On the computational complexity of the minimum committee problem

Two special cases of the Minimum Committee Problem are studied, the Minimum Committee Problem of Finite Sets (MCFS) and the Minimum Committee Problem of a System of Linear Inequalities. It is known that the first of these problems is NP-hard. In this paper we show the NP-hardness of two integer optimization problems connected with it. In addition, we analyze the hardness of approximation to the MCFS problem. In particular, we show that, unless NP ⊂ TIME(n^O(loglogn)), for every ϵ > 0 there are no approximation algorithms for this problem with approximation ratio (1−ϵ)ln (m−1), where m is the number of inclusions in the MCFS problem. To prove this bound we use the SET COVER problem, for which a similar result is known. We also show that the Minimum Committee of Linear Inequalities System problem is NP-hard as well and consider an approximation algorithm for this problem.