🤖 AI Summary
This work addresses the Total Dominating Set Problem (TDP)—an NP-hard combinatorial optimization problem in graph theory—requiring the smallest vertex subset ( D subseteq V ) such that ( D ) has no isolated vertices and every vertex in ( V setminus D ) is adjacent to at least one vertex in ( D ).
Method: We propose the first systematic quantum approximate optimization framework for TDP, introducing a quantum encoding scheme and a parameterized quantum circuit implementation based on the Quantum Approximate Optimization Algorithm (QAOA).
Contribution/Results: Our experiments reveal a markedly skewed distribution of effective QAOA parameters, offering new insights into parameter landscape structure and informing future parameter-learning strategies. We validate correctness across diverse graph topologies, confirming QAOA’s capability to solve TDP instances; however, solution quality critically depends on high-fidelity parameter optimization. This work establishes a novel quantum-approximate pathway for tackling NP-hard combinatorial optimization problems, advancing the application of variational quantum algorithms to graph-theoretic challenges.
📝 Abstract
Recent advancements in quantum computing have led to significant research into applying quantum algorithms to combinatorial optimization problems. Among these challenges, the Total Domination Problem (TDP) is particularly noteworthy, representing a classic and critical example in the field. Since the last century, research efforts have focused on establishing its NP-completeness and developing algorithms for its resolution, which have been fundamental to combinatorial mathematics. Despite this rich history, the application of quantum algorithms to the TDP remains largely unexplored. In this study, we present a pioneering application of the Quantum Approximate Optimization Algorithm (QAOA) to tackle the TDP, evaluating its efficacy across a diverse array of parameters. Our experimental findings indicate that QAOA is effective in addressing the TDP; under most parameter combinations, it successfully computes the correct total dominating set (TDS). However, the algorithm's performance in identifying the optimal TDS is contingent upon the specific parameter choices, revealing a significant bias in the distribution of effective parameter points. This research contributes valuable insights into the potential of quantum algorithms for addressing the TDP and lays the groundwork for future investigations in this area.