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Cover times, blanket times, and majorizing measures. (English) Zbl 1250.05098
Summary: We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand’s theory of majorizing measures. In particular, we show that the cover time of any graph $$G$$ is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on $$G$$, scaled by the number of edges in $$G$$.
This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an $$O(1)$$ factor for any graph, answering a question of D. J. Aldous and J. Fill [“Reversible Markov chains and radom walks on graphs” (to appear)]. We also positively resolve the blanket time conjectures of P. Winkler and D. Zuckerman [Random Struct. Algorithms 9, No. 4, 403–411 (1996; Zbl 0872.60054)], showing that for any graph, the blanket and cover times are within an $$O(1)$$ factor. The best previous approximation factor for both these problems was $$O((\log \log n)^2)$$ for $$n$$-vertex graphs, due to J. D. Kahn, J. H. Kim, L. Lovász, and V. H. Vu [“The cover time, the blanket time, and the Matthews bound”, 41st Annual Symposium on Foundations of Computer Science, Redondo Beach: IEEE Comput. Soc. Press (2000)].

##### MSC:
 05C81 Random walks on graphs 05C85 Graph algorithms (graph-theoretic aspects) 60J55 Local time and additive functionals 60G15 Gaussian processes
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##### References:
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