An FPTAS for a General Class of Parametric Optimization Problems

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 (parametric) fully-polynomial time approximation scheme for a general class of parametric optimization problems for which (i) the parameter varies on the nonnegative real line, (ii) the non-parametric problem is solvable in polynomial time, and (iii) the slopes and intercepts of the value functions of the feasible solutions are nonnegative, integer values below a polynomial-time computable upper bound. In particular, under mild assumptions, we obtain the first parametric FPTAS for each of the specific problems mentioned above.