Approximation and Parameterized Complexity of Minimax Approval Voting
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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^(2^o(d d)), unless the Exponential Time Hypothesis (ETH) fails. This means that the O^(d^2d) 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 n^O(1/ϵ^2 ·(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].
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