5 Search Results for "Thurley, Marc"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.11
Approximate Counting for Spin Systems in Sub-Quadratic Time

Authors: Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, and Jiaheng Wang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We present two randomised approximate counting algorithms with Õ(n^{2-c}/ε²) running time for some constant c > 0 and accuracy ε: 1) for the hard-core model with fugacity λ on graphs with maximum degree Δ when λ = O(Δ^{-1.5-c₁}) where c₁ = c/(2-2c); 2) for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as ℤ². For the hard-core model, Weitz’s algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when λ = o(Δ^{-2}). Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist. Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as ℤ^d, but with a running time of the form Õ(n²ε^{-2}/2^{c(log n)^{1/d}}) where d is the exponent of the polynomial growth and c > 0 is some constant.
Cite as

Konrad Anand, Weiming Feng, Graham Freifeld, Heng Guo, and Jiaheng Wang. Approximate Counting for Spin Systems in Sub-Quadratic Time. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{anand_et_al:LIPIcs.ICALP.2024.11,
  author =	{Anand, Konrad and Feng, Weiming and Freifeld, Graham and Guo, Heng and Wang, Jiaheng},
  title =	{{Approximate Counting for Spin Systems in Sub-Quadratic Time}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.11},
  URN =		{urn:nbn:de:0030-drops-201543},
  doi =		{10.4230/LIPIcs.ICALP.2024.11},
  annote =	{Keywords: Randomised algorithm, Approximate counting, Spin system, Sub-quadratic algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.54
Nearly Optimal Independence Oracle Algorithms for Edge Estimation in Hypergraphs

Authors: Holger Dell, John Lapinskas, and Kitty Meeks

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Consider a query model of computation in which an n-vertex k-hypergraph can be accessed only via its independence oracle or via its colourful independence oracle, and each oracle query may incur a cost depending on the size of the query. Several recent results (Dell and Lapinskas, STOC 2018; Dell, Lapinskas, and Meeks, SODA 2020) give efficient algorithms to approximately count the hypergraph’s edges in the colourful setting. These algorithms immediately imply fine-grained reductions from approximate counting to decision, with overhead only log^Θ(k) n over the running time n^α of the original decision algorithm, for many well-studied problems including k-Orthogonal Vectors, k-SUM, subgraph isomorphism problems including k-Clique and colourful-H, graph motifs, and k-variable first-order model checking. We explore the limits of what is achievable in this setting, obtaining unconditional lower bounds on the oracle cost of algorithms to approximately count the hypergraph’s edges in both the colourful and uncoloured settings. In both settings, we also obtain algorithms which essentially match these lower bounds; in the colourful setting, this requires significant changes to the algorithm of Dell, Lapinskas, and Meeks (SODA 2020) and reduces the total overhead to log^{Θ(k-α)}n. Our lower bound for the uncoloured setting shows that there is no fine-grained reduction from approximate counting to the corresponding uncoloured decision problem (except in the case α ≥ k-1): without an algorithm for the colourful decision problem, we cannot hope to avoid the much larger overhead of roughly n^{(k-α)²/4}. The uncoloured setting has previously been studied for the special case k = 2 (Peled, Ramamoorthy, Rashtchian, Sinha, ITCS 2018; Chen, Levi, and Waingarten, SODA 2020), and our work generalises the existing algorithms and lower bounds for this special case to k > 2 and to oracles with cost.
Cite as

Holger Dell, John Lapinskas, and Kitty Meeks. Nearly Optimal Independence Oracle Algorithms for Edge Estimation in Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dell_et_al:LIPIcs.ICALP.2024.54,
  author =	{Dell, Holger and Lapinskas, John and Meeks, Kitty},
  title =	{{Nearly Optimal Independence Oracle Algorithms for Edge Estimation in Hypergraphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{54:1--54:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.54},
  URN =		{urn:nbn:de:0030-drops-201977},
  doi =		{10.4230/LIPIcs.ICALP.2024.54},
  annote =	{Keywords: Graph oracles, Fine-grained complexity, Approximate counting, Hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.CSL.2013.149
Descriptive complexity of approximate counting CSPs

Authors: Andrei Bulatov, Victor Dalmau, and Marc Thurley

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

Abstract
Motivated by Fagin's characterization of NP, Saluja et al. have introduced a logic based frame- work for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to non-negative integers) is `defined’ by a first-order sentence phi if for every instance A of the problem, the number of possible satisfying assignments of the variables of phi in A is equal to C(A). The logic RHPI_1 has been introduced by Dyer et al. in their study of the counting complexity class #BIS. The interest in the class #BIS stems from the fact that, it is quite plausible that the problems in #BIS are not #P-hard, nor they admit a fully polynomial randomized approximation scheme. In the present paper we investigate which counting constraint satisfaction problems #CSP(H) are definable in the monotone fragment of RHPI_1. We prove that #CSP(H) is definable in monotone RHPI_1 whenever H is invariant under meet and join operations of a distributive lattice. We prove that the converse also holds if H contains the equality relation. We also prove similar results for counting CSPs expressible by linear Datalog. The results in this case are very similar to those for monotone RHPI1, with the addition that H has, additionally, \top (the greatest element of the lattice) as a polymorphism.
Cite as

Andrei Bulatov, Victor Dalmau, and Marc Thurley. Descriptive complexity of approximate counting CSPs. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 149-164, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{bulatov_et_al:LIPIcs.CSL.2013.149,
  author =	{Bulatov, Andrei and Dalmau, Victor and Thurley, Marc},
  title =	{{Descriptive complexity of approximate counting CSPs}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{149--164},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.149},
  URN =		{urn:nbn:de:0030-drops-41955},
  doi =		{10.4230/LIPIcs.CSL.2013.149},
  annote =	{Keywords: Constraint Satisfaction Problems, Approximate Counting, Descriptive Complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2012.78
An Approximation Algorithm for #k-SAT

Authors: Marc Thurley

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

Abstract
We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k>=3 within a running time that is not only non-trivial, but also significantly better than that of the currently fastest exact algorithms for the problem. More precisely, our algorithm is a randomized approximation scheme whose running time depends polynomially on the error tolerance and is mildly exponential in the number n of variables of the input formula. For example, even stipulating sub-exponentially small error tolerance, the number of solutions to 3-CNF input formulas can be approximated in time O(1.5366^n). For 4-CNF input the bound increases to O(1.6155^n). We further show how to obtain upper and lower bounds on the number of solutions to a CNF formula in a controllable way. Relaxing the requirements on the quality of the approximation, on k-CNF input we obtain significantly reduced running times in comparison to the above bounds.
Cite as

Marc Thurley. An Approximation Algorithm for #k-SAT. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 78-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{thurley:LIPIcs.STACS.2012.78,
  author =	{Thurley, Marc},
  title =	{{An Approximation Algorithm for #k-SAT}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{78--87},
  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.2012.78},
  URN =		{urn:nbn:de:0030-drops-34400},
  doi =		{10.4230/LIPIcs.STACS.2012.78},
  annote =	{Keywords: #k-SAT, approximate counting, exponential algorithms}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1821
A Complexity Dichotomy for Partition Functions with Mixed Signs

Authors: Leslie Ann Goldberg, Martin Grohe, Mark Jerrum, and Marc Thurley

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract
\emph{Partition functions}, also known as \emph{homomorphism functions}, form a rich family of graph invariants that contain combinatorial invariants such as the number of $k$-colourings or the number of independent sets of a graph and also the partition functions of certain ``spin glass'' models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill (2000) and Bulatov and Grohe (2005), we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or \#P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or \#P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices --- these turn out to be central in our proofs --- we obtain a simple algebraic tractability criterion, which says that the tractable cases are those ``representable'' by a quadratic polynomial over the field $\ensuremath{\mathbb{F}_2}$.
Cite as

Leslie Ann Goldberg, Martin Grohe, Mark Jerrum, and Marc Thurley. A Complexity Dichotomy for Partition Functions with Mixed Signs. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 493-504, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{goldberg_et_al:LIPIcs.STACS.2009.1821,
  author =	{Goldberg, Leslie Ann and Grohe, Martin and Jerrum, Mark and Thurley, Marc},
  title =	{{A Complexity Dichotomy for Partition Functions with Mixed Signs}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{493--504},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1821},
  URN =		{urn:nbn:de:0030-drops-18217},
  doi =		{10.4230/LIPIcs.STACS.2009.1821},
  annote =	{Keywords: Computational complexity}
}
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