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arXiv:2111.14390

Withdrawn: A Measurement-based Algorithm for Graph Colouring

Michael Epping, Tobias Stollenwerk

Published Nov 29, 2021• 0 citations
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Abstract

In a previous version of this document we misinterpreted the runtime of a part of the described algorithm. Indeed, the runtime is not better than the Grover-Algorithm. We therefor withdraw this work. We present a novel algorithmic approach to find a proper vertex colouring of a graph with $d$ colours, if it exists. We associate a $d$-dimensional quantum system with each vertex and the initial state is a mixture of all possible colourings, from which we obtain a random proper colouring of the graph by measurements. The non-deterministic nature of the quantum measurement is tackled by a reset operation, which can revert the effect of unwanted projections. As in the classical case, we find that the runtime scales exponentially with the number of vertices. However, we provide numerical evidence that the average runtime of the problem-specific part of the algorithm scales polynomially in the number of edges.

DOI
10.48550/arxiv.2111.14390

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In a previous version of this document we misinterpreted the runtime of a part of the described algorithm. Indeed, the runtime is not better than the Grover-Algorithm. We therefor withdraw this work. We present a novel algorithmic approach to find a proper vertex colouring of a graph with ddd colours, if it exists. We associate a ddd-dimensional quantum system with each vertex and the initial state is a mixture of all possible colourings, from which we obtain a random proper colouring of the graph by measurements. The non-deterministic nature of the quantum measurement is tackled by a reset operation, which can revert the effect of unwanted projections. As in the classical case, we find that the runtime scales exponentially with the number of vertices. However, we provide numerical evidence that the average runtime of the problem-specific part of the algorithm scales polynomially in the number of edges.
We present a novel algorithmic approach to find a proper vertex colouring of a graph with d colours, if it exists. We associate a d-dimensional quantum system with each vertex and the initial state is a mixture of all possible colourings, from which we obtain a random proper colouring of the graph by measurements. The non-deterministic nature of the quantum measurement is tackled by a reset operation, which can revert the effect of unwanted projections. As in the classical case, we find that the runtime scales exponentially with the number of vertices. However, we provide numerical evidence that the average runtime for random graphs scales polynomially in the number of edges.
Abstract Graph colouring with minimum colours is a well-known NP-complete problem in computer science and mathematics, having diverse real-life applications such as scheduling, register allocation, and frequency assignment. However, a classical computer requires exponential time to solve the graph colouring problem with the minimum number of colours. Therefore, classical approaches become infeasible for solving this problem on large graphs. In contrast, quantum computing offers the potential for exponential speed up on specific problems, although solutions are typically obtained with a certain probability of correctness. This work proposes a quantum-classical method for solving the graph colouring problem with guaranteed correctness. The proposed method is inspired by Grover’s quantum search algorithm, the Clifford circuit, and the classical binary search algorithm. Here, Grover’s algorithm helps to achieve a quadratic speed up, Clifford circuit helps to mitigate the errors, and binary search helps to find the chromatic number with guaranteed correctness in polynomial time. We also conducted several experiments to evaluate the proposed method's performance and correctness, and the experimental results corroborate our claims regarding the proposed method. Additionally, a mathematical analysis is performed to assess its time and qubit complexities, and to compare its efficiency with existing methods.
Graph coloring is a computationally difficult problem, and currently the best known classical algorithm forkkk-coloring of graphs onnnnvertices has runtimesΩ(2n)\Omega (2^n)Ω(2n)fork≥5k\geq 5k≥5. The list coloring problem asks the following more general question: given alistof available colors for each vertex in a graph, does it admit a proper coloring? We propose a hybrid classical-quantum algorithm based on Grover search 12 to quadratically speed up exhaustive search. Our algorithm loses in complexity to classical ones in specific restricted cases, but improves exhaustive search for cases, where the lists and graphs considered are arbitrary in nature.
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.
Quantum annealing is the quantum equivalent of the well known classical simulated annealing algorithm for combinatorial optimization problems. Despite the appeal of the approach, quantum annealing algorithms competitive with the state of the art for specific problems hardly exist in the literature. Graph colouring is a difficult problem of practical significance that can be formulated as combinatorial optimization. By introducing a symmetry-breaking problem representation, and finding fast incremental techniques to calculate energy changes, a competitive graph colouring algorithm based on quantum annealing is derived. This algorithm is further enhanced by tuning simplification techniques; replica spacing techniques to increase robustness; and a messaging protocol, which enables quantum annealing to efficiently take advantage of multiprocessor environments. Additionally, observations of some patterns in the tuning for random graphs led to a more effective algorithm able to find new upper bounds for several widely-used benchmark graphs, some of which had resisted improvement in the last two decades.