Generalized Pyramidal Tours for the Generalized Traveling Salesman Problem

In this paper, we introduce the notion of l-quasi-pyramidal and l-pseudo-pyramidal tours extending the classic notion of pyramidal tours to the case of the Generalized Traveling Salesman Problem (GTSP). We show that, for the instance of GTSP on n cities and k clusters with arbitrary weights, l-quasi-pyramidal and l-pseudo-pyramidal optimal tours can be found in time O(4(l)n(3)) and O(2(l) k(l+4)n(3)), respectively. Consequently, we show that, in the most general setting, GTSP belongs to FPT for parametrizations induced by these special kinds of tours. Also, we describe a non-trivial polynomially solvable subclass of GTSP, for which the existence of l-quasi-pyramidal optimal tour (for some fixed value of l) is proved.