Quantum Speedups for Exponential-Time Dynamic Programming Algorithms

Abstract

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 * (1.817 n ). 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 * (1.817 n ), and graph bandwidth in time O * (2.946 n ). Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time O * (1.728 n ).

Computational Speedups Using Small Quantum Devices

“…3SAT is the canonical example of the so-called NP-complete problems, believed to be exponentially difficult even for quantum computers. Nevertheless, quantum computers can still accelerate their solving [1,2], and given their ubiquity, they may become one of the most important applications of quantum computers. However, the best quantum algorithms, which "quantum-enhance" classical SAT solvers [1], require many qubits and are not directly applicable given small quantum computers.…”

On the Embed and Project Algorithm for the Graph Bandwidth Problem

“…Quantum algorithms for GBP and some other NP-hard problems were studied in [38]. Theoretical results of speedups were presented, albeit without any numerical results.…”

A Grover Search-Based Algorithm for the List Coloring Problem

“…While exhaustive search is believed to be the fastest classical approach for several NP-complete problems including satisfiability and hitting-set [8], there are much better classical algorithms using dynamic programming, inclusionexclusion and other structural approaches for problems such as graph coloring [4], [10], [18], the traveling salesman problem [13], [24], set cover [15] etc. Several authors have obtained quantum speedup on these classical algorithms [2], [30], [33]; however, all of these algorithms have the limitation that they cannot be easily generalized to the list coloring problem.…”