A Modern View on Stability of Approximation
Résumé
In order to attack hard optimization problems that do not admit any polynomial-time approximation scheme (PTAS) or α-approximation algorithm for a reasonable constant α (or even with a worse approximability), Hromkovi£ et al. [12, 31] introduced the notion of Stability of Approximation. The main idea of the stability concept is to try to split the set of all input instances into (potentially innitely many) classes with respect to the achievable approximation ratio. The crucial point in applying this concept is to nd parameters that well capture the diculty of problem instances of the particular hard problem. The concept of stability of approximation turns out to be ubiquitous in the research of approximation algorithms and is applicable beyond approximation. In the literature, however, the relation to stability of approximation has stayed implicit. The purpose of this survey is to take a broader view and to explicitly analyze algorithmic problems and their algorithms with respect to stability of approximation. I would like to express my belief that, in order to make essential progress in algorithmics, one has to move from measuring the problem hardness in a worst-case manner to classifying the hardness of the instances of the investigated algorithmic problems.
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