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Documents authored by Kleer, Pieter

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DOI: 10.4230/LIPIcs.STACS.2023.7
Approximate Sampling and Counting of Graphs with Near-Regular Degree Intervals

Authors: Georgios Amanatidis and Pieter Kleer

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
The approximate uniform sampling of graphs with a given degree sequence is a well-known, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. In this work we study a generalization of the problem, where degree intervals are specified instead of a single degree sequence. We are interested in sampling and counting graphs whose degree sequences satisfy the corresponding degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on networks that are only partially observed. We provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with near-regular degree intervals, i.e., graphs in which every node has a degree from an interval not too far away from a given r ∈ ℕ. In order to design our FPAUS, we rely on various state-of-the-art tools from Markov chain theory and combinatorics. In particular, by carefully using Markov chain decomposition and comparison arguments, we reduce part of our problem to the recent breakthrough of Anari, Liu, Oveis Gharan, and Vinzant (2019) on sampling a base of a matroid under a strongly log-concave probability distribution, and we provide the first non-trivial algorithmic application of a breakthrough asymptotic enumeration formula of Liebenau and Wormald (2017). As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and Müller-Hannemann (2018), based on three local operations - switches, hinge flips, and additions/deletions of an edge. We obtain the first theoretical results for this Markov chain, showing it is rapidly mixing for the case of near-regular degree intervals of size at most one.
Cite as

Georgios Amanatidis and Pieter Kleer. Approximate Sampling and Counting of Graphs with Near-Regular Degree Intervals. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{amanatidis_et_al:LIPIcs.STACS.2023.7,
  author =	{Amanatidis, Georgios and Kleer, Pieter},
  title =	{{Approximate Sampling and Counting of Graphs with Near-Regular Degree Intervals}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.7},
  URN =		{urn:nbn:de:0030-drops-176596},
  doi =		{10.4230/LIPIcs.STACS.2023.7},
  annote =	{Keywords: graph sampling, degree interval, degree sequence, Markov Chain Monte Carlo method, switch Markov chain}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.36
Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence

Authors: Corrie Jacobien Carstens and Pieter Kleer

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
We consider the well-studied problem of uniformly sampling (bipartite) graphs with a given degree sequence, or equivalently, the uniform sampling of binary matrices with fixed row and column sums. In particular, we focus on Markov Chain Monte Carlo (MCMC) approaches, which proceed by making small changes that preserve the degree sequence to a given graph. Such Markov chains converge to the uniform distribution, but the challenge is to show that they do so quickly, i.e., that they are rapidly mixing. The standard example of this Markov chain approach for sampling bipartite graphs is the switch algorithm, that proceeds by locally switching two edges while preserving the degree sequence. The Curveball algorithm is a variation on this approach in which essentially multiple switches (trades) are performed simultaneously, with the goal of speeding up switch-based algorithms. Even though the Curveball algorithm is expected to mix faster than switch-based algorithms for many degree sequences, nothing is currently known about its mixing time. On the other hand, the switch algorithm has been proven to be rapidly mixing for several classes of degree sequences. In this work we present the first results regarding the mixing time of the Curveball algorithm. We give a theoretical comparison between the switch and Curveball algorithms in terms of their underlying Markov chains. As our main result, we show that the Curveball chain is rapidly mixing whenever a switch-based chain is rapidly mixing. We do this using a novel state space graph decomposition of the switch chain into Johnson graphs. This decomposition is of independent interest.
Cite as

Corrie Jacobien Carstens and Pieter Kleer. Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carstens_et_al:LIPIcs.APPROX-RANDOM.2018.36,
  author =	{Carstens, Corrie Jacobien and Kleer, Pieter},
  title =	{{Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.36},
  URN =		{urn:nbn:de:0030-drops-94403},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.36},
  annote =	{Keywords: Binary matrix, graph sampling, Curveball, switch, Markov chain decomposition, Johnson graph}
}
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