6 Search Results for "Perarnau, Guillem"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.68
Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree

Authors: Xiaoyu Chen and Weiming Feng

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both 2-spin and multi-spin systems. As applications for this approach: - We prove the optimal O(n) relaxation time for the Glauber dynamics of random q-list-coloring on an n-vertices triangle-tree graph with maximum degree Δ such that q/Δ > α^⋆, where α^⋆ ≈ 1.763 is the unique positive solution of the equation α = exp(1/α). This improves the n^{1+o(1)} relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition. - We prove the optimal O(n) relaxation time and near-optimal Õ(n) mixing time for the Glauber dynamics on hardcore models with parameter λ in balanced bipartite graphs such that λ < λ_c(Δ_L) for the max degree Δ_L in left part and the max degree Δ_R of right part satisfies Δ_R = O(Δ_L). This improves the previous result by Chen, Liu, and Yin (2023). At the heart of our proof is the notion of coupling independence which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics.
Cite as

Xiaoyu Chen and Weiming Feng. Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 68:1-68:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.68,
  author =	{Chen, Xiaoyu and Feng, Weiming},
  title =	{{Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{68:1--68:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.68},
  URN =		{urn:nbn:de:0030-drops-244345},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.68},
  annote =	{Keywords: coupling independence, Glauber dynamics, mixing times, relaxation times, spin systems}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.54
Decay of Correlation for Edge Colorings When q > 3Δ

Authors: Zejia Chen, Yulin Wang, Chihao Zhang, and Zihan Zhang

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We examine various perspectives on the decay of correlation for the uniform distribution over proper q-edge colorings of graphs with maximum degree Δ. First, we establish the coupling independence property when q ≥ 3Δ for general graphs. Together with the recent work of Chen, Feng, Guo, Zhang and Zou (2024), this result implies a fully polynomial-time approximation scheme (FPTAS) for counting the number of proper q-edge colorings. Next, we prove the strong spatial mixing property on trees, provided that q > (3+o(1))Δ. The strong spatial mixing property is derived from the spectral independence property of a version of the weighted edge coloring distribution, which is established using the matrix trickle-down method developed in Abdolazimi, Liu and Oveis Gharan (FOCS, 2021) and Wang, Zhang and Zhang (STOC, 2024). Finally, we show that the weak spatial mixing property holds on trees with maximum degree Δ if and only if q ≥ 2Δ-1.
Cite as

Zejia Chen, Yulin Wang, Chihao Zhang, and Zihan Zhang. Decay of Correlation for Edge Colorings When q > 3Δ. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 54:1-54:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2025.54,
  author =	{Chen, Zejia and Wang, Yulin and Zhang, Chihao and Zhang, Zihan},
  title =	{{Decay of Correlation for Edge Colorings When q \rangle 3\Delta}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{54:1--54:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.54},
  URN =		{urn:nbn:de:0030-drops-234314},
  doi =		{10.4230/LIPIcs.ICALP.2025.54},
  annote =	{Keywords: Strong Spatial Mixing, Edge Coloring, Approximate Counting}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.62
Faster Algorithms on Linear Delta-Matroids

Authors: Tomohiro Koana and Magnus Wahlström

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We present new algorithms and constructions for linear delta-matroids. Delta-matroids are generalizations of matroids that also capture structures such as matchable vertex sets in graphs and path-packing problems. As with matroids, an important class of delta-matroids is given by linear delta-matroids, which generalize linear matroids and are represented via a "twist" of a skew-symmetric matrix. We observe an alternative representation, termed a contraction representation over a skew-symmetric matrix. This representation is equivalent to the more standard twist representation up to O(n^ω)-time transformations (where n is the dimension of the delta-matroid and ω < 2.372 the matrix multiplication exponent), but it is much more convenient for algorithmic tasks. For instance, the problem of finding a max-weight feasible set now reduces directly to finding a max-weight basis in a linear matroid. Supported by this representation, we provide new algorithms and constructions for linear delta-matroids. In particular, we show that the union and delta-sum of linear delta-matroids are again linear delta-matroids, and that a representation for the resulting delta-matroid can be constructed in randomized time O(n^ω) (or more precisely, in O(n^ω) field operations, over a field of size at least Ω(n⋅(1/ε)), where ε > 0 is an error parameter). Previously, it was only known that these operations define delta-matroids. We also note that every projected linear delta-matroid can be represented as an elementary projection. This implies that several optimization problems over (projected) linear delta-matroids, including the coverage, delta-coverage, and parity problems, reduce (in their decision versions) to a single O(n^ω)-time matrix rank computation. Using the methods of Harvey, previously applied by Cheung, Lao and Leung for linear matroid parity, we furthermore show how to solve the search versions in the same time. This improves on the O(n⁴)-time augmenting path algorithm of Geelen, Iwata and Murota, albeit with randomization. Finally, we consider the maximum-cardinality delta-matroid intersection problem (equivalently, the maximum-cardinality delta-matroid matching problem). Using Storjohann’s algorithms for symbolic determinants, we show that such a solution can be found in O(n^{ω+1}) time. This provides the first (randomized) polynomial-time solution for the problem, thereby solving an open question of Kakimura and Takamatsu.
Cite as

Tomohiro Koana and Magnus Wahlström. Faster Algorithms on Linear Delta-Matroids. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 62:1-62:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{koana_et_al:LIPIcs.STACS.2025.62,
  author =	{Koana, Tomohiro and Wahlstr\"{o}m, Magnus},
  title =	{{Faster Algorithms on Linear Delta-Matroids}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{62:1--62:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.62},
  URN =		{urn:nbn:de:0030-drops-228876},
  doi =		{10.4230/LIPIcs.STACS.2025.62},
  annote =	{Keywords: Delta-matroids, Randomized algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.24
Sampling List Packings

Authors: Evan Camrud, Ewan Davies, Alex Karduna, and Holden Lee

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We initiate the study of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted substantial attention. For list packing the setup is similar, but we seek a full decomposition of the lists of colors into pairwise-disjoint proper list colorings. The existence of a list packing implies the existence of a list coloring, but the converse is false. Recent works on list packing have focused on existence or extremal results of on the number of list packings, but here we turn to the algorithmic aspects of counting and sampling. In graphs of maximum degree Δ and when the number of colors is at least Ω(Δ²), we give a fully polynomial-time randomized approximation scheme (FPRAS) based on rapid mixing of a natural Markov chain (the Glauber dynamics) which we analyze with the path coupling technique. Some motivation for our work is the investigation of an atypical spin system, one where the number of spins for each vertex is much larger than the graph degree.
Cite as

Evan Camrud, Ewan Davies, Alex Karduna, and Holden Lee. Sampling List Packings. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{camrud_et_al:LIPIcs.ITCS.2025.24,
  author =	{Camrud, Evan and Davies, Ewan and Karduna, Alex and Lee, Holden},
  title =	{{Sampling List Packings}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.24},
  URN =		{urn:nbn:de:0030-drops-226528},
  doi =		{10.4230/LIPIcs.ITCS.2025.24},
  annote =	{Keywords: List packing, Graph colouring, Markov chains, Path coupling}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.26
Local Convergence and Stability of Tight Bridge-Addable Graph Classes

Authors: Guillaume Chapuy and Guillem Perarnau

Published in: LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Abstract
A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and G_n is a uniform n-vertex graph from G, then G_n is connected with probability at least (1+o(1))e^{-1/2}. The constant e^{-1/2} is best possible since it is reached for the class of forests. In this paper we prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph G_n is connected with probability close to e^{-1/2}, then G_n is asymptotically close to a uniform forest in some "local" sense. For example, if the probability converges to e^{-1/2}, then G_n converges for the Benjamini-Schramm topology, to the uniform infinite random forest F_infinity. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the "stable" extremum is not a graph but a graph class.
Cite as

Guillaume Chapuy and Guillem Perarnau. Local Convergence and Stability of Tight Bridge-Addable Graph Classes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 26:1-26:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chapuy_et_al:LIPIcs.APPROX-RANDOM.2016.26,
  author =	{Chapuy, Guillaume and Perarnau, Guillem},
  title =	{{Local Convergence and Stability of Tight Bridge-Addable Graph Classes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{26:1--26:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.26},
  URN =		{urn:nbn:de:0030-drops-66494},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.26},
  annote =	{Keywords: bridge-addable classes, random graphs, stability, local convergence, random forests}
}
Document
DOI: 10.4230/LIPIcs.STACS.2012.408
On the treewidth and related parameters of random geometric graphs

Authors: Dieter Mitsche and Guillem Perarnau

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Abstract
We give asymptotically exact values for the treewidth tw(G) of a random geometric graph G(n,r) in [0,sqrt(n)]^2. More precisely, we show that there exists some c_1 > 0, such that for any constant 0 < r < c_1, tw(G)=Theta(log(n)/loglog(n)), and also, there exists some c_2 > c_1, such that for any r=r(n)> c_2, tw(G)=Theta(r sqrt(n)). Our proofs show that for the corresponding values of r the same asymptotic bounds also hold for the pathwidth and treedepth of a random geometric graph.
Cite as

Dieter Mitsche and Guillem Perarnau. On the treewidth and related parameters of random geometric graphs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 408-419, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mitsche_et_al:LIPIcs.STACS.2012.408,
  author =	{Mitsche, Dieter and Perarnau, Guillem},
  title =	{{On the treewidth and related parameters of random geometric graphs}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{408--419},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.408},
  URN =		{urn:nbn:de:0030-drops-34280},
  doi =		{10.4230/LIPIcs.STACS.2012.408},
  annote =	{Keywords: Random geometric graphs, treewidth, treedepth}
}
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