Approximation and Parameterized Complexity of Minimax Approval Voting

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 log d ) , unless the Exponential Time Hypothesis (ETH) fails. This means that the O ⋆ ( d 2 d ) 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) 2 d ), 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/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).