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 \((3/2 + \varepsilon )\)-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.

在一个（线性）参数优化问题中，每个可行解的目标值都是实值参数的仿射函数，并且有兴趣为该参数的每个可能值计算一个解。对于许多重要的参数优化问题，包括最短路径问题，分配问题和最小成本流问题的参数版本，但是，分段线性函数将参数映射到相应非参数实例的最优目标值（最优值函数）可以具有多个多项式的断点（斜率变化点）。这意味着，针对此问题的任何最佳算法都必须输出超多项式解。我们提供了一种将非参数优化问题的近似算法提升到其对应参数的方法，该方法适用于一般类别的参数优化问题。通过这种方法对参数问题的逼近保证可以任意接近算法对相应非参数问题的逼近保证。如果非参数算法具有多项式运行时间，则它输出多项式许多解并且具有多项式运行时间。在非参数问题可以在多项式时间内准确解决或FPTAS可用的情况下，该方法将得出FPTAS。\（（（3/2 + \ varepsilon）\） -用于参数度量旅行商问题的近似算法。此外，我们描述了一种后处理程序，如果可以在多项式时间内准确解决非参数问题，则进一步减少返回的解的数量，以使该方法输出的解最多为实现该解所需的最小值的两倍。所需的近似保证。