The.Algorithm.Design.Manual.Springer-Verlag.1998

Combinatorial Search and Heuristic Methods
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Combinatorial Search and Heuristic
Methods
We have seen how clever algorithms can reduce the complexity of sorting from to , which is good.
However, the algorithmic stakes can be even higher for combinatorially explosive problems, whose time
grows exponentially in the size of the problem. Looking back at Figure will make clear the
limitations of exponential-time algorithms on even modest-sized problems.
By using exhaustive search techniques, we can solve small problems to optimality, although the time
complexity may be enormous. For certain applications, it may well pay to spend extra time to be certain
of the optimal solution. A good example occurs in testing a circuit or a program on all possible inputs.
You can prove the correctness of the device by trying all possible inputs and verifying that they give the
correct answer. Proving such correctness is a property to be proud of. However, claiming that it works
correctly on all the inputs you tried is worth much, much less.
In this section, we present backtracking as a technique for listing all configurations representing possible
solutions for a combinatorial algorithm problem. We then discuss techniques for pruning search that
significantly improve efficiency by eliminating irrelevant configurations from consideration. We
illustrate the power of clever pruning techniques to speed up real search applications. For problems that
are too large to contemplate using brute-force combinatorial search, we introduce heuristic methods such
as simulated annealing. Such heuristic methods are an important weapon in the practical algorist's
arsenal.
The take-home lessons from this chapter are:
● Combinatorial search, augmented with tree pruning techniques, can be used to find the optimal
solution of small optimization problems. How small depends upon the specific problem, but the
size limit is likely to be somewhere between items.
● Clever pruning techniques can speed up combinatorial search to an amazing extent. Proper
pruning will have a greater impact on search time than any other factor.
● Simulated annealing is a simple but effective technique to efficiently obtain good but not optimal
solutions to combinatorial search problems.
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