Asynchronous P systems for hard graph problems

Abstract

In the present paper, we consider fully asynchronous parallelism in membrane computing and propose asynchronous P systems for the following four graph problems: minimum coloring, maximum independent set, minimum vertex cover, and maximum clique. We first propose an asynchronous P system that solves the minimum graph coloring for a graph with n nodes and show that the proposed P system works in O(nⁿ⁺²) sequential steps or O(n²) parallel steps by using O(n²) kinds of objects. Second, we 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 by using O(n²) kinds of objects. We next propose two asynchronous P systems that solve the minimum vertex cover and the maximum clique for the same input graph 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 by using O(n²) kinds of objects.