@MISC{Feldman_optimalhardness, author = {Vitaly Feldman}, title = {Optimal Hardness Results for Maximizing . . . }, year = {} }

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Abstract

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 [12] and Kearns et al. [17]. Finding a monomial with the highest agreement rate was proved to be NP-hard by Kearns and Li [15]. 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 − ɛ, for any constant ɛ> 0 [5]. The strongest known hardness of approximation result is due to Bshouty and Burroughs, who proved an inapproximability factor of 59 −ɛ [8]. We show that the agreement rate is NP-58 hard to approximate within 2 − ɛ for any constant ɛ> 0. This is optimal up to the second order terms and resolves an open question due to Blum [6]. We extend this result to ɛ = 2 − log1−λ n for any constant λ> 0 under the assumption that NP � ⊆ RTIME(npoly log(n)), thus also obtaining an inapproximability factor of 2log1−λ n for the symmetric problem of minimizing disagreements. This improves on the log n hardness of approximation factor due to Kearns et al. [17] and Hoffgen et al. [13].