Computational Complexity

Abstract

Until recently, Reti-like (RL) problems, as well as many other problems represented here as Complex Systems, were considered as computationally hard. A general opinion was that these problems belong to the class of NP-hard problems (Gary and Johnson, 1991). In this chapter we will show that the running time of the conventional search algorithms, when applied to these problems is O(3^dn), with n being the size of a side of the equilateral d-dimensional operational district. Some of the RL problems have been solved employing the LG approach with very small search trees (Chapters 3, 4, and 6). Moreover, a new version of the LG approach, the LG no-search algorithm, introduced in Chapter 13 allowed us to solve RL problems without any search at all and simultaneously prove optimality of the solutions (Theorem 13.1). In this chapter we will prove that the running time of the LG no-search algorithm for these problems is bounded by a low-degree polynomial. To prove this we will evaluate the running time of two LG generating grammars: the Grammar of Trajectories and the Grammar of Zones. We will also show how to avoid the direct usage of two procedures that cause exponential growth in the running time of the LG algorithms: unfolding bundles of trajectories and cloning Zones.