Fully Polynomial Time Approximation Schemes

Recall that the approximation ratio for an approximation algorithm is a measure to evaluate the approximation performance of the algorithm. The closer the ratio to 1 the better the approximation performance of the algorithm. It is notable that there is a class of NP-hard optimization problems, most originating from scheduling problems, for which there are polynomial time approximation algorithms whose approximation ratio 1 + can be as close to 1 as desired. Of course, the running time of such an algorithm increases with the reciprocal of the error bound , but in a very reasonable way: it is bounded by a polynomial of 1/ . Such an approximation algorithm is called a fully polynomial time approximation scheme (or shortly FPTAS) for the NP-hard optimization problem. A fully polynomial time approximation scheme seems the best possible we can expect for approximating an NP-hard optimization problem. In this chapter, we introduce the main techniques for constructing fully polynomial time approximation schemes for NP-hard optimization problems. These techniques include pseudo-polynomial time algorithms and approximation by scaling. Two NP-hard optimization problems, the Knapsack problem and the c-Makespan problem, are used as examples to illustrate the techniques. In the last section of this chapter, we also give a detailed discussion on what NP-hard optimization problems may not have a fully polynomial time approximation scheme. An important concept, the strong NP-hardness, is introduced, and we prove that in most cases, a strongly NP-hard optimization problem has no fully polynomial time approximation scheme unless our working conjecture P 6= NP fails.