Application of the Quantum Approximate Optimization Algorithm in Solving the Total Domination Problem

Recent advancements in quantum computing have spurred substantial research into the application of quantum algorithms to combinatorial optimization problems. Among these challenges, the Total Domination Problem (TDP) emerges as a classic and critical paradigm in the field. For a graph G(V, E), TDP entails finding a minimal subset D subset of V that contains no isolated vertices, where every vertex not in D has at least one neighbor in D. TDP finds extensive applications across domains such as computer networks, social networks, and communications. Since the latter half of the last century, research efforts have focused on establishing its NP-completeness and developing solution algorithms, which have become foundational to combinatorial mathematics. Despite this rich history, the application of quantum algorithms to TDP remains largely underexplored. In this study, we present a pioneering application of the Quantum Approximate Optimization Algorithm (QAOA) to tackle TDP, evaluating its efficacy across a diverse set of parameters. This paper proves that the upper bound on the number of qubits required to solve TDP is 2|V| + |V| log_2( (2|E|)/|V| - 1 ). Our experimental findings demonstrate that QAOA is effective in addressing TDP: under most parameter combinations, it successfully computes a valid total dominating set (TDS). However, the algorithm's performance in identifying the optimal TDS is contingent upon 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 solving TDP and lays a solid groundwork for future investigations in this area.