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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

In the Fastest Mixing Markov Chain problem, we are given a graph G = (V, E) and desire the discrete-time Markov chain with smallest mixing time τ subject to having equilibrium distribution uniform on V and non-zero transition probabilities only across edges of the graph [Boyd et al., 2004].
It is well-known that the mixing time τ_RW of the lazy random walk on G is characterised by the edge conductance Φ of G via Cheeger’s inequality: Φ^{-1} ≲ τ_RW ≲ Φ^{-2} log|V|. Edge conductance, however, fails to characterise the fastest mixing time τ^⋆ of G. Take, for example, a graph consisting of two n-vertex cliques connected by a perfect matching: its edge conductance is Θ(1/n), while τ^⋆ can be shown to be O(log n). We show, instead, that is possible to characterise the fastest mixing time τ^⋆ via a Cheeger-type inequality but for a different geometric quantity, namely the vertex conductance Ψ of G: Ψ^{-1} ≲ τ^* ≲ Ψ^{-2}(log|V|)^2. We prove this result by first relating vertex conductance to a new expansion measure, which we call matching conductance. We then relate matching conductance to a variational characterisation of τ^⋆ (or, more precisely, of the fastest relaxation time) due to Roch [Roch, 2005]. This is done by interpreting Roch’s characterisation as a particular instance of fractional vertex cover, which is dual to fractional matching. We believe matching conductance to be of independent interest and might have further applications in studying connectivity properties of graphs.
This characterisation forbids fast mixing for graphs with small vertex conductance. To bypass this fundamental barrier, we consider Markov chains on G with equilibrium distribution which need not be uniform, but rather only ε-close to uniform in total variation. We call such chains almost mixing. We show that it is always possible to construct an almost mixing chain with mixing time τ ≲ ε^{-1} (diam G)² log |V|. Our proof is based on carefully constructing a reweighted spanning tree of G with good expansion properties and superimposing it over a simple "base" chain.
In summary, our work together with known results shows that three fundamental geometric quantities characterise the mixing time on a graph according to three different notions of mixing: edge conductance characterises the mixing time of the lazy random walk, vertex conductance the fastest mixing time, while the diameter characterises the almost mixing time.
Finally, we also discuss analogous questions for continuous-time and time-inhomogeneous chains.

Sam Olesker-Taylor and Luca Zanetti. Geometric Bounds on the Fastest Mixing Markov Chain. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, p. 109:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{oleskertaylor_et_al:LIPIcs.ITCS.2022.109, author = {Olesker-Taylor, Sam and Zanetti, Luca}, title = {{Geometric Bounds on the Fastest Mixing Markov Chain}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {109:1--109:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.109}, URN = {urn:nbn:de:0030-drops-157051}, doi = {10.4230/LIPIcs.ITCS.2022.109}, annote = {Keywords: mixing time, random walks, conductance, fastest mixing Markov chain} }

Document

**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We study spectral approaches for the MAX-2-LIN(k) problem, in which we are given a system of m linear equations of the form x_i - x_j is equivalent to c_{ij} mod k, and required to find an assignment to the n variables {x_i} that maximises the total number of satisfied equations.
We consider Hermitian Laplacians related to this problem, and prove a Cheeger inequality that relates the smallest eigenvalue of a Hermitian Laplacian to the maximum number of satisfied equations of a MAX-2-LIN(k) instance I. We develop an O~(kn^2) time algorithm that, for any (1-epsilon)-satisfiable instance, produces an assignment satisfying a (1 - O(k)sqrt{epsilon})-fraction of equations. We also present a subquadratic-time algorithm that, when the graph associated with I is an expander, produces an assignment satisfying a (1- O(k^2)epsilon)-fraction of the equations. Our Cheeger inequality and first algorithm can be seen as generalisations of the Cheeger inequality and algorithm for MAX-CUT developed by Trevisan.

Huan Li, He Sun, and Luca Zanetti. Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{li_et_al:LIPIcs.ESA.2019.71, author = {Li, Huan and Sun, He and Zanetti, Luca}, title = {{Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {71:1--71:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.71}, URN = {urn:nbn:de:0030-drops-111926}, doi = {10.4230/LIPIcs.ESA.2019.71}, annote = {Keywords: Spectral methods, Hermitian Laplacians, the Max-2-Lin problem, Unique Games} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion properties which allow us to capture the progress the random walk makes through t-step probabilities.
We apply our framework to dynamically changing graphs, where the set of vertices is fixed while the set of edges changes in each round. For random walks on dynamic connected graphs for which the stationary distribution does not change over time, we show that their behaviour is in a certain sense similar to static graphs. For example, we show that the mixing and hitting times of any sequence of d-regular connected graphs is O(n^2), generalising a well-known result for static graphs. We also provide refined bounds depending on the isoperimetric dimension of the graph, matching again known results for static graphs. Finally, we investigate properties of random walks on dynamic graphs that are not always connected: we relate their convergence to stationarity to the spectral properties of an average of transition matrices and provide some examples that demonstrate strong discrepancies between static and dynamic graphs.

Thomas Sauerwald and Luca Zanetti. Random Walks on Dynamic Graphs: Mixing Times, Hitting Times, and Return Probabilities. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 93:1-93:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sauerwald_et_al:LIPIcs.ICALP.2019.93, author = {Sauerwald, Thomas and Zanetti, Luca}, title = {{Random Walks on Dynamic Graphs: Mixing Times, Hitting Times, and Return Probabilities}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {93:1--93:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.93}, URN = {urn:nbn:de:0030-drops-106696}, doi = {10.4230/LIPIcs.ICALP.2019.93}, annote = {Keywords: random walks, dynamic graphs, hitting times} }

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