An Approximation Algorithm for a General Class of Multi-Parametric Optimization Problems
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In a widely studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. For many important multi-parametric optimization problems, an optimal solutions set with minimum cardinality can contain super-polynomially many solutions. Consequently, any exact algorithm for such problems must output a super-polynomial number of solutions. We propose an approximation algorithm that is applicable to a general class of multi-parametric optimization problems and outputs a number of solutions that is bounded polynomially in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric formulations, providing an approximation guarantee that is arbitrarily close to the approximation guarantee for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, the method yields an FPTAS. We discuss implications to important multi-parametric combinatorial optimizations problems. Remarkably, we obtain a (3/2 + ε)-approximation algorithm for the multi-parametric metric travelling salesman problem, whereas the non-parametric version is known to be APX-complete. Furthermore, we show that the cardinality of a minimal size approximation set is in general not ℓ-approximable for any natural number ℓ.
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