The Hamiltonian Cycle and Travelling Salesman Problems in cP Systems

The Hamiltonian Cycle Problem (HCP) and Travelling Salesman Problem (TSP) are long-standing and well-known NP-hard problems. The HCP is concerned with finding paths through a given graph such that those paths visit each node exactly once after the start, and end where they began (i.e., Hamiltonian cycles). The TSP builds on the HCP and is concerned with computing the lowest cost Hamiltonian cycle on a weighted (di)graph. Many solutions to these problems exist, including some from the perspective of P systems. For the TSP however, almost all these papers have combined membrane computing with other approaches for approximate solution algorithms, which is surprising given the plethora of P systems solutions to the HCP. A recent paper presented a bruteforce style P systems solution to the TSP with a time complexity of O(n), exploiting the ability of P systems to reduce time complexity in exchange for space complexity, but the resultant system had a fairly high number of rules, around 50. Inspired by this paper, and seeking a more concise representation of an exact brute-force TSP algorithm, we have devised a P systems algorithm based on cP systems (P systems with Complex Objects) which requires five rules and takes n + 3 steps. We first provide some background on cP systems and demonstrate a fast new cP systems method to Address for correspondence: Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Corresponding author 1002 J. Cooper, R. Nicolescu / The HCP & TSP in cP systems find the minimum of a multiset, then describe our solution to the HCP, and build on that for our TSP algorithm. This paper describes said algorithms, and provides an example application of our TSP algorithm to a given graph and a digraph variant.