Optimal hardness results for maximizing agreements with monomials

We consider the problem of finding a monomial (or a term) that maximizes the agreement rate with a given set of examples over the Boolean hypercube. The problem is motivated by learning of monomials in the agnostic framework of Haussler (Hastad, 2001) and Kearns et al. (1994). Finding a monomial with the highest agreement rate was proved to be NP-hard by Kearns and Li (1993). Ben-David et al. gave the first inapproximability result for this problem, proving that the maximum agreement rate is NP-hard to approximate within 770/767 - epsi, for any constant epsi > 0 (Ben-David et al., 2003). The strongest known hardness of approximation result is due to Bshouty and Burroughs, who proved an inapproximability factor of 59/58 - epsi (2002). We show that the agreement rate NP-hard to approximate within 2 - epsi for any constant epsi > 0. This is optimal up to the second order terms and resolves an open question due to Blum (2002). We extend this result to epsi = 2^{-log1-lambda;}n for any constant lambda > 0 under the assumption that NP nsube RTIME(n^{poly log(n)}), thus also obtaining an inapproximability factor of 2^log1-lambda n for the symmetric problem of minimizing disagreements. This improves on the log n hardness of approximation factor due to Kearns et al. (1994) and Hoffgen et al. (1995)