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Results for the close-enough traveling salesman problem with a branch-and-bound algorithm

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Abstract

The Close-Enough Traveling Salesman Problem is a generalization of the Traveling Salesman Problem that requires a salesman to just get close enough to each customer instead of visiting the exact location of each customer. In this paper, we propose improvements to an existing branch-and-bound (B &B) algorithm for this problem that finds and proves optimality of solutions by examining partial sequences. The proposed improvements include a new search strategy, a simplified branching vertex selection scheme, a method to avoid unnecessary computation, a method to improve the quality of feasible solutions, and a method to reduce the space requirement of the algorithm. Numerical experiments show that the improved B &B algorithm proves optimality faster on some instances, finds good feasible solutions faster than the best known existing algorithm on instances that cannot be solved to optimality, and uses less space during the solving process.

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Data availibility

The datasets analyzed during the current study are available in the Github repository https://github.com/UranusR/CETSP_Data. instances.

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Funding

This research has been supported in part by the Air Force Office of Scientific Research (FA9550-19-1-0106). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Government, or the Air Force Office of Scientific Research.

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Illinois at Urbana-Champaign, Champaign, USA

    Wenda Zhang

  2. Department of Computer Science, University of Wisconsin-La Crosse, La Crosse, USA

    Jason J. Sauppe

  3. Department of Computer Science, University of Illinois at Urbana-Champaign, Champaign, USA

    Sheldon H. Jacobson

Authors
  1. Wenda Zhang
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  2. Jason J. Sauppe
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  3. Sheldon H. Jacobson
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Correspondence to Wenda Zhang.

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Appendices

Appendix A Implementation details

In addition to the algorithmic improvements proposed in Sects. 3 and 4, we have also introduced implementation improvements that have substantial impact on the performance. First, the process of checking whether a vertex is covered by the optimal tour of a partial sequence is improved to reduce redundant computation. Second, because the SOCP problems for the child nodes of the same parent have a large number of constraints in common, we reduced the number of constraints that have to be created for each SOCP problem by reusing as many constraints from the previous SOCP problem as possible. Finally, other code changes, including using more suitable data structures for node storage and retrieval, and reducing communication overhead between functions, also contribute to an improvement in performance.

To demonstrate the impact of the implementation changes above, let Cout-Orig represent the original implementation of Cout and let Cout represent our improved implementation. Table 4 shows the results of the implementation improvements on several of the TSPLIB Instances with overlap ratio 0.1 that can be solved to optimality by both Cout-Orig and Cout. As can be seen, the implementation improvements substantially reduced the overall running time (column “Running Time”) as well as the total SOCP solve time (column “SOCP Time”).

Table 4 Implementation improvement comparison
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Appendix B Full computational results

Table 5 presents the results of all algorithms on 62 2D instances. The column “Instance” contains the name of the instances, and the column “Known” contains the best known feasible solution for each instance in the literature. The column “UB” contains the best upper bound (incumbent solution) found during the search. Given that the best known solution is used as an upper bound for these instances, if the best feasible solution found is not strictly better than the best known solution, the corresponding “UB” is marked by -. If the number in column “UB” is marked by *, it means that the corresponding instance is solved to optimality; when the instance is not solved to optimality, the number in column “UB” is bold when that solution is the best feasible solution found compared with all algorithms as well as the best known solution. The column “LB” contains the best lower bound when the algorithm terminates.

The column “Time” reports the running times for each algorithm and the column “Unexp” reports the maximum number of unexplored nodes that each algorithm has to store. The instances that are not marked by * are not solved to optimality. They are either terminated due to reaching the time limit (if “Time” column entry is not less than 14400) or reaching the space limit.

Table 6 presents the results of all algorithms on 42 3D instances. The column notations are the same as the ones in Table 5.

Table 7 presents the results of all algorithms on the 56 TSPLIB instances with no known feasible solution. The column notations are the same as the ones in Table 5 with the exception that there is no column for best known solutions.

Table 8 presents the results of all algorithms on 90 Behdani instances. The instances are separated into 3 groups based on the radius r used (0.25, 0.5 and 1), and the size of an instance is represented by the first number in an instance name while the second number is an index for a particular instance. For example, “CETSP-50-2” indicates that this instance has 50 vertices with covering regions, and it is the second one out of 10 instances of the same size.

Table 5 Results on 62 2D instances with known feasible solution
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Table 6 Results on 42 3D instances with known feasible solution
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Table 7 Results on 56 instances with constant radii with no known feasible solution
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Table 8 Results on 90 Behdani instances with no known feasible solution
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Zhang, W., Sauppe, J.J. & Jacobson, S.H. Results for the close-enough traveling salesman problem with a branch-and-bound algorithm. Comput Optim Appl 85, 369–407 (2023). https://doi.org/10.1007/s10589-023-00474-3

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