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# Improving Viterbi is Hard: Better Runtimes Imply Faster Clique Algorithms

*Proceedings of the 34th International Conference on Machine Learning*, PMLR 70:311-321, 2017.

#### Abstract

The classic algorithm of Viterbi computes the most likely path in a Hidden Markov Model (HMM) that results in a given sequence of observations. It runs in time $O(Tn^2)$ given a sequence of T observations from a HMM with n states. Despite significant interest in the problem and prolonged effort by different communities, no known algorithm achieves more than a polylogarithmic speedup. In this paper, we explain this difficulty by providing matching conditional lower bounds. Our lower bounds are based on assumptions that the best known algorithms for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique problem in edge-weighted graphs are essentially tight. Finally, using a recent algorithm by Green Larsen and Williams for online Boolean matrix-vector multiplication, we get a $2^{\Omega(\sqrt{\log n})}$ speedup for the Viterbi algorithm when there are few distinct transition probabilities in the HMM.