Quantum Speedups for Exponential-Time Dynamic Programming Algorithms

In this paper we study quantum algorithms for NP-complete problems whose best classical algorithm is an exponential time application of dynamic programming. We introduce the path in the hypercube problem that models many of these dynamic programming algorithms. In this problem we are asked whether there is a path from $0^n$ to $1^n$ in a given subgraph of the Boolean hypercube, where the edges are all directed from smaller to larger Hamming weight. We give a quantum algorithm that solves path in the hypercube in time $O^*(1.817n)$. The technique combines Grover's search with computing a partial dynamic programming table. We use this approach to solve a variety of vertex ordering problems on graphs in the same time $O^*(1.817n)$, and graph bandwidth in time $O^*(2.946n)$. Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time $O^*(1.728n)$.