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arXiv:2409.01942 · Theoretical Computer Science

Quantum algorithms for one-sided crossing minimization

Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista

Published Jun 20, 2025• 1 citation
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Abstract

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.

DOI
10.1016/j.tcs.2025.115424

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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.
In this paper, we initiate the study of quantum algorithms in the Graph Drawing research area. We focus on two foundational drawing standards: 2-level drawings and book layouts. Concerning 222-level drawings, we consider the problems of obtaining drawings with the minimum number of crossings, kkk-planar drawings, quasi-planar drawings, and the problem of removing the minimum number of edges to obtain a 222-level planar graph. Concerning book layouts, we consider the problems of obtaining 111-page book layouts with the minimum number of crossings, book embeddings with the minimum number of pages, and the problem of removing the minimum number of edges to obtain an outerplanar graph. We explore both the quantum circuit and the quantum annealing models of computation. In the quantum circuit model, we provide an algorithmic framework based on Grover's quantum search, which allows us to obtain, ignoring polynomial terms, a quadratic speedup on the best known classical exact algorithms for all the considered problems. In the quantum annealing model, we perform experiments on the quantum processing unit provided by D-Wave, focusing on the classical 222-level crossing minimization problem, demonstrating that quantum annealing is competitive with respect to classical algorithms.
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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).
Abstract The fastest known classical algorithm deciding the k-colorability of n-vertex graph requires running time Ω(2n)\varOmega (2^n)Ω(2n) Ω ( 2 n ) for k≥5k\ge 5k≥5 k ≥ 5 . In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time O(1.9140n)O(1.9140^n)O(1.9140n) O ( 1 . 9140 n ) using quantum random access memory (QRAM). Our approach is based on Ambainis et al’s quantum dynamic programming with applications of Grover’s search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 20-coloring problem with running time O(1.9575n)O(1.9575^n)O(1.9575n) O ( 1 . 9575 n ) . For the polynomial-space quantum algorithm, we essentially develop (4−ϵ)n(4-\epsilon )^n(4−ϵ)n ( 4 - ϵ ) n -time classical algorithms that can be improved quadratically by Grover’s search.
The fastest known classical algorithm deciding the kkk-colorability of nnn-vertex graph requires running time Ω(2n)Ω(2^n)Ω(2n) for k≥5k\ge 5k≥5. In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time O(1.9140n)O(1.9140^n)O(1.9140n) using quantum random access memory (QRAM). Our approach is based on Ambainis et al's quantum dynamic programming with applications of Grover's search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 202020-coloring problem with running time O(1.9575n)O(1.9575^n)O(1.9575n). In the polynomial-space quantum algorithm, we essentially show (4−ε)n(4-ε)^n(4−ε)n-time classical algorithms that can be improved quadratically by Grover's search.
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 we can improve the space complexity 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 O~(1.859n)\widetilde O(1.859^n)O(1.859n) time and O~(1.315n)\widetilde O(1.315^n)O(1.315n) QRAM space.