Unless P=NP, we cannot obtain a polynomial-time algorithm solving hard combinatorial problems. One practical approach in solving this kind of problem is to relax the condition of always finding the optimal solution for an instance and settle for “good enough” solutions. The kind of algorithms which are guaranteed to obtain a solution with a certain quality are called approximative algorithms. However, not all hard problems are approximable, i.e., we can obtain a polynomial-time algorithm that can guarantee the goodness of the solution for a problem.
In this lecture, we will present the concept of reoptimization. In this approach, given an instance I of some problem Π, an optimal solution OPT for Π in I, and a modified instance I’ resulting from a local perturbation of I, we wish to use OPT in order to solve Π in I’. With this additional information, reoptimization may help to improve the approximability of the problem or the running time of the solution to it. In fact, we can obtain a polynomial-time approximation scheme (PTAS) for a reoptimization variant of a problem given that the unmodified problem is approximable.