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Documents authored by Diaz, Josep

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Diaz, Josep

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.48
Improved Reconstruction of Random Geometric Graphs

Authors: Varsha Dani, Josep Díaz, Thomas P. Hayes, and Cristopher Moore

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r = n^α for α > 0, with high probability reconstructs the vertex positions with a maximum error of O(n^β) where β = 1/2-(4/3)α, until α ≥ 3/8 where β = 0 and the error becomes O(√{log n}). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in ℝ³ and to hypercubes in any constant dimension.
Cite as

Varsha Dani, Josep Díaz, Thomas P. Hayes, and Cristopher Moore. Improved Reconstruction of Random Geometric Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dani_et_al:LIPIcs.ICALP.2022.48,
  author =	{Dani, Varsha and D{\'\i}az, Josep and Hayes, Thomas P. and Moore, Cristopher},
  title =	{{Improved Reconstruction of Random Geometric Graphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{48:1--48:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.48},
  URN =		{urn:nbn:de:0030-drops-163897},
  doi =		{10.4230/LIPIcs.ICALP.2022.48},
  annote =	{Keywords: Reconstruction algorithm, distances in RGG, d-dimensional hypercube, 3 dimensional sphere}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.35
The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions

Authors: Josep Diaz and Mordecai Golin

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case. More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0. For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors.
Cite as

Josep Diaz and Mordecai Golin. The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{diaz_et_al:LIPIcs.APPROX-RANDOM.2019.35,
  author =	{Diaz, Josep and Golin, Mordecai},
  title =	{{The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{35:1--35:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.35},
  URN =		{urn:nbn:de:0030-drops-112501},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.35},
  annote =	{Keywords: maximal points, probabilistic geometry, perturbations, Minkowski sum}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.50
Algorithmically Efficient Syntactic Characterization of Possibility Domains

Authors: Josep Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos

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

Abstract
We call domain any arbitrary subset of a Cartesian power of the set {0,1} when we think of it as reflecting abstract rationality restrictions on vectors of two-valued judgments on a number of issues. In Computational Social Choice Theory, and in particular in the theory of judgment aggregation, a domain is called a possibility domain if it admits a non-dictatorial aggregator, i.e. if for some k there exists a unanimous (idempotent) function F:D^k - > D which is not a projection function. We prove that a domain is a possibility domain if and only if there is a propositional formula of a certain syntactic form, sometimes called an integrity constraint, whose set of satisfying truth assignments, or models, comprise the domain. We call possibility integrity constraints the formulas of the specific syntactic type we define. Given a possibility domain D, we show how to construct a possibility integrity constraint for D efficiently, i.e, in polynomial time in the size of the domain. We also show how to distinguish formulas that are possibility integrity constraints in linear time in the size of the input formula. Finally, we prove the analogous results for local possibility domains, i.e. domains that admit an aggregator which is not a projection function, even when restricted to any given issue. Our result falls in the realm of classical results that give syntactic characterizations of logical relations that have certain closure properties, like e.g. the result that logical relations component-wise closed under logical AND are precisely the models of Horn formulas. However, our techniques draw from results in judgment aggregation theory as well from results about propositional formulas and logical relations.
Cite as

Josep Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos. Algorithmically Efficient Syntactic Characterization of Possibility Domains. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{diaz_et_al:LIPIcs.ICALP.2019.50,
  author =	{D{\'\i}az, Josep and Kirousis, Lefteris and Kokonezi, Sofia and Livieratos, John},
  title =	{{Algorithmically Efficient Syntactic Characterization of Possibility Domains}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{50:1--50:13},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.50},
  URN =		{urn:nbn:de:0030-drops-106269},
  doi =		{10.4230/LIPIcs.ICALP.2019.50},
  annote =	{Keywords: collective decision making, computational social choice, judgment aggregation, logical relations, algorithm complexity}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.630
Absorption Time of the Moran Process

Authors: Josep Díaz, Leslie Ann Goldberg, David Richerby, and Maria Serna

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is polynomial on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we give the expected absorption time is blog n lower bound and an explicit quadratic upper bound. We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.
Cite as

Josep Díaz, Leslie Ann Goldberg, David Richerby, and Maria Serna. Absorption Time of the Moran Process. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 630-642, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{diaz_et_al:LIPIcs.APPROX-RANDOM.2014.630,
  author =	{D{\'\i}az, Josep and Goldberg, Leslie Ann and Richerby, David and Serna, Maria},
  title =	{{Absorption Time of the Moran Process}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{630--642},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.630},
  URN =		{urn:nbn:de:0030-drops-47279},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.630},
  annote =	{Keywords: Moran Process}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2008.1750
A new upper bound for 3-SAT

Authors: Josep Diaz, Lefteris Kirousis, Dieter Mitsche, and Xavier Perez-Gimenez

Published in: LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

Abstract
We show that a randomly chosen $3$-CNF formula over $n$ variables with clauses-to-variables ratio at least $4.4898$ is asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was $4.506$. The first such bound, independently discovered by many groups of researchers since 1983, was $5.19$. Several decreasing values between $5.19$ and $4.506$ were published in the years between. The probabilistic techniques we use for the proof are, we believe, of independent interest.
Cite as

Josep Diaz, Lefteris Kirousis, Dieter Mitsche, and Xavier Perez-Gimenez. A new upper bound for 3-SAT. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 163-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{diaz_et_al:LIPIcs.FSTTCS.2008.1750,
  author =	{Diaz, Josep and Kirousis, Lefteris and Mitsche, Dieter and Perez-Gimenez, Xavier},
  title =	{{A new upper bound for 3-SAT}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{163--174},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-08-8},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{2},
  editor =	{Hariharan, Ramesh and Mukund, Madhavan and Vinay, V},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1750},
  URN =		{urn:nbn:de:0030-drops-17507},
  doi =		{10.4230/LIPIcs.FSTTCS.2008.1750},
  annote =	{Keywords: Satisfiability, 3-SAT, random, threshold}
}

Díaz, Josep

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.48
Improved Reconstruction of Random Geometric Graphs

Authors: Varsha Dani, Josep Díaz, Thomas P. Hayes, and Cristopher Moore

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r = n^α for α > 0, with high probability reconstructs the vertex positions with a maximum error of O(n^β) where β = 1/2-(4/3)α, until α ≥ 3/8 where β = 0 and the error becomes O(√{log n}). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in ℝ³ and to hypercubes in any constant dimension.
Cite as

Varsha Dani, Josep Díaz, Thomas P. Hayes, and Cristopher Moore. Improved Reconstruction of Random Geometric Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dani_et_al:LIPIcs.ICALP.2022.48,
  author =	{Dani, Varsha and D{\'\i}az, Josep and Hayes, Thomas P. and Moore, Cristopher},
  title =	{{Improved Reconstruction of Random Geometric Graphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{48:1--48:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.48},
  URN =		{urn:nbn:de:0030-drops-163897},
  doi =		{10.4230/LIPIcs.ICALP.2022.48},
  annote =	{Keywords: Reconstruction algorithm, distances in RGG, d-dimensional hypercube, 3 dimensional sphere}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.35
The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions

Authors: Josep Diaz and Mordecai Golin

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case. More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0. For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors.
Cite as

Josep Diaz and Mordecai Golin. The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{diaz_et_al:LIPIcs.APPROX-RANDOM.2019.35,
  author =	{Diaz, Josep and Golin, Mordecai},
  title =	{{The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{35:1--35:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.35},
  URN =		{urn:nbn:de:0030-drops-112501},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.35},
  annote =	{Keywords: maximal points, probabilistic geometry, perturbations, Minkowski sum}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.50
Algorithmically Efficient Syntactic Characterization of Possibility Domains

Authors: Josep Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos

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

Abstract
We call domain any arbitrary subset of a Cartesian power of the set {0,1} when we think of it as reflecting abstract rationality restrictions on vectors of two-valued judgments on a number of issues. In Computational Social Choice Theory, and in particular in the theory of judgment aggregation, a domain is called a possibility domain if it admits a non-dictatorial aggregator, i.e. if for some k there exists a unanimous (idempotent) function F:D^k - > D which is not a projection function. We prove that a domain is a possibility domain if and only if there is a propositional formula of a certain syntactic form, sometimes called an integrity constraint, whose set of satisfying truth assignments, or models, comprise the domain. We call possibility integrity constraints the formulas of the specific syntactic type we define. Given a possibility domain D, we show how to construct a possibility integrity constraint for D efficiently, i.e, in polynomial time in the size of the domain. We also show how to distinguish formulas that are possibility integrity constraints in linear time in the size of the input formula. Finally, we prove the analogous results for local possibility domains, i.e. domains that admit an aggregator which is not a projection function, even when restricted to any given issue. Our result falls in the realm of classical results that give syntactic characterizations of logical relations that have certain closure properties, like e.g. the result that logical relations component-wise closed under logical AND are precisely the models of Horn formulas. However, our techniques draw from results in judgment aggregation theory as well from results about propositional formulas and logical relations.
Cite as

Josep Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos. Algorithmically Efficient Syntactic Characterization of Possibility Domains. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{diaz_et_al:LIPIcs.ICALP.2019.50,
  author =	{D{\'\i}az, Josep and Kirousis, Lefteris and Kokonezi, Sofia and Livieratos, John},
  title =	{{Algorithmically Efficient Syntactic Characterization of Possibility Domains}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{50:1--50:13},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.50},
  URN =		{urn:nbn:de:0030-drops-106269},
  doi =		{10.4230/LIPIcs.ICALP.2019.50},
  annote =	{Keywords: collective decision making, computational social choice, judgment aggregation, logical relations, algorithm complexity}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.630
Absorption Time of the Moran Process

Authors: Josep Díaz, Leslie Ann Goldberg, David Richerby, and Maria Serna

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is polynomial on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we give the expected absorption time is blog n lower bound and an explicit quadratic upper bound. We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.
Cite as

Josep Díaz, Leslie Ann Goldberg, David Richerby, and Maria Serna. Absorption Time of the Moran Process. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 630-642, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

Copy BibTex To Clipboard

@InProceedings{diaz_et_al:LIPIcs.APPROX-RANDOM.2014.630,
  author =	{D{\'\i}az, Josep and Goldberg, Leslie Ann and Richerby, David and Serna, Maria},
  title =	{{Absorption Time of the Moran Process}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{630--642},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.630},
  URN =		{urn:nbn:de:0030-drops-47279},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.630},
  annote =	{Keywords: Moran Process}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2008.1750
A new upper bound for 3-SAT

Authors: Josep Diaz, Lefteris Kirousis, Dieter Mitsche, and Xavier Perez-Gimenez

Published in: LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

Abstract
We show that a randomly chosen $3$-CNF formula over $n$ variables with clauses-to-variables ratio at least $4.4898$ is asymptotically almost surely unsatisfiable. The previous best such bound, due to Dubois in 1999, was $4.506$. The first such bound, independently discovered by many groups of researchers since 1983, was $5.19$. Several decreasing values between $5.19$ and $4.506$ were published in the years between. The probabilistic techniques we use for the proof are, we believe, of independent interest.
Cite as

Josep Diaz, Lefteris Kirousis, Dieter Mitsche, and Xavier Perez-Gimenez. A new upper bound for 3-SAT. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 163-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

Copy BibTex To Clipboard

@InProceedings{diaz_et_al:LIPIcs.FSTTCS.2008.1750,
  author =	{Diaz, Josep and Kirousis, Lefteris and Mitsche, Dieter and Perez-Gimenez, Xavier},
  title =	{{A new upper bound for 3-SAT}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{163--174},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-08-8},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{2},
  editor =	{Hariharan, Ramesh and Mukund, Madhavan and Vinay, V},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1750},
  URN =		{urn:nbn:de:0030-drops-17507},
  doi =		{10.4230/LIPIcs.FSTTCS.2008.1750},
  annote =	{Keywords: Satisfiability, 3-SAT, random, threshold}
}
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