# Geometric and LP-based heuristics for the quadratic travelling salesman problem

A generalization of the classical TSP is the so-called quadratic travelling salesman problem (QTSP), in which a cost coefficient is associated with the transition in every vertex, i.e. with every pair of edges traversed in succession. In this paper we consider two geometrically motivated special cases of the QTSP known from the literature, namely the angular-metric TSP, where transition costs correspond to turning angles in every vertex, and the angular-distance-metric TSP, where a linear combination of turning angles and Euclidean distances is considered. At first we introduce a wide range of heuristic approaches, motivated by the typical geometric structure of optimal solutions. In particular, we exploit lens-shaped neighborhoods of edges and a decomposition of the graph into layers of convex hulls, which are then merged into a tour by a greedy-type procedure or by utilizing an ILP model. Secondly, we consider an ILP model for a standard linearization of QTSP and compute fractional solutions of a relaxation. By rounding we obtain a collection of subtours, paths and isolated points, which are combined into a tour by various strategies, all of them involving auxiliary ILP models. Finally, different improvement heuristics are proposed, most notably a matheuristic which locally reoptimizes the solution for rectangular sectors of the given point set by an ILP approach. Extensive computational experiments for benchmark instances from the literature and extensions thereof illustrate the Pareto-efficient frontier of algorithms in a (running time, objective value)-space. It turns out that our new methods clearly dominate the previously published heuristics.

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