The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal solution for graph coloring problems. However, this naive approach can increase the computation cost exponentially as the graph size and the number of colors increase. To mitigate such intolerable overhead, we investigate the methods to take the advantages of quantum computing properties to find a solution for multi-coloring graph problems in polynomial time. We utilize the variational quantum eigensolver (VQE) technique and quantum approximate optimization algorithm (QAOA) to find solutions for three combinatorial applications by both transferring each problem model to the corresponding Ising model and by using the calculated Hamiltonian matrices. Our results demonstrate that VQE and QAOA algorithms can find one of the best solutions for each application. Therefore, our modeling approach with hybrid quantum algorithms can be applicable for combinatorial problems in various fields to find an optimal solution in polynomial time.
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal solution for graph coloring problems. However, this naive approach can increase the computation cost exponentially as the graph size and the number of colors increase. To mitigate such intolerable overhead, we investigate the methods to take the advantages of quantum computing properties to find a solution for multi-coloring graph problems in polynomial time. We utilize the variational quantum eigensolver (VQE) technique and quantum approximate optimization algorithm (QAOA) to find solutions for three combinatorial applications by both transferring each problem model to the corresponding Ising model and by using the calculated Hamiltonian matrices. Our results demonstrate that VQE and QAOA algorithms can find one of the best solutions for each application. Therefore, our modeling approach with hybrid quantum algorithms can be applicable for combinatorial problems in various fields to find an optimal solution in polynomial time.
Graph coloring is a well-known combinatorial optimization problem with broad applications in network security, scheduling, and register allocation. Being an NP-hard problem, exact solutions for large instances are computationally infeasible, calling for other solution strategies. In this study, parameterized and approximation methods to tackle the graph coloring problem are investigated. While approximation algorithms yield near-optimal solutions with theoretical guarantees of performance that are well-suited for large-scale applications, parameterized algorithms utilize graph-specific structural properties to manage constrained instances efficiently. We examine the trade-offs in these approaches, highlighting their strengths and weaknesses. In addition, we introduce a hybrid approach that combines both parameterized and approximation methods to best optimize computational resources and balance accuracy versus feasibility. Furthermore, we discuss the newly realized potential of quantum computing—specifically quantum annealing and the Quantum Approximate Optimization Algorithm (QAOA)—to speed up graph coloring by leveraging quantum parallelism and state superposition. Such quantum-based techniques represent a promising line of attack for breaking up large and hard instances that overwhelm classical approaches. Future research will aim to tighten approximation guarantees, create quantum-augmented hybrid models, and test theoretical results against real-world datasets.
In the era of Noisy Intermediate Scale Quantum (NISQ) computing, available quantum resources are limited. Many NP-hard problems can be efficiently addressed using hybrid classical and quantum computational methods. This paper proposes a hybrid variational quantum algorithm designed to solve the k-coloring problem of graph vertices. The hybrid classical and quantum algorithms primarily partition the graph into multiple subgraphs through hierarchical techniques. The Quantum Approximate Optimization Algorithm (QAOA) is employed to determine the coloring within the subgraphs, while a classical greedy algorithm is utilized to find the coloring of the interaction graph. Fixed coloring is applied to the interaction graph, and feedback is provided to correct any conflicting colorings within the subgraphs. The merging process into the original graph is iteratively optimized to resolve any arising conflicts. We employ a hierarchical framework that integrates feedback correction and conflict resolution to achieve k-coloring of arbitrary graph vertices. Through experimental analysis, we demonstrate the effectiveness of the algorithm, highlighting the rapid convergence of conflict evolution and the fact that iterative optimization allows the classical algorithm to approximate the number of colorings. Finally, we apply the proposed algorithm to optimize the scheduling of a subway transportation network, demonstrating a high degree of fairness.
We propose a technique to solve the problem of graph multi coloring, which is a problem corresponding to channel allocation problem, with the quantum approximate optimization algorithm(QAOA). QAOA is an algorithm that solve an optimization problem. Among the optimization problems, this algorithm is specialized as solving the quadratic unconstrained binary optimization(QUBO) problems. It is designed to find approximate solutions to combinatorial optimization problems by leveraging quantum resources. In general, it is evaluated to have strengths in optimizing the base of optimal path exploration, small factor resolution, and mass data exploration. The study on coloring problem using digital algorithm has solved the problem of coloring by replacing the graph with simple graph or applying algorithm or reducing the number of nodes. Despite these efforts, however, the digital algorithm that solves the coloring problem studied so far has not achieved satisfactory performance by some limitations. Therefore, in order to overcome these limitations, this paper will show the result of solving multi coloring problem by reducing initial state dimension of the QAOA
Graph coloring is a well-known NP-complete problem with applications in scheduling, register allocation, frequency assignment, and network optimization. Quantum computing offers the potential for polynomial or even super-polynomial speedups in certain problem instances, yet practical quantum graph coloring methods must carefully balance qubit resources, circuit depth, and constraint enforcement. We proposed a solver agnostic quantum framework for K-coloring. Along with a novel encoding and efficient constraint implementation for its exact coloring exploiting Grover search and almost optimal coloring by the Quantum Approximate Optimization Algorithm \emph{(QAOA)}, and \emph{Quantum Annealing}. Unlike using O(NK) qubits as in \emph{SOTA}, we reduced the qubit requirements to O(Nlog2K), along with optimized comparator circuits enforcing adjacency constraints. Further symmetry-based graph reduction is incorporated as an optional preprocessing step to further reduce the instance size before quantum execution.
ABSTRACT: Even if the quantum computing popularity has recently grown, the currently-available technology is not ready for a fundamental revolution of informatics. In this transition phase, hybrid quantum-classical algorithms can help in understanding the real power that this novel paradigm can reach in the future. In this work, we tested the capability of two such algorithms to solve combinatorial optimization problems: the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA). We used them to solve max-cut problems, consisting of separating the vertices of an undirected graph in two sub-groups so that the number of edges connecting vertices of different sub-groups is maximal. To do that, we mapped the problem into a quantum system, using the Ising formalism. In particular, we investigated the effect of changing the number of measurements of the quantum circuit associated with the system on the algorithms convergence. A large number of measurements (or shots) allows a better knowledge of the quantum state, but on the other hand, increases the convergence time, making the algorithms less competitive compared to classical techniques. The results showed that VQE needs a mínimum number of measurements to start converging towards the optimal problem solution. Once this threshold is overcome, the solution improves as 1/√shots. QAOA, on the other hand, does not seem to converge, independently of the number of shots.
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.