Parallel Heuristic Search - Introductions and ANew

In the previous chapters of this book, we have seen techniques for solving NP-hard combinatorial optimization problems exactly, i.e. methods that nd a solution which is guaranteed to be optimal. These methods carry out an implicit search through the entire space of solutions, thereby guaranteeing the best solution found during this process to be an optimal solution. The price paid for this guarantee is that the running time may increase exponentially with problem size. The unfortunate implication is thus that for many problem types, we are only able to solve small or medium-sized problem instances to proven optimality. A wealth of interesting problems can therefore not be handled by exact methods. A as consequence of this fact, we must in those cases settle for a less ambitious goal: nding solutions which are "good" by some standard, but not necessarily optimal. The techniques for achieving this are either approximation algorithms or heuristics. 1 What are heuristics ? It may be diicult to tell the diierence between approximation algorithms and heuristics, since there is no clear deenition of either of the classes of methods. In this context, we informally deene the bordering line between the two classes as whether or not the algorithm in question provides a performance guarantee with respect to the quality of the produced solution. We thus deene approximation algorithms as algorithms which provide a feasible solution to a problem, with a cost guaranteed to be no more than a given factor higher than the cost of an optimal solution. In 6], Christoodes gives a simple approximation algorithm for the Traveling Salesman Problem (TSP), which produces a tour of length at most a factor 1.5 higher than the cost of an optimal tour. Approximation algorithms are often quite fast, and it is indeed appealing that a guarantee on the solution quality can be given. Still, the bound is seldom particularly tight, as compared to results achieved by heuristics. Contrary to approximation algorithms, heuristics usually only have empirical evidence of their problem solving abilities. Wilf 30] characterizes heuristics as "...methods that seem to work well in practice, for reasons nobody understand...", and this is admittedly true in many cases. Numerous heuristics are tailored to a speciic problem type, one of the most well-known being the Lin-Kernighan