007_s10878-005-5485-2
    DOI: 10.1007/s10878-005-5485-2
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    References listed on IDEAS

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    1. Nimrod Megiddo, 1979. "Combinatorial Optimization with Rational Objective Functions," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 414-424, November.
    2. Danny Z. Chen & Ovidiu Daescu & Xiaobo (Sharon) Hu & Xiaodong Wu & Jinhui Xu, 2001. "Determining an Optimal Penetration Among Weighted Regions in Two and Three Dimensions," Journal of Combinatorial Optimization, Springer, vol. 5(1), pages 59-79, March.
    3. Gabriel R. Bitran & Thomas L. Magnanti, 1976. "Duality and Sensitivity Analysis for Fractional Programs," Operations Research, INFORMS, vol. 24(4), pages 675-699, August.
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    Cited by:

    1. Viviane Köhler & Marcia Fampa & Olinto Araújo, 2013. "Mixed-Integer Linear Programming Formulations for the Software Clustering Problem," Computational Optimization and Applications, Springer, vol. 55(1), pages 113-135, May.
    2. Takahito Kuno & Toshiyuki Masaki, 2013. "A practical but rigorous approach to sum-of-ratios optimization in geometric applications," Computational Optimization and Applications, Springer, vol. 54(1), pages 93-109, January.
    3. T Drezner & Z Drezner & P Kalczynski, 2011. "A cover-based competitive location model," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(1), pages 100-113, January.

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