text: We consider the robust version of items selection problem, in which the goal is to choose representatives from a family of sets, preserving constraints on the allowed items' combinations. We prove NP-hardness of the deterministic version, and establish polynomially solvable special cases. Next, we consider the robust version in which we aim at minimizing the maximum regret of the solution under interval parameter uncertainty. We show that this problem is hard for the second level of polynomial-time hierarchy. We develop an exact solution algorithm for the robust problem, based on cut generation, and present the results of computational experiments.
  @cite_2: In this paper new complexity and approximation results on the robust versions of the representatives selection problem, under the scenario uncertainty representation, are provided, which extend the results obtained in the recent papers by Dolgui and Kovalev (2012) and Deineko and Woeginger (2013). Namely, it is shown that if the number of scenarios is a part of input, then the min-max (regret) representatives selection problem is not approximable within a ratio of O ( log 1 - ? K ) for any ? > 0 , where K is the number of scenarios, unless the problems in NP have quasi-polynomial time algorithms. An approximation algorithm with an approximation ratio of O ( log K log log K ) for the min-max version of the problem is also provided.
  @cite_6: We establish strong NP-hardness and in-approximability of the so-called representatives selection problem, a tool selection problem in the area of robust optimization. Our results answer a recent question of Dolgui and Kovalev (4OR Q J Oper Res 10:181–192, 2012).
  @cite_8: The following optimization problem is studied. There are several sets of integer positive numbers whose values are uncertain. The problem is to select one representative of each set such that the sum of the selected numbers is minimum. The uncertainty is modeled by discrete and interval scenarios, and the min–max and min–max (relative) regret approaches are used for making a selection decision. The arising min–max, min–max regret and min–max relative regret optimization problems are shown to be polynomially solvable for interval scenarios. For discrete scenarios, they are proved to be NP-hard in the strong sense if the number of scenarios is part of the input. If it is part of the problem type, then they are NP-hard in the ordinary sense, pseudo-polynomially solvable by a dynamic programming algorithm and possess an FPTAS. This study is motivated by the problem of selecting tools of minimum total cost in the design of a production line.
summary: Note that deterministic version of Representatives Selection Problem is easily solvable in polynomial time. For interval uncertainty representation of cost parameters the problem can still be solved in polynomial time, both in case of minimizing the maximum regret and the relative regret @cite_8 . However, in case of discrete set of scenarios, the problem becomes NP-hard even for 2 scenarios, and strongly NP-hard when the number of scenarios @math is a part of the input. In @cite_6 authors prove that strong NP-hardness holds also when sets of eligible items are bounded. In @cite_2 an @math -approximation algorithm for this variant was given.