Cooperative TSP

  • Amitai Armon
  • Adi Avidor
  • Oded Schwartz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)


In this paper we introduce and study cooperative variants of the Traveling Salesperson Problem. In these problems a salesperson has to make deliveries to customers who are willing to help in the process. Customer cooperativeness may be manifested in several modes: they may assist by approaching the salesperson, by reselling the goods they purchased to other customers, or by doing both.

Several objectives are of interest: minimizing the total distance traveled by all the participants, minimizing the maximal distance traveled by a participant and minimizing the total time until all the deliveries are made.

All the combinations of cooperation-modes and objective functions are considered, both in weighted undirected graphs and in Euclidean space. We show that most of the problems have a constant approximation algorithm, many of the others admit a PTAS, and a few are solvable in polynomial time. On the intractability side we provide NP-hardness proofs and inapproximability factors, some of which are tight.


Pixel Center Goal Function Travel Salesperson Problem Weighted Undirected Graph Cooperation Mode 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ABF+02]
    Arkin, E.M., Bender, M.A., Fekete, S.P., Mitchell, J.S.B., Skutella, M.: The Freeze-Tag Problem: How to Wake Up a Swarm of Robots. In: Proc. of SODA 2002, pp. 568–577 (2002)Google Scholar
  2. [ABG+03]
    Arkin, E.M., Bender, M.A., Ge, D., He, S., Mitchell, J.S.B.: Improved Approximation Algorithms or the Freeze-Tag Problem. In: SPAA 2003, pp. 295–303 (2003)Google Scholar
  3. [AC04]
    Arora, S., Chang, K.L.: Approximation Schemes for Degree-Restricted MST and Red-Blue Separation Problems. Algorithmica 40(3), 189–210 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. [AH94]
    Arkin, E., Hassin, R.: Approximation Algorithms for the Geometric Covering Salesman Problem. Discrete Applied Math. 55, 197–218 (1994)MATHCrossRefMathSciNetGoogle Scholar
  5. [Aro98]
    Arora, S.: Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems. Journal of the ACM 45(5), 753–782 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. [BNGNS98]
    Bar-Noy, A., Guha, S., Naor, J., Schieber, B.: Multicasting in Heterogeneous Networks. In: Proc. of STOC 1998, pp. 448–453 (1998)Google Scholar
  7. [Cha03]
    Chan, T.M.: Euclidean Bounded-Degree Spanning Tree Ratios. In: Proc. 19th ACM SoCG, pp. 11–19 (2003)Google Scholar
  8. [Chr76]
    Christofides, N.: Worst-Case Analysis of a New Heuristic for the Traveling Salesman Problem. Technical report, Graduate School of Industrial Administration, Carnegy–Mellon University (1976)Google Scholar
  9. [dBGK+05]
    de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with Neighborhoods of Varying Size. Journal of Algorithms 57, 22–36 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. [DM01]
    Dumitrescu, A., Mitchell, J.S.B.: Approximation Algorithms for TSP with Neighborhoods in the Plane. In: Proc. of SODA 2001, pp. 38–46 (2001)Google Scholar
  11. [EK01]
    Engebretsen, L., Karpinski, M.: Approximation Hardness of TSP with Bounded Metrics. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 201–212. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. [FKK+97]
    Fekete, S.P., Khuller, S., Klemmstein, M., Raghavachari, B., Young, N.: A Network Flow Technique for Finding Low-Weight Bounded-Degree Trees. Journal of Algorithms 24, 310–324 (1997)MATHCrossRefMathSciNetGoogle Scholar
  13. [HHL88]
    Hedetniemi, S.M., Hedetniemi, S.T., Liestman, A.L.: A Survey of Gossiping and Broadcasting in Communication Networks. Networks 18(4), 319–359 (1988)MATHCrossRefMathSciNetGoogle Scholar
  14. [KLS04]
    Könemann, J., Levin, A., Sinha, A.: Approximating the Degree-Bounded Minimum Diameter Spanning Tree Problem. Algorithmica 41(2), 117–129 (2004)CrossRefGoogle Scholar
  15. [KRY96]
    Khuller, S., Raghavachari, B., Young, N.: Low Degree Spanning Trees of Small Weight. SIAM Journal of Computing 25(2), 355–368 (1996)MATHCrossRefMathSciNetGoogle Scholar
  16. [Mit99]
    Mitchell, J.S.B.: Guillotine Subdivisions Approximate Polygonal Subdivisions: Part II – A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal of Computing 28(4), 1298–1309 (1999)MATHCrossRefGoogle Scholar
  17. [Mit06]
    Mitchell, J.S.B.: A PTAS for TSP with Neighborhoods Among Fat Regions in the Plane. Private communication (2006)Google Scholar
  18. [MM95]
    Mata, C., Mitchell, J.S.B.: Approximation Algorithms for Geometric Tour and Network Design Problems. In: SCG 1995, pp. 360–369 (1995)Google Scholar
  19. [PV84]
    Papadimitriou, C.H., Vazirani, U.V.: On Two Geometric Problems Related to the Traveling Salesman Problem. J. of Alg. 5, 231–246 (1984)MATHCrossRefMathSciNetGoogle Scholar
  20. [Rav94]
    Ravi, R.: Rapid Rumor Ramification: Approximating the Minimum Broadcast Time. In: Proc. of FOCS 1994, pp. 202–213 (1994)Google Scholar
  21. [SABM02]
    Sztainberg, M.O., Arkin, E.M., Bender, M.A., Mitchell, J.S.B.: Analysis of Heuristics for the Freeze-Tag Problem. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 270–279. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. [SS05]
    Safra, S., Schwartz, O.: On the Complexity of Approximating TSP with Neighborhoods and Related Problems. Computational Complexity 14, 281–307 (2005)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Amitai Armon
    • 1
  • Adi Avidor
    • 1
  • Oded Schwartz
    • 1
  1. 1.School of Computer ScienceTel-Aviv UniversityIsrael

Personalised recommendations