Searching a fixed graph

  • Elias Koutsoupias
  • Christos Papadimitriou
  • Mihalis Yannakakis
Session 6: Graph Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1099)

Abstract

We study three combinatorial optimization problems related to searching a graph that is known in advance, for an item that resides at an unknown node. The search ratio of a graph is the optimum competitive ratio (the worst-case ratio of the distance traveled before the unknown node is visited, over the distance between the node and a fixed root, minimized over all Hamiltonian walks of the graph). We also define the randomized search ratio (we minimize over all distributions of permutations). Finally, the traveling repairman problem seeks to minimize the expected time of visit to the unknown node, given some distribution on the nodes. All three of these novel graph-theoretic parameters are NP-complete —and MAXSNP-hard — to compute exactly; we present interesting approximation algorithms for each. We also show that the randomized search ratio and the traveling repairman problem are related via duality and polyhedral separation.

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References

  1. [ACP+86]
    F. Afrati, S. Cosmadakis, C. H. Papadimitriou, G. Papageorgiou, and N. Papakostantinou. The complexity of the travelling repairman problem. Informatique Theórique et Applications, 20(1):79–87, 1986.Google Scholar
  2. [BCR88]
    R.A. Baeza-Yates, J.C. Culberson, and G.J.E. Rawlins. Searching with uncertainty. SWAT 88. 1st Scandinavian Workshop on Algorithm Theory. Proceedings, pages 176–89, 1988.Google Scholar
  3. [BCC+94]
    A. Blum, P. Chalasani, D. Coppersmith, W. Pulleyblank, P. Raghavan, and M. Sudan. The minimum latency problem. Proceedings 26th Annual Symposium on Theory of Computing, pages 163–171, 1994.Google Scholar
  4. [Chr76]
    N. Christofides. Worst-case analysis of a new heuristic for the traveling salesman problem. Technical report, GSIA, Carnegie-Mellon University, 1976.Google Scholar
  5. [DKP91]
    X. Deng, T. Kameda, and C. Papadimitriou. How to learn an unknown environment. Proceedings 32nd Annual Symposium on Foundations of Computer Science, pages 298–303, 1991.Google Scholar
  6. [DP90]
    X. Deng and C. H. Papadimitriou. Exploring an unknown graph. Proceedings 31st Annual Symposium on Foundations of Computer Science, pages 355–361 vol. 1, 1990.Google Scholar
  7. [FLM93]
    M. Fischetti, G. Laporte, and M. Martello. The delivery man problem and cumulative matroids. Operations Research, vol. 41, pages 1055–1064, 1993.MathSciNetGoogle Scholar
  8. [GK96]
    M. Goemans and J. Kleinberg. An improved approximation ratio for the minimum latency problem. Proceedings Annual Symposium on Discrete Algorithms, to appear, 1996.Google Scholar
  9. [GLS88]
    M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimization. Springer-Verlag, 1988.Google Scholar
  10. [Min89]
    E. Minieka. The delivery man problem on a tree network. Annals of Operations Research, vol. 18, pages 261–266, 1989.Google Scholar
  11. [PST91]
    S.A. Plotkin, D.B. Shmoys, and É. Tardos. Fast approximation algorithms for fractional packing and covering problems. Proceedings 32nd Annual Symposium on Foundations of Computer Science, pages 495–504, 1991.Google Scholar
  12. [PY91]
    C. H. Papadimitriou and M. Yannakakis. Shortest paths without a map. Theoretical Computer Science, 84(1):127–50, July 1991.CrossRefGoogle Scholar
  13. [PY93]
    C. H. Papadimitriou and M. Yannakakis. The traveling salesman problem with distances one and two. Mathematics of Operations Research, 18(1):1–11, February 1993.Google Scholar
  14. [ST85]
    D. D. Sleator and R. E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28(2):202–8, February 1985.CrossRefGoogle Scholar
  15. [Tar95]
    É. Tardos. Private communication, 1995.Google Scholar
  16. [Wes95]
    D. West. Private communication, 1995.Google Scholar
  17. [Wil93]
    T. G. Will. Extremal Results and Algorithms for Degree Sequences of Graphs. PhD thesis, U. of Illinois at Urbana-Champaign, 1993.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Elias Koutsoupias
    • 1
  • Christos Papadimitriou
    • 2
  • Mihalis Yannakakis
    • 3
  1. 1.CS DepartmentUCLAUSA
  2. 2.EECS DepartmentUC BerkeleyUSA
  3. 3.Bell LaboratoriesMurray Hill

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