Existence of approximable NP-hard problems without robust approximations

Determine whether there exist NP-hard optimization problems that admit a standard approximation algorithm but do not admit any robust approximation algorithm under interval uncertainty, or prove that every approximation algorithm can be transformed into a robust approximation algorithm.

Background

The paper develops a framework for converting certain approximation algorithms into robust approximation algorithms for linear selection problems under interval uncertainty. Its results apply in particular to algorithms based on local search satisfying well-behaved potential, minimal move covering, and efficient good move retrieval properties, and they yield a robust approximation for Weighted $k$-Set Cover.

The conclusion identifies a broader unresolved issue: whether the existence of an ordinary approximation algorithm is sufficient for the existence of a robust approximation algorithm. The authors explicitly state that they do not know of any NP-hard problem separating these two notions, and present either finding such a separation or proving a general transformation theorem as a major open direction.