Improving Viterbi is Hard: Better Runtimes Imply Faster Clique Algorithms A. Hardness of VITERBI PATH with small alphabet: Proof

We will perform a reduction from the MIN-WEIGHT k-CLIQUE problem for k = p+ 2 to the VITERBI PATH problem. In the instance of the MIN-WEIGHT k-CLIQUE problem, we are given a k-partite graphG = (V1∪V2∪U1 . . .∪Up, E) such that |V1| = |V2| = n and |U1| = . . . = |Up| = m = Θ(n). We want to find a clique of minimum weight in the graph G. Before describing our final VITERBI PATH instance, we first define a weighted directed graphG′ = ({1, 2, 3}∪V1∪V2, E′) similar to the graph in the proof of Theorem 1. E′ contains all the edges ofG between V1 and V2, directed from V1 towards V2, edges from node 1 towards all nodes in V1 of weight 0 and edges from all nodes in V2 towards node 2 of weight 0. We also add a self-loop at nodes 1 and 3 of weight 0 as well as an edge of weight 0 from node 2 towards node 3. We obtain the final graph G′′ as follows: