Next: Approximating Vertex Cover Up: Intractable Problems and Approximations Previous: War Story: Hard Against

# Approximation Algorithms

For the practical person, demonstrating that a problem is NP-complete is never the end of the line. Presumably, there was a reason why you wanted to solve it in the first place. That reason for wanting the solve it will not have gone away on being told that there is no polynomial-time algorithm. You still seek a program that solves the problem of interest. All you know is that you won't find one that quickly solves the problem to optimality in the worst case. You still have the following options:

• Algorithms fast in the average case - Examples of such algorithms include backtracking algorithms with substantial pruning.
• Heuristics - Heuristic methods like simulated annealing or greedy approaches can be used to find a solution with no requirement that it be the best one.
• Approximation algorithms - The theory of NP-completeness does not stipulate that it is hard to get close to the answer, only that it is hard to get the optimal answer. With clever, problem-specific heuristics, we can often get provably close to the optimal answer.

Approximation algorithms return solutions with a guarantee attached, namely that the optimal solution can never be much better than this given solution. Thus you can never go too far wrong in using an approximation algorithm. No matter what your input instance is and how lucky you are, you are doomed to do all right. Further, approximation algorithms realizing provably good bounds often are conceptually simple, very fast, and easy to program.

One thing that is usually not clear, however, is how well the solution from an approximation algorithm compares to what you might get from a heuristic that gives you no guarantees. The answer could be worse or it could be better. Leaving your money in a savings account in a bank guarantees you 3% interest without risk. Still, you likely will do much better putting your money in stocks than in the bank, even though performance is not guaranteed.

One way to get the best of approximation algorithms and heuristics is to run both of them on the problem instance and pick the solution giving the better answer. This way, you get a solution that comes with a guarantee and a second chance to do even better. When it comes to heuristics for hard problems, sometimes you can have it both ways.

Next: Approximating Vertex Cover Up: Intractable Problems and Approximations Previous: War Story: Hard Against

Algorithms
Mon Jun 2 23:33:50 EDT 1997