Asynchronous P Systems for the Maximum Independent Set and Related Graph Problems

In the present paper, we consider fully asynchronous parallelism in membrane computing, and propose asynchronous P systems for three graph problems, which are the maximum independent set, the minimum vertex cover, and the maximum clique. We first propose an asynchronous P system that solves the maximum independent set for a graph with n nodes, and show that the proposed P system works in O(n² · 2ⁿ) sequential steps or O(n²) parallel steps using O(n²) kinds of objects. We next propose an asynchronous P system that solves the minimum vertex cover and the maximum clique for a graph with n nodes by reduction to the maximum independent set, and show that the proposed P system works in O(n² · 2ⁿ) sequential steps or O(n²) parallel steps using O(n²) kinds of objects.