A class of exponential neighbourhoods for the quadratic travelling salesman problem

The Quadratic Travelling Salesman Problem (QTSP) is to find a least-cost Hamiltonian cycle in an edge-weighted graph, where costs are defined on all pairs of edges such that each edge in the pair is contained in the Hamiltonian cycle. This is a more general version than the one that appears in the literature as the QTSP, denoted here as the adjacent quadratic TSP, which only considers costs for pairs of adjacent edges. Major directions of research work on the linear TSP include exact algorithms, heuristics, approximation algorithms, polynomially solvable special cases and exponential neighbourhoods [20] among others. In this paper we explore the complexity of searching exponential neighbourhoods for QTSP, the fixedrank QTSP, and the adjacent quadratic TSP. The fixed-rank QTSP is introduced as a restricted version of the QTSP where the cost matrix has fixed rank p. When c = 0, it is referred to as the homogenous rank p QTSP. It is shown that the homogeneous fixed-rank QTSP is solvable in pseudopolynomial time and admits FPTAS for each of the special cases studied, except for the case of matching edge ejection tours. The adjacent quadratic TSP is shown to be polynomially-solvable in many of the cases for which the linear TSP is polynomially-solvable. Interestingly, optimizing over the matching edge ejection tour neighbourhood is shown to be pseudopolynomial for the homogeneous rank 1 case, but NP-hard for the adjacent quadratic TSP case.