A Hard Problem: The TSP

Abstract

Up to now, we designed algorithms only for those optimization problems which allow an efficient (that is, polynomial) solution. This chapter is devoted to NP-complete problems; we use the Travelling Salesman Problem introduced in Chapter 1 and shown to be NP-complete in Chapter 2 as the standard example. We saw in Chapter 2 that NP-complete problems are a class of problems for which no good algorithms are known and it is quite likely even that no such algorithms exist. Now we consider the question how such ‘hard’ problems might be handled. Possible approaches include approximation techniques, heuristics, relaxations, post-optimization, local optimums, complete enumeration and several more. We explain all the techniques by using the TSP which can serve as a paradigm for a difficult problem.