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arXiv:2108.09061 · IEEE Transactions on Quantum Engineering

A Grover Search-Based Algorithm for the List Coloring Problem

Sayan Mukherjee

Published Jan 1, 2022• 25 citations
arXivHTMLPDF

Abstract

Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for$k$-coloring of graphs on$n$vertices has runtimes$\Omega (2^n)$for$k\geq 5$. 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.

DOI
10.1109/tqe.2022.3151137

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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 Graph coloring with minimum colors 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 coloring problem with the minimum number of colors. Therefore, classical approaches become infeasible for solving this problem on large graphs. In contrast, quantum computing offers the potential for speedup 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 coloring problem and finding the chromatic number of a graph with guaranteed correctness, and also reduces the time and qubit complexities. The proposed method is inspired by Randomized Grover’s search algorithm, and the classical binary search algorithm. Here, Randomized Grover’s search algorithm provides quadratic speedup and this algorithm is useful in real situations when the number of solutions is not known in advance in contrast to the Grover’s search algorithm. The binary search algorithm helps to achieve logarithmic speedup, which reduces overall complexity. The experimental results strongly validate our claims and demonstrate the superiority of the approach. Furthermore, a comprehensive mathematical analysis is carried out to examine its complexities, as well as to compare its computational efficiency with existing methods.
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 computing based Grover's algorithm performs unstructured search problems in unsorted data to structured data in validating bit string through graph coloring. The quantum Grover's algorithm and graph theory was looked at in graph coloring. In graph coloring algorithm can be leveraged to validate coloring assignments for graph with arbitrary structures. It's accuracy and efficiency has been achieved by the iteration of square root (N) By adapting Grover's search mechanism, we aim to optimize the process of finding valid coloring in a graph, which is a crucial NP-complete problem in classical computation. This algorithm determines shortest node to see if the gates can be colored rightly for detect validation bit string. To get a validated a significant advantage of the hybrid classical-quantum approach reduces time complexity and exhaustive search. Grover's algorithm reduces the number of iterations needed for a solution, providing a quantum advantage in efficiency.
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.
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.
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.