Online Graph Exploration: New Results on Old and New Algorithms
We study the problem of exploring an unknown undirected connected graph. Beginning in some start vertex, a searcher must visit each node of the graph by traversing edges. Upon visiting a vertex for the first time, the searcher learns all incident edges and their respective traversal costs. The goal is to find a tour of minimum total cost. Kalyanasundaram and Pruhs  proposed a sophisticated generalization of a Depth First Search that is 16-competitive on planar graphs. While the algorithm is feasible on arbitrary graphs, the question whether it has constant competitive ratio in general has remained open. Our main result is an involved lower bound construction that answers this question negatively. On the positive side, we prove that the algorithm has constant competitive ratio on any class of graphs with bounded genus. Furthermore, we provide a constant competitive algorithm for general graphs with a bounded number of distinct weights.
KeywordsPlanar Graph Minimum Span Tree Travel Salesman Problem Competitive Ratio Online Algorithm
Unable to display preview. Download preview PDF.
- 6.Bender, M.A., Slonim, D.K.: The power of team exploration: Two robots can learn unlabeled directed graphs. In: Proceedings of FOCS, pp. 75–85 (1994)Google Scholar
- 8.Betke, M., Rivest, R.L., Singh, M.: Piecemeal learning of an unknown environment. Machine Learning 18, 231–254 (1995)Google Scholar
- 10.Deng, X., Papadimitriou, C.H.: Exploring an unknown graph (extended abstract). In: Proceedings of FOCS, pp. 355–361 (1990)Google Scholar
- 28.Rao, N., Kareti, S., Shi, W., Iyengar, S.: Robot navigation in unknown terrains: Introductory survey of nonheuristic algorithms. Report ORNL/TM-12410, Oak Ridge Nat. Lab (1993)Google Scholar