Stability of Approximation Algorithms for Hard Optimization Problems

Abstract

To specify the set of tractable (practically solvable) computing problems is one of the few main research tasks of theoretical computer science. Because of this the investigation of the possibility or the impossibility to efficiently compute approximations of hard optimization problems becomes one of the central and most fruitful areas of recent algorithm and complexity theory. The current point of view is that optimization problems are considered to be tractable if there exist polynomial-time randomized approximation algorithms that solve them with a reasonable approximation ratio. If a optimization problem does not admit such a polynomial-time algorithm, then the problem is considered to be not tractable.

The main goal of this paper is to relativize this specication of tractability. The main reason for this attempt is that we consider the requirement for the tractability to be strong because of the denition of the complexity as the “worst-case” complexity. This definition is also related to the approximation ratio of approximation algorithms and then an optimization problem is considered to be intractable because some subset of problem instances is hard. But in the practice we often have the situation that the hard problem instances do not occur. The general idea of this paper is to try to partition the set of all problem instances of a hard optimization problem into a (possibly infinite) spectrum of subclasses according to their polynomial-time approximability. Searching for a method enabling such a fine problem analysis (classification of problem instances) we introduce the concept of stability of approximation. To show that the application of this concept may lead to a “fine” characterization of the hardness of particular problem instances we consider the traveling salesperson problem and the knapsack problem.