A fast and differentiable optimization model for finding the minimum dominating set in large-scale graphs

The minimum dominating set is an important NP-hard problem in graph theory, widely used in various fields such as computer network layout and social networks. Although linear programming algorithms can obtain a global optimal solution on small graphs, they are not suitable for large graphs. To find a local optimal solution, many heuristic algorithms have been proposed. These algorithms require strong techniques and programming, and most of them still need improvement in performance on large graphs. This paper presents a cooperative- competitive model that transforms the discrete problem into a continuously differentiable problem. By using the gradient descent method, this approach can handle all vertices simultaneously, and its time complexity is linear. Compared to existing search algorithms, this method is based on a straightforward principle and is fast. For small graphs, it can achieve performance almost comparable to linear programming methods. For large graphs, this method can explore more diverse local optimal solutions in the same amount of time, making it easier to obtain better performance. In this paper, a comparison and analysis of this gradient-based algorithm and graph theory-based search algorithms are conducted on the minimum dominating set, allowing researchers to have a clearer understanding of the advantages and disadvantages of search algorithms and gradient methods in complex problems. The referenced search algorithms may potentially improve the gradient descent algorithm, leading to better solutions in other fields. At the same time, gradient-based methods can be used to solve many similar NP-hard problems in graph theory.