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.
Research supported in part by a NSF grant
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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.
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.
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.
N. Christofides. Worst-case analysis of a new heuristic for the traveling salesman problem. Technical report, GSIA, Carnegie-Mellon University, 1976.
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.
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.
M. Fischetti, G. Laporte, and M. Martello. The delivery man problem and cumulative matroids. Operations Research, vol. 41, pages 1055–1064, 1993.
M. Goemans and J. Kleinberg. An improved approximation ratio for the minimum latency problem. Proceedings Annual Symposium on Discrete Algorithms, to appear, 1996.
M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimization. Springer-Verlag, 1988.
E. Minieka. The delivery man problem on a tree network. Annals of Operations Research, vol. 18, pages 261–266, 1989.
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.
C. H. Papadimitriou and M. Yannakakis. Shortest paths without a map. Theoretical Computer Science, 84(1):127–50, July 1991.
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.
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.
É. Tardos. Private communication, 1995.
D. West. Private communication, 1995.
T. G. Will. Extremal Results and Algorithms for Degree Sequences of Graphs. PhD thesis, U. of Illinois at Urbana-Champaign, 1993.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Koutsoupias, E., Papadimitriou, C., Yannakakis, M. (1996). Searching a fixed graph. In: Meyer, F., Monien, B. (eds) Automata, Languages and Programming. ICALP 1996. Lecture Notes in Computer Science, vol 1099. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61440-0_135
Download citation
DOI: https://doi.org/10.1007/3-540-61440-0_135
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61440-1
Online ISBN: 978-3-540-68580-7
eBook Packages: Springer Book Archive