Skip to main content
SpringerLink
Log in

The m-Steiner Traveling Salesman Problem with online edge blockages

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

We consider the online multiple Steiner Traveling Salesman Problem based on the background of the delivery of packages in an urban traffic network. In this problem, given an edge-weighted undirected graph \(G = (V, E)\), a subset \(D\subset V\) of customer vertices, and m salesmen. For each edge in E, the weight w(e) is associated with the traversal time or the cost of the edge. The aim is to find m closed tours that visit each vertex of D at least once. We formulate the traffic congestion with k non-recoverable blocked edges revealed to the salesmen in real-time, meaning that the salesmen know about a blocked edge whenever it occurs. For the version to minimize the maximum cost of m salesmen (minmax mSTSP), we prove a lower bound and propose the ForestTraversal algorithm. The corresponding competitive ratio is proved to be linear with k. For the version to minimize the total cost of m salesmen (minsum mSTSP), we also propose a lower bound and the Retrace algorithm, where the competitive ratio of the algorithm is proved to be linear with k.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Arkin EM, Hassin R, Levin A (2006) Approximations for minimum and min-max vehicle routing problems. J Algorithms 59(1):1–18

    Article  MathSciNet  Google Scholar 

  • Bektas T (2006) The multiple traveling salesman problem: an overview of formulations and solution procedures. Omega 34(3):209–219

    Article  Google Scholar 

  • Christofides N (1976) Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie Mellon University

  • Even G, Garg N, KöNemann J, Ravi R, Sinha A (2004) Min-max tree covers of graphs. Oper Res Lett 32(4):309–315

    Article  MathSciNet  Google Scholar 

  • Frederickson GN, Hecht MS, Kim CE (1978) Approximation algorithms for some routing problems. SAIM J Comput 7(2):178–193

    Article  MathSciNet  Google Scholar 

  • Garey M and Johnson D (1979) Computers And Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, San Francisco

  • Gavish B, Srikanth K (1986) An optimal solution method for large-scale multiple traveling salesmen problems. Oper Res 34(5):698–717

    Article  MathSciNet  Google Scholar 

  • Iqbal AA, Kennington JL (1986) The asymmetric m -travelling salesmen problem: A duality based branch-and-bound algorithm. Discrete Appl Math 13(2):259–276

    Article  MathSciNet  Google Scholar 

  • Khani MR, Salavatipour MR (2014) Improved approximation algorithms for the min-max tree cover and bounded tree cover problems. Algorithmica 69(2):443–460

    Article  MathSciNet  Google Scholar 

  • Kulkarni RV, Bhave PR (1985) Integer programming formulations of vehicle routing problems. Eur J Oper Res 20(1):58–67

    Article  MathSciNet  Google Scholar 

  • Laporte G, Nobert Y (1980) A cutting planes algorithm for the m -salesmen problem. J Oper Res Soc 31(11):1017–1023

    Article  MathSciNet  Google Scholar 

  • Liao C-S, Huang Y (2014) The covering canadian traveller problem. Theor Comput Sci 530:80–88

    Article  MathSciNet  Google Scholar 

  • Malik W, Rathinam S, Darbha S (2007) An approximation algorithm for a symmetric generalized multiple depot, multiple travelling salesman problem. Oper Res Lett 35(6):747–753

    Article  MathSciNet  Google Scholar 

  • Nagamochi H (2005) Approximating the minmax rooted-subtree cover problem. IEICE Trans Fundam Electron Commun Comput Sci 88(5):1335–1338

    Article  Google Scholar 

  • Nemhauser GL, Wolsey LA (1988) Integer and Combinatorial Optimization. Wiley-Interscience, New York, NY, USA

    Book  Google Scholar 

  • Papadimitriou CH, Yannakakis M (1991) Shortest paths without a map. Theor Comput Sci 84(1):127–150

    Article  MathSciNet  Google Scholar 

  • Papadimitriou CH, Yannakakis M (1993) The traveling salesperson problem with distances one and two. Math Oper Res 18(1):1–11

    Article  MathSciNet  Google Scholar 

  • Westphal S (2008) A note on the k-canadian traveller problem. Inf Process Lett 106(3):87–89

    Article  MathSciNet  Google Scholar 

  • Xu Y, Hu M, Su B, Zhu B, Zhu Z (2009) The canadian traveller problem and its competitive analysis. J Comb Optim 18(2):195–205

    Article  MathSciNet  Google Scholar 

  • Xu Z, Rodrigues B (2017) An extension of the christofides heuristic for the generalized multiple depot multiple traveling salesmen problem. Eur J Oper Res 257(3):735–745

    Article  MathSciNet  Google Scholar 

  • Xu Z, Wen Q (2010) Approximation hardness of minmax tree covers. Oper Res Lett 38(3):169–173

    Article  MathSciNet  Google Scholar 

  • Xu Z, Xu L, Rodrigues B (2011) An analysis of the extended christofides heuristic for the k-depot tsp. Oper Res Lett 39(3):218–223

    Article  MathSciNet  Google Scholar 

  • Yu W, Liu Z (2016) Improved approximation algorithms for some min-max and minimum cycle cover problems. Theor Comput Sci 654:45–58

    Article  MathSciNet  Google Scholar 

  • Zhang H, Tong W, Lin G, Xu Y (2019) Online minimum latency problem with edge uncertainty. Eur J Oper Res 273(2):418–429

    Article  MathSciNet  Google Scholar 

  • Zhang H, Tong W, Xu Y, Lin G (2015) The steiner traveling salesman problem with online edge blockages. Eur J Oper Res 243(1):30–40

    Article  MathSciNet  Google Scholar 

  • Zhang H, Tong W, Xu Y, Lin G (2016) The steiner traveling salesman problem with online advanced edge blockages. Comput Oper Res 70:26–38

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We first acknowledge the insightful comments and suggestions of the anonymous reviewers to improve this paper substantially. This work was partially supported by the NSFC (Grant No. 72071157, 71601152, 71732006); the China Postdoctoral Science Foundation (Grant No. 2016M592811); and the Natural Science Basic Research Program of Shaanxi (Grant No. 2020JQ-654).

Author information

Authors and Affiliations

  1. School of Management, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Henan Liu & Huili Zhang

  2. State Key Lab for Manufacturing Systems Engineering, Xi’an, 710049, People’s Republic of China

    Huili Zhang

  3. School of Economics and Management, Xi’an University of Technology, Xi’an, 710054, People’s Republic of China

    Yi Xu

Authors
  1. Henan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huili Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huili Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Zhang, H. & Xu, Y. The m-Steiner Traveling Salesman Problem with online edge blockages. J Comb Optim 41, 844–860 (2021). https://doi.org/10.1007/s10878-021-00720-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-021-00720-6

Keywords

Search

Navigation