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We present three results on the complexity of MINIMAX APPROVAL VOTING. First, we study MINIMAX APPROVAL VOTING parameterized by the Hamming distance d from the solution to the votes. We show MINIMAX APPROVAL VOTING admits no algorithm running in time O⋆(2o(d log d), unless the Exponential Time Hypothesis (ETH) fails. This means that the O⋆(d2d) algorithm of Misra et al. (AAMAS 2015) is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O⋆((3/ε)2d), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for MINIMAX APPROVAL VOTING, which runs in time nO(1/ε2·log(1/ε))· poly(m), almost matching the running time of the fastest known PTAS for CLOSEST STRING due to Ma and Sun (SIAM J. Comp. 2009).