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.
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.
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 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.
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.
The integration of quantum algorithms into machine learning pipelines offers a promising direction for addressing complex combinatorial optimization problems inherent in graph-structured data. This study investigates the use of the Quantum Approximate Optimization Algorithm (QAOA) as a quantum subroutine within a hybrid quantum–classical framework to enhance machine learning solutions for graph-based tasks. QAOA is employed to solve NP-hard graph optimization problems such as Max-Cut, graph coloring, and community detection, which are reformulated as Quadratic Unconstrained Binary Optimization (QUBO) problems. The solutions provided by QAOA are then incorporated into machine learning models, such as Graph Neural Networks (GNNs) or clustering algorithms, as enriched features or optimized graph partitions. The hybrid approach is evaluated on benchmark datasets, demonstrating competitive accuracy and improved scalability in certain problem instances compared to purely classical methods. The findings highlight the potential of leveraging near-term quantum hardware for augmenting classical learning systems in domains such as logistics, social network analysis, and computational biology.
Quantum computing holds the potential to efficiently solve certain problems believed to be intractable for classical computers. Realizing this potential, however, requires substantial physical resources, including large qubit counts, sufficient connectivity, and long coherence times. As current quantum devices are still far from this regime, the efficient compilation of quantum algorithms into hardware-compatible instructions remains a central challenge in quantum computing, especially for near-term applications. A representative example of such applications is combinatorial optimization, where exponentially large search spaces motivate quantum algorithms that leverage interference to approximate solutions more efficiently than classical heuristics, even at modest system sizes. Two prominent paradigms in this context are quantum annealing and the Quantum Approximate Optimization Algorithm (QAOA): Quantum annealing realizes optimization via continuous-time quantum dynamics implemented on analog hardware, while QAOA is a variational, gate-based approach that builds parameterized quantum circuits from problem-specific Hamiltonians. Both paradigms require compilation strategies tailored to the constraints of their respective hardware architectures. Most existing compilation strategies are problem-agnostic and therefore broadly applicable, yet often suboptimal for specific tasks. This thesis uses the graph coloring problem as a case study to investigate a problem-driven approach to quantum compilation, where structural features of the problem class are exploited to design tailored compilation schemes that offer improved efficiency and scalability for the targeted application. To address hardware connectivity constraints, the presented approaches adopt the parity framework, which encodes long-range interactions into single-qubit terms. This thesis develops two novel problem-driven compilation methods for the graph coloring problem, the Bandwidth-Optimized LHZ layout and the 2D-Twine circuit. The former leverages bandwidth minimization to optimize parity encodings for analog quantum annealers, while the latter constructs QAOA unitaries for gate-based optimization using parallelized Parity Twine circuits. Both approaches are benchmarked against general-purpose methods and demonstrate substantial improvements in qubit usage or circuit complexity. These results illustrate the advantages of incorporating problem structure into the compilation process and provide a proof-of-concept for the effectiveness of problem-driven compilation strategies.
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.