Hidden Markov models (HMMs) and their variants were successfully used for several sequence annotation tasks. Traditionally, inference with HMMs is done using the Viterbi and posterior decoding algorithms. However, recently a variety of different optimization criteria and associated computational problems were proposed. In this paper, we consider three HMM decoding criteria and prove their NP hardness. These criteria consider the set of states used to generate a certain sequence, but abstract from the exact locations of regions emitted by individual states. We also illustrate experimentally that these criteria are useful for HIV recombination detection.
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