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Approximation and Parameterized Complexity of Minimax Approval Voting

26 July 2016
Marek Cygan
Lukasz Kowalik
Arkadiusz Socala
Krzysztof Sornat
ArXiv (abs)PDFHTML
Abstract

We present three results on the complexity of Minimax Approval Voting. First, we study Minimax Approval Voting parameterized by the Hamming distance ddd from the solution to the votes. We show Minimax Approval Voting admits no algorithm running in time O⋆(2o(dlog⁡d))\mathcal{O}^\star(2^{o(d\log d)})O⋆(2o(dlogd)), unless the Exponential Time Hypothesis (ETH) fails. This means that the O⋆(d2d)\mathcal{O}^\star(d^{2d})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)\mathcal{O}^\star(\left({3}/{\epsilon}\right)^{2d})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)n^{\mathcal{O}(1/\epsilon^2 \cdot \log(1/\epsilon))} \cdot \mathrm{poly}(m)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].

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