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An Algorithm for the Euclidean Bounded Multiple Traveling Salesman Problem

Víctor Pacheco-Valencia, Nodari Vakhania

TL;DR

This work addresses the Euclidean Bounded MTSP by introducing the Partition-Construction-Improvement (PCI) three-phase heuristic. PCI partitions the city set into $k$ feasible subsets, constructs initial tours via a fast Euclidean TSP solver, and then improves the solution through sub-tour relocation, vertex relocation, and swaps, with a worst-case complexity of $O(k^2 n^4)$. Empirically, PCI delivers substantial gains on 22 benchmark instances (19 BKS improvements, including the 12 largest) and outperforms a general ILP solver on 168 new instances in most cases, while remaining orders of magnitude faster. The results demonstrate practical effectiveness for large-scale BMTSP with realistic fleet sizes, and the authors publicly share the instances and code to facilitate further research and benchmarking.

Abstract

In the Bounded Multiple Traveling Salesman Problem (BMTSP), a tour for each salesman, that starts and ends at the depot and that respects the bounds on the number of cities that a feasible salesman tour should satisfy, is to be constructed. The objective is to minimize the total length of all tours. Already Euclidean traveling salesman problem is NP-hard. We propose a 3-Phase heuristic algorithm for the Euclidean BMTSP. We tested the algorithm for the 22 benchmark instances and 168 new problem instances that we created. We report 19 best known solutions for the 22 benchmark instances including the 12 largest ones. For the newly created instances, we compared the performance of our algorithm with that of an ILP-solver CPLEX, which was able to construct a feasible solution for 71% of the instances within the time limit of two hours imposed by us. For about 10% of the smallest new instances, CPLEX delivered slightly better solutions, where our algorithm took less than 180 seconds for the largest of these instances. For the remaining 61% of the instances solved by CPLEX, the solutions by our heuristic were, on average, about 21.5% better than those obtained by CPLEX.

An Algorithm for the Euclidean Bounded Multiple Traveling Salesman Problem

TL;DR

This work addresses the Euclidean Bounded MTSP by introducing the Partition-Construction-Improvement (PCI) three-phase heuristic. PCI partitions the city set into feasible subsets, constructs initial tours via a fast Euclidean TSP solver, and then improves the solution through sub-tour relocation, vertex relocation, and swaps, with a worst-case complexity of . Empirically, PCI delivers substantial gains on 22 benchmark instances (19 BKS improvements, including the 12 largest) and outperforms a general ILP solver on 168 new instances in most cases, while remaining orders of magnitude faster. The results demonstrate practical effectiveness for large-scale BMTSP with realistic fleet sizes, and the authors publicly share the instances and code to facilitate further research and benchmarking.

Abstract

In the Bounded Multiple Traveling Salesman Problem (BMTSP), a tour for each salesman, that starts and ends at the depot and that respects the bounds on the number of cities that a feasible salesman tour should satisfy, is to be constructed. The objective is to minimize the total length of all tours. Already Euclidean traveling salesman problem is NP-hard. We propose a 3-Phase heuristic algorithm for the Euclidean BMTSP. We tested the algorithm for the 22 benchmark instances and 168 new problem instances that we created. We report 19 best known solutions for the 22 benchmark instances including the 12 largest ones. For the newly created instances, we compared the performance of our algorithm with that of an ILP-solver CPLEX, which was able to construct a feasible solution for 71% of the instances within the time limit of two hours imposed by us. For about 10% of the smallest new instances, CPLEX delivered slightly better solutions, where our algorithm took less than 180 seconds for the largest of these instances. For the remaining 61% of the instances solved by CPLEX, the solutions by our heuristic were, on average, about 21.5% better than those obtained by CPLEX.
Paper Structure (11 sections, 6 theorems, 14 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 11 sections, 6 theorems, 14 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Phase 1: The partition
  4. Phase 2: Construction
  5. Phase 3: Improvement
  6. Procedure RELOCATE_SUBTOURS
  7. Procedure RELOCATE_A_VERTEX
  8. Procedure SWAP_VERTICES
  9. Procedure MAIN
  10. The implementation and results of our experiments
  11. Conclusions

Key Result

Lemma 1

The time complexity of Procedure pF1Partition is $O(kn^3)$.

Figures (5)

  • Figure 1: Example of removing of a sub-tour
  • Figure 2: Example of inserting a sub-tour $(i^j_p, \cdots, i^j_{p+m-1})$ with $m$ consecutive vertices into tour $T^l$
  • Figure 3: Costs for the 22 benchmarks instances from junjie2006necula2015
  • Figure 4: Quality of our solutions compared to their BKS for the 22 benchmark instances
  • Figure 5: $GAP(PCI,CPLEX)$ between CPLEX and PCI-algorithm for the 168 new instances

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6