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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)].

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