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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We study a number of graph exploration problems in the following natural scenario: an algorithm starts exploring an undirected graph from some seed vertex; the algorithm, for an arbitrary vertex v that it is aware of, can ask an oracle to return the set of the neighbors of v. (In the case of social networks, a call to this oracle corresponds to downloading the profile page of user v.) The goal of the algorithm is to either learn something (e.g., average degree) about the graph, or to return some random function of the graph (e.g., a uniform-at-random vertex), while accessing/downloading as few vertices of the graph as possible.
Motivated by practical applications, we study the complexities of a variety of problems in terms of the graph's mixing time t_{mix} and average degree d_{avg} - two measures that are believed to be quite small in real-world social networks, and that have often been used in the applied literature to bound the performance of online exploration algorithms.
Our main result is that the algorithm has to access Omega (t_{mix} d_{avg} epsilon^{-2} ln delta^{-1}) vertices to obtain, with probability at least 1-delta, an epsilon additive approximation of the average of a bounded function on the vertices of a graph - this lower bound matches the performance of an algorithm that was proposed in the literature.
We also give tight bounds for the problem of returning a close-to-uniform-at-random vertex from the graph. Finally, we give lower bounds for the problems of estimating the average degree of the graph, and the number of vertices of the graph.

Flavio Chierichetti and Shahrzad Haddadan. On the Complexity of Sampling Vertices Uniformly from a Graph. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 149:1-149:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chierichetti_et_al:LIPIcs.ICALP.2018.149, author = {Chierichetti, Flavio and Haddadan, Shahrzad}, title = {{On the Complexity of Sampling Vertices Uniformly from a Graph}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {149:1--149:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.149}, URN = {urn:nbn:de:0030-drops-91538}, doi = {10.4230/LIPIcs.ICALP.2018.149}, annote = {Keywords: Social Networks, Sampling, Graph Exploration, Lower Bounds} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Markov chains defined on the set of permutations of n elements have been studied widely by mathematicians and theoretical computer scientists. We consider chains in which a position i<n is chosen uniformly at random, and then sigma(i) and sigma(i+1) are swapped with probability depending on sigma(i) and sigma(i+1). Our objective is to identify some conditions that assure rapid mixing.
One case of particular interest is what we call the "gladiator chain," in which each number g is assigned a "strength" s_g and when g and g' are swapped, g comes out on top with probability s_g / ( s_g + s_g' ). The stationary probability of this chain is the same as that of the slow-mixing "move ahead one" chain for self-organizing lists, but an open conjecture of Jim Fill's implies that all gladiator chains mix rapidly. Here we obtain some positive partial results by considering cases where the gladiators fall into only a few strength classes.

Shahrzad Haddadan and Peter Winkler. Mixing of Permutations by Biased Transposition. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{haddadan_et_al:LIPIcs.STACS.2017.41, author = {Haddadan, Shahrzad and Winkler, Peter}, title = {{Mixing of Permutations by Biased Transposition}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {41:1--41:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.41}, URN = {urn:nbn:de:0030-drops-69928}, doi = {10.4230/LIPIcs.STACS.2017.41}, annote = {Keywords: Markov chains, permutations, self organizing lists, mixing time} }