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(dlogd)), 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].Comment: 14 pages, 3 figures, 2 pseudocode