# Searching a fixed graph

- First Online:

DOI: 10.1007/3-540-61440-0_135

- Cite this paper as:
- 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

## 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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