The intermittent travelling salesman problem

The continuous Hopfield network (CHN) can be used to solve, among other combinatorial optimization problems, the traveling salesman problem (TSP). In order to improve the performance of this heuristic technique, a divide-and-conquer strategy based on two phases is proposed. The first phase involves linking cities with the most neighbors to define a set of chains of cities and, secondly, to join these with isolated cities to define the final tour. Both problems are solved by mapping the two TSPs onto their respective CHNs. The associated parameter-setting procedures are deduced; these procedures ensure the feasibility of the obtained tours, and the quality of the solutions is compared with the pure CHN approach using some traveling salesman problem library (TSPLIB) instances. By means of this strategy, solving TSP instances with sizes of up to 13,509 cities are allowed with the computational resources we had available. Finally, the new divide-and-conquer procedure is improved by tuning the parameter which controls the first phase.

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP.

This paper addresses a MinMax variant of the Dubins multiple traveling salesman problem (mTSP). This routing problem arises naturally in mission planning applications involving fixed-wing unmanned vehicles and ground robots. We first formulate the routing problem, referred to as the one-in-a-set Dubins mTSP problem (MD-GmTSP), as a mixed-integer linear program (MILP). We then develop heuristic-based search methods for the MD-GmTSP using tour construction algorithms to generate initial feasible solutions relatively fast and then improve on these solutions using variants of the variable neighborhood search (VNS) metaheuristic. Finally, we also explore a graph neural network to implicitly learn policies for the MD-GmTSP using a learning-based approach; specifically, we employ an S-sample batch reinforcement learning method on a shared graph neural network architecture and distributed policy networks to solve the MD-GMTSP. All the proposed algorithms are implemented on modified TSPLIB instances, and the performance of all the proposed algorithms is corroborated. The results show that learning based approaches work well for smaller sized instances, while the VNS based heuristics find the best solutions for larger instances.

In this paper, we present some new results obtained for the traveling salesman problem (TSP) by using the iterated tabu search (ITS) meta-heuristic. ITS is a promising extension to the ordinary tabu search scheme. It seeks near-optimal solutions by combining intensification (standard tabu search) and diversification (perturbation of solutions) in a proper way. For the TSP, the main effect is achieved due to decomposition of the solution neighbourhood structure and considerably speeding-up the tabu search process, which is used, namely, in the role of intensification. This fast-iterated tabu search (FITS) technique resulted in quite encouraging solutions for the TSP instances from the TSP instance library TSPLIB. FITS obviously outperformed the other heuristic algorithms used in the experimentation, especially, on the smaller TSP instances.

Recent work applying deep reinforcement learning (DRL) to solve traveling salesman problems (TSP) has shown that DRL-based solvers can be fast and competitive with TSP heuristics for small instances, but do not generalize well to larger instances. In this work, we propose a novel approach named MAGIC that includes a deep learning architecture and a DRL training method. Our architecture, which integrates a multilayer perceptron, a graph neural network, and an attention model, defines a stochastic policy that sequentially generates a TSP solution. Our training method includes several innovations: (1) we interleave DRL policy gradient updates with local search (using a new local search technique), (2) we use a novel simple baseline, and (3) we apply curriculum learning. Finally, we empirically demonstrate that MAGIC is superior to other DRL-based methods on random TSP instances, both in terms of performance and generalizability. Moreover, our method compares favorably against TSP heuristics and other state-of-the-art approach in terms of performance and computational time.

We show how, by a constructive process, we can generate arbitrarily large instances of the Traveling Salesman Problem (TSP) using standard fractals such as those of Peano, Koch, or Sierpinski. We show that optimal solutions for these TSPs can be known a priori, and thus, they provide us with new nontrivial TSP instances offering the possibility of testing heuristics well beyond the scope of testbed instances which have been solved by exact numerical methods. Furthermore, instances may be constructed with different features, for example, with different fractal dimensions. To four of these fractal TSPs we apply three standard constructive heuristics, namely Multiple Fragment, Nearest Neighbor, and Farthest Insertion from Convex-Hull, which have efficient general-purpose implementations. The ability of different algorithms to solve these different fractal TSPs gives us significant insight into the nature of TSP heuristics in a way which is complementary to approaches such as worst-case or average-case analysis.

We report for the first time on finding shortest path solutions for the travelling salesman problem (TSP) using hybrid “in materio” computation: a technique that uses search algorithms to configure materials for computation. A single-walled carbon nanotube (SWCNT) / polymer composite material deposited on a micro-electrode array is configured using static voltages so that voltage output readings determine the path order in which to visit cities in a TSP. Our initial results suggest that the hybrid computation with the SWCNT material is able to solve small instances of the TSP as efficiently as a comparable evolutionary search algorithm performing the same computation in software. Interestingly the results indicate that the hybrid system’s search performance on TSPs scales linearly rather than exponentially on these smaller instances. This exploratory work represents the first step towards building SWCNT-based electrode arrays in parallel so that they can solve much larger problems.KeywordsShort PathTravelling Salesman ProblemTravel Salesman ProblemElectrode ArraySmall InstanceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the traveling salesman problem, there are a collection of cities and travel cost between each pair of them. The aim is to find the minimum cost way of visiting all cities and returning to the starting point. This kind of problem is deceptive and one of the most intensely studied problems in computational mathematics. No effective solution method is known for the general case. Variable Neighborhood Search (VNS) is a recent metaheuristic for solving combinatorial and global optimization problems whose basic idea is systematic change of neighborhood within a local search. Its development has been rapid, with several dozen papers already published or to appear. Many extensions have been made, mainly to allow solution of large problem instances. In most of them, an effort has been made to keep the simplicity of the basic scheme. In this study, variable neighborhood search structure as a metaheuristic optimization technique and neighborhood approximation is developed. K-opt neighborhood structure is generated. This new structure’s solvability in benchmark and symmetric traveling salesman problem instances is tested, and results are listed.

A three-phase algorithm for the pollution traveling Salesman problem

The traveling salesman problem (TSP) is one of the most studied problems in combinatorial optimization. Given a set of nodes and the distances between them, it consists in finding the shortest route that visits each node exactly once and returns to the first. Nevertheless, more flexible and applicable formulations of this problem exist and can be considered. The Steiner TSP (STSP) is a variant of the TSP that assumes that only a given subset of nodes must be visited by the shortest route, eventually visiting some nodes and edges more than once. In this paper, we adapt some classical TSP constructive heuristics and neighborhood structures to the STSP variant. In particular, we propose a reduced 2‐opt neighborhood and we show that it leads to better results in smaller computation times. Computational results with an implementation of a GRASP heuristic using path‐relinking and restarts are reported. In addition, ten large test instances are generated. All instances and their best‐known solutions are made available for download and benchmarking purposes.

For the traveling salesman problem (TSP), the existing supervised learning based algorithms suffer seriously from the lack of generalization ability. To overcome this drawback, this paper tries to train (in supervised manner) a small-scale model, which could be repetitively used to build heat maps for TSP instances of arbitrarily large size, based on a series of techniques such as graph sampling, graph converting and heat maps merging. Furthermore, the heat maps are fed into a reinforcement learning approach (Monte Carlo tree search), to guide the search of high-quality solutions. Experimental results based on a large number of instances (with up to 10,000 vertices) show that, this new approach clearly outperforms the existing machine learning based TSP algorithms, and significantly improves the generalization ability of the trained model.

Given a Traveling Salesman Problem solution, the best 3-OPT move requires us to remove three edges and replace them with three new ones so as to shorten the tour as much as possible. No worst-case algorithm better than the Θ(n3) enumeration of all triples is likely to exist for this problem, but algorithms with average case O(n3−ϵ) are not ruled out. In this paper we describe a strategy for 3-OPT optimization which can find the best move by looking only at a fraction of all possible moves. We extend our approach also to some other types of cubic moves, such as some special 6-OPT and 5-OPT moves. Empirical evidence shows that our algorithm runs in average subcubic time (upper bounded by O(n2.5)) on a wide class of random graphs as well as Traveling Salesman Problem Library (TSPLIB) instances.

Heuristic approaches for the cutting path problem

We propose an evolutionary algorithm (EA) that applies to the traveling salesman problem (TSP). The EA uses edge assembly crossover (EAX), which is known to be efficient and effective for solving TSPs. Recently, a fast implementation of EAX and an effective technique for preserving population diversity were proposed. This makes it possible to compare the EA with EAX comparable to state-of-the-art TSP heuristics based on Lin-Karnighan heuristics. We further improved the performance of EAs with EAX, especially for large instances of more than 10,000 cities. Our method can find optimal solutions for instances of up to 24978 cities within a day using a single Itanium 2 1.3-GHz processor. Moreover, our EA found three new best tours for unsolved national TSP instances in a reasonable computation time.

Parallel techniques for solving large scale travelling salesperson problems