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arXiv:2104.14384 · Leibniz international proceedings in informatics

Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs

Adam Glos, Martins Kokainis, Ryuhei Mori, Jevgēnijs Vihrovs

Published Jan 1, 2021• 6 citations
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Abstract

Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the $n$-dimensional lattice graph $Q(D,n)$ with vertices in $\{0,1,\ldots,D\}^n$. We study the complexity of the following problem: given a subgraph $G$ of $Q(D,n)$ via query access to the edges, determine whether there is a path from $0^n$ to $D^n$. While the classical query complexity is $\widetildeΘ((D+1)^n)$, we show a quantum algorithm with complexity $\widetilde O(T_D^n)$, where $T_D < D+1$. The first few values of $T_D$ are $T_1 \approx 1.817$, $T_2 \approx 2.660$, $T_3 \approx 3.529$, $T_4 \approx 4.421$, $T_5 \approx 5.332$. We also prove that $T_D \geq \frac{D+1}{\mathrm e}$, thus for general $D$, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice. While the presented quantum algorithm is a natural generalization of the known quantum algorithm for $D=1$ by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddle-point method, which is a common tool in analytic combinatorics, but has not been widely used in this field. We then show an implementation of this algorithm with time complexity $\text{poly}(n)^{\log n} T_D^n$, and apply it to the Set Multicover problem. In this problem, $m$ subsets of $[n]$ are given, and the task is to find the smallest number of these subsets that cover each element of $[n]$ at least $D$ times. While the time complexity of the best known classical algorithm is $O(m(D+1)^n)$, the time complexity of our quantum algorithm is $\text{poly}(m,n)^{\log n} T_D^n$.

DOI
10.4230/lipics.mfcs.2021.50

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Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the nnn-dimensional lattice graph Q(D,n)Q(D,n)Q(D,n) with vertices in {0,1,…,D}n\{0,1,\ldots,D\}^n{0,1,…,D}n. We study the complexity of the following problem: given a subgraph GGG of Q(D,n)Q(D,n)Q(D,n) via query access to the edges, determine whether there is a path from 0n0^n0n to DnD^nDn. While the classical query complexity is Θ~((D+1)n)\widetildeΘ((D+1)^n)Θ((D+1)n), we show a quantum algorithm with complexity O~(TDn)\widetilde O(T_D^n)O(TDn​), where TD<D+1T_D < D+1TD​<D+1. The first few values of TDT_DTD​ are T1≈1.817T_1 \approx 1.817T1​≈1.817, T2≈2.660T_2 \approx 2.660T2​≈2.660, T3≈3.529T_3 \approx 3.529T3​≈3.529, T4≈4.421T_4 \approx 4.421T4​≈4.421, T5≈5.332T_5 \approx 5.332T5​≈5.332. We also prove that TD≥D+1eT_D \geq \frac{D+1}{\mathrm e}TD​≥eD+1​, thus for general DDD, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice. While the presented quantum algorithm is a natural generalization of the known quantum algorithm for D=1D=1D=1 by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddle-point method, which is a common tool in analytic combinatorics, but has not been widely used in this field. We then show an implementation of this algorithm with time complexity poly(n)log⁡nTDn\text{poly}(n)^{\log n} T_D^npoly(n)lognTDn​, and apply it to the Set Multicover problem. In this problem, mmm subsets of [n][n][n] are given, and the task is to find the smallest number of these subsets that cover each element of [n][n][n] at least DDD times. While the time complexity of the best known classical algorithm is O(m(D+1)n)O(m(D+1)^n)O(m(D+1)n), the time complexity of our quantum algorithm is poly(m,n)log⁡nTDn\text{poly}(m,n)^{\log n} T_D^npoly(m,n)lognTDn​.
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 0n0^n0n to 1n1^n1n 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)O^*(1.817^n)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)O^*(1.817^n)O∗(1.817n), and graph bandwidth in time O∗(2.946n)O^*(2.946^n)O∗(2.946n). Then we use similar ideas to solve the travelling salesman problem and minimum set cover in time O∗(1.728n)O^*(1.728^n)O∗(1.728n).
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 0n to 1n 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).
We present a general quantum algorithm for solving finite-horizon dynamic programming problems. Up to polylogarithmic factors, our algorithm provides a quadratic quantum advantage in terms of the number of states of a given dynamic programming problem. This speedup comes at the expense of the appearance of other polynomial factors representative of the number of actions of the dynamic programming problem, the maximum value of the instantaneous reward, and the time horizon of the problem. Our algorithm can be applied to combinatorial optimization problems solved classically using dynamic programming techniques. As one application, we show that the travelling salesperson problem can be solved in O∗(⌈c⌉42n)O^*(\lceil c \rceil^4 \sqrt{2^n})O∗(⌈c⌉42n​) on a quantum computer, where nnn is the number of vertices of the underlying graph and ⌈c⌉\lceil c \rceil⌈c⌉ is its maximum edge-weight. As another example, we show that the minimum set-cover problem can be solved in O(2npoly⁡(m,n))O(\sqrt{2^n} \operatorname{poly}(m, n))O(2n​poly(m,n)), where mmm is the number of sets used to cover a universe of size nnn. Finally, we prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for solving dynamic programming problems, and show that no greater-than-quadratic speedup in either the number of states or number of actions can be achieved for solving dynamic programming problems using quantum algorithms.
In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string x∈{−1,1}nx \in \{-1,1\}^nx∈{−1,1}n. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(nlog⁡n)O(\sqrt{n} \log n)O(n​logn) and a near-matching lower bound of Ω(n)Ω(\sqrt{n})Ω(n​). Belovs [Comput.~Comp.'15] subsequently showed a tight O(n)O(\sqrt{n})O(n​) upper bound. We consider a natural generalization of this problem, parametrized by a subset Q⊆2[n]\cal{Q} \subseteq 2^{[n]}Q⊆2[n], where an algorithm may test whether xS=bx_S = bxS​=b for an arbitrary S∈QS \in \cal{Q}S∈Q and b∈{−1,1}Sb \in \{-1,1\}^Sb∈{−1,1}S of its choice, at unit cost. We show near-tight bounds when Q\cal{Q}Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set. All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning xxx is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions f:{−1,1}n→Rf : \{-1,1\}^n \to \mathbb{R}f:{−1,1}n→R the ratio of the maximum value of fff to the maximum (over T∈QT \in \cal{Q}T∈Q) standard deviation of fff on a subcube whose free variables are exactly TTT.' To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.
Let U be a universe on n elements, let k be a positive integer, and let ℱ be a family of (implicitly defined) subsets of U. We consider the problems of partitioning U into k sets from ℱ, covering U with k sets from ℱ, and packing k non-intersecting sets from ℱ into U. Classically, these problems can be solved via inclusion-exclusion in 2ⁿ n^O(1) time [Andreas Björklund et al., 2009]. Quantumly, there are faster algorithms for graph coloring with running time O(1.9140ⁿ) [Kazuya Shimizu and Ryuhei Mori, 2022] and for Set Cover with a small number of sets with running time O(1.7274ⁿ |ℱ|^O(1)) [Andris Ambainis et al., 2019]. In this paper, we give a quantum speedup for Set Partition, Set Cover, and Set Packing whenever there is a classical enumeration algorithm that lends itself to a quadratic quantum speedup, which, for any subinstance on a set X ⊆ U, enumerates at least one member of a k-partition, k-cover, or k-packing (if one exists) restricted to (or projected onto, in the case of k-cover) the set X in c^|X| n^O(1) time with c < 2. Our bounded-error quantum algorithm runs in time (2+c)^{n/2} n^O(1) for Set Partition, Set Cover, and Set Packing. It is obtained by combining three algorithms that have the best running time for some values of c. When c ≤ 1.147899, our algorithm is slightly faster than (2+c)^{n/2} n^O(1); when c approaches 1, it matches the O(1.7274ⁿ |ℱ|^O(1)) running time of [Andris Ambainis et al., 2019] for Set Cover when |ℱ| is subexponential in n. For covering, packing, and partitioning into maximal independent sets, maximal cliques, maximal bicliques, maximal cluster graphs, maximal triangle-free graphs, maximal cographs, maximal claw-free graphs, maximal trivially-perfect graphs, maximal threshold graphs, maximal split graphs, maximal line graphs, and maximal induced forests, we obtain bounded-error quantum algorithms with running times ranging from O(1.8554ⁿ) to O(1.9629ⁿ). Packing and covering by maximal induced matchings can be done quantumly in O(1.8934ⁿ) time. For Graph Coloring (covering with k maximal independent sets), we further improve the running time to O(1.7956ⁿ) by leveraging faster algorithms for coloring with a small number of colors to better balance our divide-and-conquer steps. For Domatic Number (packing k minimal dominating sets), we obtain a O((2-ε)ⁿ) running time for some ε > 0. Several of our results should be of interest to proponents of classical computing: - We present an inclusion-exclusion algorithm with running time O^*(∑_{i=0}^⌊αn⌋ binom(n,i)), which determines, for each X ⊆ U of size at most α n, 0 ≤ α ≤ 1, whether (X,ℱ) has a k-cover, k-partition, or k-packing. This running time is best-possible, up to polynomial factors. - We prove that for any linear-sized vertex subset X ⊆ V of a graph G = (V,E), the number of minimal dominating sets of G that are subsets of X is O((2-ε)^|X|) for some ε > 0.
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\mathcal PP and an instance III of P\mathcal PP, such a speedup can be expressed in terms of the average degree δδδ of the dependency digraph GP(I)G_{\mathcal{P}}(I)GP​(I) of III, determined by a recursive formulation of P\mathcal PP. The nodes of this graph are the subproblems of P\mathcal PP induced by III 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 O~(∣V(GP(I))∣δ)\tilde{O}(|V(G_{\mathcal{P}}(I))| \sqrtδ)O~(∣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 nnn-vertex digraph with mmm edges that runs in O~(nnm)\tilde{O}(n\sqrt{nm})O~(nnm​) time, which improves the best known classical upper bound when m∈Ω(n1.4)m \in Ω(n^{1.4})m∈Ω(n1.4).