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A (3/2 + ε)-Approximation for Multiple TSP with a Variable Number of Depots

Authors Max Deppert , Matthias Kaul , Matthias Mnich

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Author Details

Max Deppert
  • Institute for Algorithms and Complexity, Hamburg University of Technology, Germany
Matthias Kaul
  • Institute for Algorithms and Complexity, Hamburg University of Technology, Germany
Matthias Mnich
  • Institute for Algorithms and Complexity, Hamburg University of Technology, Germany


The third author thanks L{á}szló Végh for inspiring discussions on multi-depot TSP and feedback on an earlier version.

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Max Deppert, Matthias Kaul, and Matthias Mnich. A (3/2 + ε)-Approximation for Multiple TSP with a Variable Number of Depots. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 39:1-39:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


One of the most studied extensions of the famous Traveling Salesperson Problem (TSP) is the Multiple TSP: a set of m ≥ 1 salespersons collectively traverses a set of n cities by m non-trivial tours, to minimize the total length of their tours. This problem can also be considered to be a variant of Uncapacitated Vehicle Routing, where the objective is to minimize the sum of all tour lengths. When all m tours start from and end at a single common depot v₀, then the metric Multiple TSP can be approximated equally well as the standard metric TSP, as shown by Frieze (1983). The metric Multiple TSP becomes significantly harder to approximate when there is a set D of d ≥ 1 depots that form the starting and end points of the m tours. For this case, only a (2-1/d)-approximation in polynomial time is known, as well as a 3/2-approximation for constant d which requires a prohibitive run time of n^Θ(d) (Xu and Rodrigues, INFORMS J. Comput., 2015). A recent work of Traub, Vygen and Zenklusen (STOC 2020) gives another approximation algorithm for metric Multiple TSP with run time n^Θ(d), which reduces the problem to approximating metric TSP. In this paper we overcome the n^Θ(d) time barrier: we give the first efficient approximation algorithm for Multiple TSP with a variable number d of depots that yields a better-than-2 approximation. Our algorithm runs in time (1/ε)^O(dlog d) ⋅ n^O(1), and produces a (3/2+ε)-approximation with constant probability. For the graphic case, we obtain a deterministic 3/2-approximation in time 2^d ⋅ n^O(1).

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Traveling salesperson problem
  • rural postperson problem
  • multiple TSP
  • vehicle routing


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