Approximation Algorithms for Right-Hand-Side Parametric Linear Programming Problems

Abstract

In parametric linear optimization problems, the objective function or the feasible set depends linearly on one or several real-valued parameters. The goal is to provide an optimal solution set, i.e., a set containing an optimal solution for each non-parametric problem induced by a particular choice of parameter values. In this paper, we consider right-hand-side single-parametric linear programming problems (RHS-PLPPs) that are usually very difficult to solve exactly. An optimal solution set of minimum cardinality can be infinite, while a representation by optimal bases may still require super-polynomially many different bases. Hence, we develop approximation methods for RHS-PLPPs that compute approximation sets of polynomially bounded cardinality, each containing, for every parameter value, a feasible solution whose objective value is at most a prescribed multiplicative factor times the optimum. Our first approach constructs an approximation set using a greedy staircase procedure and yields an FPTAS under general sufficient conditions. We show that these conditions hold for some important classes of RHS-PLPPs. For monotone instances, in which the feasible sets are nested as the parameter varies, our algorithm yields an approximation set of minimum cardinality. Beyond monotonicity, we establish near-minimum cardinality guarantees under more general sufficient conditions. Finally, we introduce a dual approximation algorithm that returns solutions with objective value no greater than the optimum while allowing a controlled multiplicative violation of the parameter-dependent right-hand sides.