New Exact Methods for Solving Quadratic Traveling Salesman Problem

Yuxiao Chen, Anubhav Singh, Ryo Kuroiwa, Chris Beck

Abstract: The Quadratic Traveling Salesman Problem (QTSP) is a variant of the Traveling Salesman Problem (TSP) in which the objective function depends on pairs of consecutive edges in the tour; hence, it is quadratic and generally hard to optimize. While various exact-solving approaches have been explored, many rely on specialized procedures and struggle to scale to large instances. Carefully crafted metaheuristics have demonstrated better primal bounds and scalability than the exact approaches, but, of course, cannot provide guarantees of solution quality nor prove optimality. In this work, we define encodings of QTSP in domain-independent dynamic programming (DIDP), constraint programming (CP), mixed integer quadratic programming (MIQP), and mixed integer linear programming (MILP), and compare them with the best-known exact method, a branch and cut (B&C) algorithm, and the state-of-the-art metaheuristic, a hybrid genetic algorithm (HGA). Our experimental results demonstrate that a DIDP model finds the best feasible solutions and the smallest optimality gaps on average among all exact solvers, including the B&C algorithm, for sufficiently large problems. HGA finds the best feasible solution among all approaches, with DIDP within 15% of the HGA cost on all experimental instances. Interestingly, our MILP model with the subtour elimination constraints generally finds better feasible solutions than the B&C algorithm while matching it in proving optimality, suggesting that lazily adding sub-tour elimination cuts is not particularly helpful in QTSP.