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An approximation algorithm for a general class of parametric optimization problems

Author

Listed:
  • Cristina Bazgan

    (Université Paris-Dauphine)

  • Arne Herzel

    (University of Kaiserslautern)

  • Stefan Ruzika

    (University of Kaiserslautern)

  • Clemens Thielen

    (Technical University of Munich)

  • Daniel Vanderpooten

    (Université Paris-Dauphine)

Abstract

In a (linear) parametric optimization problem, the objective value of each feasible solution is an affine function of a real-valued parameter and one is interested in computing a solution for each possible value of the parameter. For many important parametric optimization problems including the parametric versions of the shortest path problem, the assignment problem, and the minimum cost flow problem, however, the piecewise linear function mapping the parameter to the optimal objective value of the corresponding non-parametric instance (the optimal value function) can have super-polynomially many breakpoints (points of slope change). This implies that any optimal algorithm for such a problem must output a super-polynomial number of solutions. We provide a method for lifting approximation algorithms for non-parametric optimization problems to their parametric counterparts that is applicable to a general class of parametric optimization problems. The approximation guarantee achieved by this method for a parametric problem is arbitrarily close to the approximation guarantee of the algorithm for the corresponding non-parametric problem. It outputs polynomially many solutions and has polynomial running time if the non-parametric algorithm has polynomial running time. In the case that the non-parametric problem can be solved exactly in polynomial time or that an FPTAS is available, the method yields an FPTAS. In particular, under mild assumptions, we obtain the first parametric FPTAS for each of the specific problems mentioned above and a $$(3/2 + \varepsilon )$$ ( 3 / 2 + ε ) -approximation algorithm for the parametric metric traveling salesman problem. Moreover, we describe a post-processing procedure that, if the non-parametric problem can be solved exactly in polynomial time, further decreases the number of returned solutions such that the method outputs at most twice as many solutions as needed at minimum for achieving the desired approximation guarantee.

Suggested Citation

  • Cristina Bazgan & Arne Herzel & Stefan Ruzika & Clemens Thielen & Daniel Vanderpooten, 2022. "An approximation algorithm for a general class of parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 43(5), pages 1328-1358, July.
    • Handle: RePEc:spr:jcomop:v:43:y:2022:i:5:d:10.1007_s10878-020-00646-5
    DOI: 10.1007/s10878-020-00646-5
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    References listed on IDEAS

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    1. S. Thomas McCormick, 1999. "Fast Algorithms for Parametric Scheduling Come From Extensions to Parametric Maximum Flow," Operations Research, INFORMS, vol. 47(5), pages 744-756, October.
    2. Christos H. Papadimitriou & Mihalis Yannakakis, 1993. "The Traveling Salesman Problem with Distances One and Two," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 1-11, February.
    3. Hans Schneider & Michael H. Schneider, 1991. "Max-Balancing Weighted Directed Graphs and Matrix Scaling," Mathematics of Operations Research, INFORMS, vol. 16(1), pages 208-222, February.
    4. Mitsos, Alexander & Barton, Paul I., 2009. "Parametric mixed-integer 0-1 linear programming: The general case for a single parameter," European Journal of Operational Research, Elsevier, vol. 194(3), pages 663-686, May.
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    1. Stephan Helfrich & Arne Herzel & Stefan Ruzika & Clemens Thielen, 2022. "An approximation algorithm for a general class of multi-parametric optimization problems," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1459-1494, October.

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