Abstract
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(MCLE). It is known that the first of these problems is NP-hard (see (Mazurov et al., Proc. Steklov Inst. Math., 1:67–101, 2002)). 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 (Feige, J. ACM, 45:634–652, 1998). We also show that the Minimum Committee of Linear Inequalities System (MCLE) problem is NP-hard as well and consider an approximation algorithm for this problem.
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Khachay, M.Y. On the Computational Complexity of the Minimum Committee Problem. J Math Model Algor 6, 547–561 (2007). https://doi.org/10.1007/s10852-006-9056-z
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DOI: https://doi.org/10.1007/s10852-006-9056-z
Key words
- computational complexity
- NP-completeness
- set cover problem
- graph 3-colorability problem
- minimum committee problem
- approximation algorithms