A Cellular Solution to Subset Sum Using Division of Non-elementary Membranes and Dissolution, with Time and Initial Resources Bounded by log k

Abstract

The aim of our paper is twofold. On one hand we prove the ability of polar- izationless P systems with dissolution and with division rules for non-elementary mem- branes to solve NP-complete problems in a polynomial number of steps, and we do this by presenting a solution to the Subset Sum problem. On the other hand, we improve some similar results obtained for di®erent models of P systems by reducing the number of steps and the necessary resources to be of a logarithmic order with respect to k (recall that n and k are the two parameters used to indicate the size of an instance of the Subset Sum problem). As the model we work with does not allow cooperative rules and does not consider the membranes to have an associated polarization, the strategy that we will follow consists on using objects to represent the weights of the subsets through their multiplicities, and comparing the number of objects against a ¯xed number of membranes. More precisely, we will generate k membranes in log k steps.