Quantum Algorithm for Dynamic Programming Approach for DAGs and Applications

TL;DR

This paper introduces a quantum algorithm that accelerates dynamic programming on DAGs, enabling faster solutions for problems like Boolean formula evaluation, longest path search, and diameter computation.

Contribution

It presents a novel quantum algorithm with improved runtime for dynamic programming on DAGs, applicable to various Boolean and graph problems.

Findings

Quantum algorithm runs in $O(\sqrt{\hat{n}m}\log \hat{n})$ time

Applicable to Boolean functions like OR, AND, NAND, MAX, MIN

Enables faster longest path and diameter calculations in DAGs

Abstract

In this paper, we present a quantum algorithm for the dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is $O(\sqrt{\hat{n}m}\log \hat{n})$, and the running time of the best known deterministic algorithm is $O(n+m)$, where $n$ is the number of vertices, $\hat{n}$ is the number of vertices with at least one outgoing edge; $m$ is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX, and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluation. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.