TSP Instance Research Articles

Learning to optimise general TSP instances

The Travelling Salesman Problem is a classical combinatorial optimisation problem (COP). In recent years, learning to optimise approaches have shown success in solving TSP problems. However, they focus on one type of TSP instance, where the points are uniformly distributed in Euclidean spaces (easy instances). Such approaches cannot generalise to other embedding spaces that represent various levels of difficult instances, e.g., TSP instances where the points are distributed in a non-uniform manner and spherical spaces. Obtain optimal solutions for easy instances is achievable and can be used as training data to solve various TSP instances. However, acquire optimal solutions for complex TSP instances is difficult and time-consuming. Hence, this paper introduces a new learning-based approach based on a convolutional neural network combined with a Long Short-Term Memory, referred to as the non-Euclidean TSP network (NETSP), that utilises randomly generated instances (easy instances) to solve various common TSP instances (complex TSP instances). We have demonstrated its superiority over state-of-the-art methods for various TSP instances. We performed extensive experiments that indicate our approach generalises across many instances and scales to larger instances.

A Smart Structural Algorithm (SSA) Based on Infeasible Region to Solve Mixed Integer Problems

Optimization is an important fields of study in science where researchers seek to make the best and most practical decisions. Solving real optimization problems is an intractable issue which calls for generating an approximate using meta-heuristic algorithms. This study proposes a meta-heuristic algorithm which mainly searches the infeasible region. In this approach, the authors start from an infeasible solution, and while they try to get near to the feasible region, they ensure that the best value is kept for the objective function. The algorithm examines the space in such terms as Infeasibility and Objective Functions, Neighborhood Limited Area, Random Smart Points, and the calculation of new solutions. The algorithm can convert an infeasible solution to an appropriate corresponding feasible solution by applying a simple mathematical methodology. Finally, to test the efficiency of our algorithm, a sample random MIP problem and a hard benchmark TSP instance are solved and discussed in detail.

Simulated Annealing versus Metropolis for a TSP instance

In a recent paper [I. Wegener, Simulated Annealing beats Metropolis in combinatorial optimization, in: L. Caires, G.F. Italiano, L. Monteiro, C. Palamidessi, M. Yung (Eds.), Proc. ICALP 2005, in: LNCS, vol. 3580, 2005, pp. 589–601] Wegener gave a first natural example of a combinatorial optimization problem where for certain instances a Simulated Annealing algorithm provably performs better than the Metropolis algorithm for any fixed temperature. Wegener's example deals with a special instance of the Minimum Spanning Tree problem. In this short note we show that Wegener's technique as well can be used to prove a similar result for another important problem in combinatorial optimization, namely the Traveling Salesman Problem. The main task is to construct a suitable TSP instance for which SA outperforms MA when using the well known 2-Opt local search heuristic.

Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number

Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!/ p( n) tours for some polynomial p( n) for every TSP instance on n cities. They conjectured that, unless P=NP, the answer to this question is negative. We prove that the answer to this question is, in fact, positive. A generalization of the TSP, the quadratic assignment problem, is also considered with respect to the analogous question. Probabilistic, graph-theoretical, group-theoretical and number-theoretical methods and results are used.