Hardness of Consensus Problems for Circular Strings and Time Series Averaging
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Consensus problems for strings and sequences appear in numerous application contexts, ranging from bioinformatics over data mining to speech recognition. Closing some gaps in the literature, we show that some fundamental problems in this context are NP-hard, W[1]-hard, and the known (partially brute-force) exact algorithms are essentially close to optimality assuming the Exponential Time Hypothesis. Our classified problems include the Circular Consensus String problem, whose non-circular version is trivially linear-time solvable. We introduce a much more general family of circular consensus string problems that also serves as a key for our hardness reductions, and proves to be of independent (also practical) interest on its own. In particular, our work answers the main open question concerning the computational complexity of exact mean computation in dynamic time warping spaces [Brill et al., SDM 2018].
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