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, Nabeel and Singh 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), where n is a number of voters and m is a number of alternatives. It almost matches the running time of the fastest known PTAS for Closest String due to Ma and Sun.

We report (to our knowledge) the first evaluation of Constraint Satisfaction as a computational framework for solving closest string problems. We show that careful consideration of symbol occurrences can provide search heuristics that provide several orders of magnitude speedup at and above the optimal distance. We also report (to our knowledge) the first analysis and evaluation -- using any technique -- of the computational difficulties involved in the identification of all closest strings for a given input set. We describe algorithms for web-scale distributed solution of closest string problems, both purely based on AI backtrack search and also hybrid numeric-AI methods.