The aim of a ranking aggregation problem is to combine several rankings into a single one that best represents them. A common method for solving this problem is due to Kemeny and selects as the aggregated ranking the one that minimizes the sum of the Kendall distances to the rankings to be aggregated. Unfortunately, the identification of the said ranking—called the Kemeny ranking—is known to be a computationally expensive task. In this paper, we study different ways of computing the Kemeny ranking with quantum optimization algorithms, and in particular, we provide some alternative formulations for the search for the Kemeny ranking as an optimization problem. To the best of our knowledge, this is the first time that this problem is addressed with quantum techniques. We propose four different ways of formulating the problem, one novel to this work. Two different quantum optimization algorithms—Quantum Approximate Optimization Algorithm and Quantum Adiabatic Computing—are used to evaluate each of the different formulations. The experimental results show that the choice of the formulation plays a big role on the performance of the quantum optimization algorithms.
The aim of a ranking aggregation problem is to combine several rankings into a single one that best represents them. A common method for solving this problem is due to Kemeny and selects as the aggregated ranking the one that minimizes the sum of the Kendall distances to the rankings to be aggregated. Unfortunately, the identification of the said ranking—called the Kemeny ranking—is known to be a computationally expensive task. In this paper, we study different ways of computing the Kemeny ranking with quantum optimization algorithms, and in particular, we provide some alternative formulations for the search for the Kemeny ranking as an optimization problem. To the best of our knowledge, this is the first time that this problem is addressed with quantum techniques. We propose four different ways of formulating the problem, one novel to this work. Two different quantum optimization algorithms—Quantum Approximate Optimization Algorithm and Quantum Adiabatic Computing—are used to evaluate each of the different formulations. The experimental results show that the choice of the formulation plays a big role on the performance of the quantum optimization algorithms.
Consensus ranking is a technique used to derive a single ranking that best represents the preferences of multiple individuals or systems. It aims to aggregate different rankings into one that minimizes overall disagreement or distance from each of the individual rankings. Kemeny ranking aggregation, in particular, is a widely used method in decision-making and social choice, with applications ranging from search engines to music recommendation systems. It seeks to determine a consensus ranking of a set of candidates based on the preferences of a group of individuals. However, existing quantum annealing algorithms face challenges in efficiently processing large datasets with many candidates. In this paper, we propose a method to improve the performance of quantum annealing for Kemeny rank aggregation. Our approach identifies the pairwise preference matrix that represents the solution list and subsequently reconstructs the ranking using classical methods. This method already yields better results than existing approaches. Furthermore, we present a range of enhancements that significantly improve the proposed method's performance, thereby increasing the number of candidates that can be effectively handled. Finally, we evaluate the efficiency of our approach by comparing its performance and execution time with that of KwikSort, a well-known approximate algorithm.
The Kemeny Rank Aggregation (KRA) problem is a well-studied problem in the field of Social Choice with a variety of applications in many different areas like databases and search engines. Intuitively, given a set of votes over a set of candidates, the problem asks to find an aggregated ranking of candidates that minimizes the overall dissatisfaction concerning the votes. Recently, a diverse version of KRA was considered which asks for a sufficiently diverse set of sufficiently good solutions. The framework of diversity of solutions is a young and thriving topic in the field of artificial intelligence. The main idea is to provide the user with not just one, but with a set of different solutions, enabling her to pick a sufficiently good solution that satisfies additional subjective criteria that are hard or impossible to model. In this work, we use a quantum annealer to solve the KRA problem and to compute a representative set of solutions. Quantum annealing is a meta search heuristic that does not only show promising runtime behavior on currently existing prototypes but also samples the solutions space in an inherently different way, making use of quantum effects. We describe how KRA instances can be solved by a quantum annealer and provide an implementation as well as experimental evaluations. As existing quantum annealers are still restricted in their number of qubits, we further implement two different data reduction rules that can split an instance into a set of smaller instances. In our evaluation, we compare classical heuristics that allow to sample multiple solutions such as simulated annealing and local search with quantum annealing performed on a physical quantum annealer. We compare runtime, quality of solution, and diversity of solutions, with and without applying preceding data reduction rules.
Abstract Solving combinatorial optimization problems represents a challenge for researchers, especially hard NP-complete problems that are commonly handled through heuristic methods providing approximate solutions. The emergence of quantum computing, which enables performing parallel computation, raises hope for solving such problems within a reasonable time frame. This paper proposes a quantum global optimization algorithm to solve combinatorial optimization problems intended to provide an improved performance. The proposed algorithm aims to amplify the probability of selecting the global optimum of the objective function defining the addressed problem using a specific variant of Grover’s search based on simultaneously exploring all comparison cases of possible solutions to recognize the right one. The paper also includes the study of precision and runtime of the presented concept and introduces a version of the proposed algorithm addressing the knapsack problem, along with its execution results and a comparison with other competing algorithms.
Abstract Solving combinatorial optimization problems represents a challenge for researchers, especially hard NP-complete problems that are commonly handled through heuristic methods providing approximate solutions. The emergence of quantum computing, which enables performing parallel computation, raises hope for solving such problems within a reasonable time frame. This paper proposes a quantum global optimization algorithm to solve combinatorial optimization problems intended to provide an improved performance. The proposed algorithm aims to amplify the probability of selecting the global optimum of the objective function defining the addressed problem using a specific variant of Grover’s search based on simultaneously exploring all comparison cases of possible solutions to recognize the global optimum. The paper also includes the study of precision and runtime of the presented concept and introduces a version of the proposed algorithm addressing the knapsack problem, along with its execution results and a comparison with other competing algorithms
Combinatorial optimization problems play a vital role in computer science and engineering, with applications in logistics, network design, finance, and resource allocation. However, many of these problems are NP-hard, making them computationally expensive to solve using classical optimization techniques, especially for large-scale instances. In recent years, quantum computing has emerged as a promising paradigm capable of addressing these limitations by leveraging quantum mechanical principles such as superposition and entanglement. This study investigates the application of quantum computing for solving combinatorial optimization problems, with a focus on algorithm design, complexity analysis, and real-world applicability. The research primarily employs the Quantum Approximate Optimization Algorithm (QAOA) and Grover's search algorithm to solve representative problems such as Max-Cut and the Knapsack problem. The performance of these algorithms is evaluated using key metrics including accuracy, execution time, and approximation ratio, and is compared with classical optimization techniques. The results demonstrate that QAOA is capable of generating near-optimal solutions, achieving accuracy levels of up to 93% for small problem instances. However, the study also highlights the challenges associated with current quantum systems, including noise, limited qubit availability, and increased execution time. Comparative analysis reveals that while classical methods remain more efficient for small-scale problems, quantum algorithms show significant potential for scalability and handling complex optimization tasks. The findings suggest that hybrid quantum-classical approaches offer a practical solution in the current Noisy Intermediate-Scale Quantum (NISQ) era. The study concludes that although quantum computing is still evolving, it holds strong potential for transforming combinatorial optimization in the future.
Abstract Solving combinatorial optimization problems represents a challenge for researchers, especially hard NP-complete problems that are commonly handled through heuristic methods providing approximate solutions. The emergence of quantum computing, which enables performing parallel computation, raises hope for solving such problems within a reasonable time frame. This paper proposes a quantum global optimization algorithm for solving combinatorial optimization problems with improved performance and runtime approaching Grover’s optimal time of quantum search. The algorithm determines the global optimum of the objective function defining the addressed problem using an enhanced application of Grover’s search by simultaneously exploring all comparison cases of possible solutions without requiring iterative operations, unlike previous approaches, which affect performance and runtime. The paper also introduces a version of the proposed algorithm for solving the knapsack problem, along with its execution results and a comparison with other competing algorithms.