On the nearest neighbor rule for the metric traveling salesman problem

Abstract. We prove that the approximation ratio of the greedy algorithm for the metric Traveling Salesman Problem is Θ(log n). Moreover, we prove that the same result also holds for graphic, euclidean, and rectilinear instances of the Traveling Salesman Problem. Finally we show that the approximation ratio of the ClarkeWright savings heuristic for the metric Traveling Salesman Problem is Θ(log n).

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“…The bold edges in Figure 1 form a partial greedy tour on V 0 . The next lemma is similar to Lemma 1 in [4] and Lemma 1 in [3]. Proof.…”

Section: The Approximation Ratio Of the Greedy Algorithmmentioning

confidence: 70%

“…In this section we describe the construction of a family of metric TSP instances G k on which the greedy algorithm can find a TSP tour that is by a factor of Ω(k) longer than an optimum tour. Our construction is similar to the approach in [3].…”

Section: The Approximation Ratio Of the Greedy Algorithmmentioning

confidence: 99%

The approximation ratio of the greedy algorithm for the metric traveling salesman problem

Brecklinghaus

Hougardy

2015

Operations Research Letters

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“…The common method for solving the TSP with a genetic algorithm is shown by a flowchart in Figure 1(a). This algorithm is used in a lot of applications in different areas of science and engineering [32].…”

Section: The Proposed Approachmentioning

confidence: 99%

Efficient Parallelization of a Genetic Algorithm Solution on the Traveling Salesman Problem with Multi-core and Many-core Systems

Abbasi

Rafiee

2020

IJE

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Efficient parallelization of genetic algorithms (GAs) on state-of-the-art multi-threading or manythreading platforms is a challenge due to the difficulty of scheduling hardware resources regarding the concurrency of threads. In this paper, for resolving the problem, a novel method is proposed, which parallelizes the GA by designing three concurrent kernels, each of which are running some dependent effective operators of GA. The proposed method can be straightforwardly adapted to run on many-core and multi-core processors by using Compute Unified Device Architecture (CUDA) and Threading Building Blocks (TBB) platforms. To efficiently use the valuable resources of such computing cores in concurrent execution of the GA, threads that run any of the triple kernels are synchronized by a considerably fast switching technique. The offered method was used for parallelizing a GA-based solution of Traveling Salesman Problem (TSP) over CUDA and TBB platforms with identical settings. The results confirm the superiority of the proposed method to state-of-the-art methods in effective parallelization of GAs on Graphics Processing Units (GPUs) as well as on multi-core Central Processing Units (CPUs). Also, for GA problems with a modest initial population, though the switching time among GPU kernels is negligible, the TBB-based parallel GA exploits the resources more efficiently.

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“…Thus, the constructional heuristic methods are in general aimed for selecting the edges of minimal length. In the case of Nearest Neighbour method [13], the algorithm starts with a random selection of a vertex. In each iteration step, the nearest free vertex is selected and it will be connected to the current node.…”

Section: Introductionmentioning

confidence: 99%

“…Greedy Vertex Insertion algorithm [15] extends the existing route with a vertex having the lowest cost increase. In the case of Nearest Neighbour method [13], the shortest free edge is selected related to the actual node. In all cases, if we extend the graph with a random new element and process this element with the mentioned methods, the resulted route will be usually suboptimal.…”

Section: Introductionmentioning

confidence: 99%

IntraClusTSP—An Incremental Intra-Cluster Refinement Heuristic Algorithm for Symmetric Travelling Salesman Problem

2018

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The Symmetric Traveling Salesman Problem (sTSP) is an intensively studied NP-hard problem. It has many important real-life applications such as logistics, planning, manufacturing of microchips and DNA sequencing. In this paper we propose a cluster level incremental tour construction method called Intra-cluster Refinement Heuristic (IntraClusTSP). The proposed method can be used both to extend the tour with a new node and to improve the existing tour. The refinement step generates a local optimal tour for a cluster of neighbouring nodes and this local optimal tour is then merged into the global optimal tour. Based on the performed evaluation tests the proposed IntraClusTSP method provides an efficient incremental tour generation and it can improve the tour efficiency for every tested state-of-the-art methods including the most efficient Chained Lin-Kernighan refinement algorithm. As an application example, we apply IntraClusTSP to automatically determine the optimal number of clusters in a cluster analysis problem. The standard methods like Silhouette index, Elbow method or Gap statistic method, to estimate the number of clusters support only partitional (single level) clustering, while in many application areas, the hierarchical (multi-level) clustering provides a better clustering model. Our proposed method can discover hierarchical clustering structure and provides an outstanding performance both in accuracy and execution time.

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On the nearest neighbor rule for the metric traveling salesman problem

Cited by 18 publications

References 11 publications

The approximation ratio of the greedy algorithm for the metric traveling salesman problem

The approximation ratio of the greedy algorithm for the metric traveling salesman problem

Efficient Parallelization of a Genetic Algorithm Solution on the Traveling Salesman Problem with Multi-core and Many-core Systems

IntraClusTSP—An Incremental Intra-Cluster Refinement Heuristic Algorithm for Symmetric Travelling Salesman Problem

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