Approximate solutions to NP-optimization problems
Most combinatorial optimization problems are NP-hard, and thus unlikely to be solvable to optimality in polynomial time. This tutorial is concerned with polynomial-time algorithms for the approximate solution of such problems. Such an algorithm is said to solve a problem within F(n) if, for every problem instance, it determines the optimal value within a multiplicative error of at most F(n). It has long been known that the knapsack and bin packing problems can be approximated within 1 + a for any positive a. We discuss recent advances in the construction of approximation algorithms for graph partitioning, multicommodity flow and Steiner tree problems. We also discuss negative results, showing that, unless P = NP, it is impossible to approximate the clique number or the chromatic number of a graph within the ratio n{sup b}, where b is a certain small positive number. These negative results stem from an unexpected connection between approximation algorithms and the theory of probabilistically checkable proofs, a branch of theoretical computer science related to cryptography. We also discuss problems such as vertex cover and maximum 2-sat that can be solved within a constant ratio, but not within an arbitrarily small constant ratio (unless P = NP).
- OSTI ID:
- 36190
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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