We investigate the quantum algorithms for dynamic programming by Ambainis et al. (SODA'19). While giving provable complexity speedups and applicable to a variety of NP-hard problems, these algorithms have a notable drawback: they require a large amount of Quantum Random Access Memory (QRAM), which potentially could be very challenging to implement in a physical quantum computer. In this work, we study how the space complexity can be improved by trading it for time, while still retaining a speedup over the classical algorithms. We show novel quantum time-space tradeoffs by combining different classical approaches with quantum techniques. For instance, we show that the Travelling Salesman Problem can be solved quantumly in Õ(1.859ⁿ) time and Õ(1.315ⁿ) QRAM space.

We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P and an instance I of P, such a speedup can be expressed in terms of the average degree δ of the dependency digraph GP(I) of I, determined by a recursive formulation of P. The nodes of this graph are the subproblems of P induced by I and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in Õ(|V (GP(I))|√δ) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted n-vertex digraph with m edges that runs in Õ(n√nm) time, which improves the best known classical upper bound when m ∈ Ω(n1.4).

We present singly-exponential quantum algorithms for the One-Sided Crossing Minimization (OSCM) problem. Given an n -vertex bipartite graph G = ( U , V , E ⊆ U × V ) , a 2 -level drawing ( π U , π V ) of G is described by a linear ordering π U : U ↔ { 1 , … , | U | } of U and linear ordering π V : V ↔ { 1 , … , | V | } of V . For a fixed linear ordering π U of U , the OSCM problem seeks to find a linear ordering π V of V that yields a 2-level drawing ( π U , π V ) of G with the minimum number of edge crossings. We show that OSCM can be viewed as a set problem over V amenable for exact algorithms with a quantum speedup with respect to their classical counterparts. First, we exploit the quantum dynamic programming framework of Ambainis et al. [ Quantum Speedups for Exponential-Time Dynamic Programming Algorithms . SODA 2019] to devise a QRAM-based algorithm that solves OSCM in ⁎ O ⁎ ( 1.728 n ) time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in ⁎ O ⁎ ( 2 n ) time and polynomial space.

We study upward pointset embeddings (UPSEs) of planar $st$-graphs. Let $G$ be a planar $st$-graph and let $S \subset \mathbb{R}^2$ be a pointset with $|S|= |V(G)|$. An UPSE of $G$ on $S$ is an upward planar straight-line drawing of $G$ that maps the vertices of $G$ to the points of $S$. We consider both the problem of testing the existence of an UPSE of $G$ on $S$ (UPSE Testing) and the problem of enumerating all UPSEs of $G$ on $S$. We prove that UPSE Testing is NP-complete even for $st$-graphs that consist of a set of directed $st$-paths sharing only $s$ and $t$. On the other hand, if $G$ is an $n$-vertex planar $st$-graph whose maximum $st$-cutset has size $k$, then UPSE Testing can be solved in $O(n^{4k})$ time with $O(n^{3k})$ space; also, all the UPSEs of $G$ on $S$ can be enumerated with $O(n)$ worst-case delay, using $O(k n^{4k} \log n)$ space, after $O(k n^{4k} \log n)$ set-up time. Moreover, for an $n$-vertex $st$-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given pointset, which can be tested in $O(n \log n)$ time. Related to this result, we give an algorithm that, for a set $S$ of $n$ points, enumerates all the non-crossing monotone Hamiltonian cycles on $S$ with $O(n)$ worst-case delay, using $O(n^2)$ space, after $O(n^2)$ set-up time.

We present singly-exponential quantum algorithms for the One-Sided Crossing Minimization (OSCM) problem. We show that OSCM can be viewed as a set problem amenable for exact algorithms with a quantum speedup with respect to their classical counterparts. First, we exploit the quantum dynamic programming framework of Ambainis et al. [Quantum Speedups for Exponential-Time Dynamic Programming Algorithms. SODA 2019] to devise a QRAM-based algorithm that solves OSCM in 𝒪^*(1.728ⁿ) time and space. Second, we use quantum divide and conquer to obtain an algorithm that solves OSCM without using QRAM in 𝒪^*(2ⁿ) time and polynomial space.