Complexity and approximability of the Euclidean generalized traveling salesman problem in grid clusters

Abstract

We consider the geometric version of the well-known Generalized Traveling Salesman Problem introduced in 2015 by Bhattacharya et al. that is called the Euclidean Generalized Traveling Salesman Problem in Grid Clusters (EGTSP-GC). They proved the intractability of the problem and proposed first polynomial time algorithms with fixed approximation factors. The extension of these results in the field of constructing the polynomial time approximation schemes (PTAS) and the description of non-trivial polynomial time solvable subclasses for the EGTSP-GC appear to be relevant in the light of the classic C. Papadimitriou result on the intractability of the Euclidean TSP and recent inapproximability results for the Traveling Salesman Problem with Neighborhoods (TSPN) in the case of discrete neighborhoods. In this paper, we propose Efficient Polynomial Time Approximation Schemes (EPTAS) for two special cases of the EGTSP-GC, when the number of clusters $k=O(\log n)$ and $k=n-O(\log n)$. Also, we show that any time, when one of the grid dimensions (height or width) is fixed, the EGTSP-GC can be solved to optimality in polynomial time. As a consequence, we specify a novel non-trivial polynomially solvable subclass of the Euclidean TSP in the plane.