An approximation algorithm for a general class of parametric optimization problems
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An approximation algorithm for a general class of parametric optimization problems Cristina Bazgan1 · Arne Herzel2 Daniel Vanderpooten1
· Stefan Ruzika2 · Clemens Thielen3 ·
© The Author(s) 2020
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 + ε)-approximation algorithm for the parametric metric traveling salesman problem. Moreover, we describe a postprocessing 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.
This work was supported by the bilateral cooperation project “Approximation methods for multiobjective optimization problems” funded by the German Academic Exchange Service (DAAD, Project-ID 57388848) and by Campus France, PHC PROCOPE 2018 (Project No. 40407WF) as well as by the DFG Grants TH 1852/4-1 and RU 1524/6-1. An earlier version of this work appeared in Bazgan et al. (2019). Extended author information available on the last page of the article
123
Journal of Combinatorial Optimization
Keywords Parametric optimization · Approximation algorithm · Parametric assignment problem · Parametric minimum cost flow problem · Parametric shortest path problem · Parametric metric TSP
1 Introduction In a linear parametric optimization problem, the objective function value of a feasible solution does not only depend on the solution itself but also on a parameter λ ∈ R, where this dependence is
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