74 Search Results for "Blais, Eric"


Volume

LIPIcs, Volume 116

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

APPROX/RANDOM 2018, August 20-22, 2018, Princeton, NJ, USA

Editors: Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer

Document
Track A: Algorithms, Complexity and Games
A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs

Authors: Charlie Carlson, Ewan Davies, Alexandra Kolla, and Aditya Potukuchi

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


Abstract
We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to "high-temperature" problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e., bounded threshold rank), which, via subspace enumeration, lets us enumerate small cuts efficiently.

Cite as

Charlie Carlson, Ewan Davies, Alexandra Kolla, and Aditya Potukuchi. A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{carlson_et_al:LIPIcs.ICALP.2024.35,
  author =	{Carlson, Charlie and Davies, Ewan and Kolla, Alexandra and Potukuchi, Aditya},
  title =	{{A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{35:1--35:18},
  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.2024.35},
  URN =		{urn:nbn:de:0030-drops-201782},
  doi =		{10.4230/LIPIcs.ICALP.2024.35},
  annote =	{Keywords: approximate counting, independent sets, bipartite graphs, graph containers}
}
Document
Testing and Learning Convex Sets in the Ternary Hypercube

Authors: Hadley Black, Eric Blais, and Nathaniel Harms

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
We study the problems of testing and learning high-dimensional discrete convex sets. The simplest high-dimensional discrete domain where convexity is a non-trivial property is the ternary hypercube, {-1,0,1}ⁿ. The goal of this work is to understand structural combinatorial properties of convex sets in this domain and to determine the complexity of the testing and learning problems. We obtain the following results. Structural: We prove nearly tight bounds on the edge boundary of convex sets in {0,±1}ⁿ, showing that the maximum edge boundary of a convex set is Õ(n^{3/4})⋅3ⁿ, or equivalently that every convex set has influence Õ(n^{3/4}) and a convex set exists with influence Ω(n^{3/4}). Learning and sample-based testing: We prove upper and lower bounds of 3^{Õ(n^{3/4})} and 3^{Ω(√n)} for the task of learning convex sets under the uniform distribution from random examples. The analysis of the learning algorithm relies on our upper bound on the influence. Both the upper and lower bound also hold for the problem of sample-based testing with two-sided error. For sample-based testing with one-sided error we show that the sample-complexity is 3^{Θ(n)}. Testing with queries: We prove nearly matching upper and lower bounds of 3^{Θ̃(√n)} for one-sided error testing of convex sets with non-adaptive queries.

Cite as

Hadley Black, Eric Blais, and Nathaniel Harms. Testing and Learning Convex Sets in the Ternary Hypercube. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{black_et_al:LIPIcs.ITCS.2024.15,
  author =	{Black, Hadley and Blais, Eric and Harms, Nathaniel},
  title =	{{Testing and Learning Convex Sets in the Ternary Hypercube}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{15:1--15:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.15},
  URN =		{urn:nbn:de:0030-drops-195435},
  doi =		{10.4230/LIPIcs.ITCS.2024.15},
  annote =	{Keywords: Property testing, learning theory, convex sets, testing convexity, fluctuation}
}
Document
Distribution Testing with a Confused Collector

Authors: Renato Ferreira Pinto Jr. and Nathaniel Harms

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
We are interested in testing properties of distributions with systematically mislabeled samples. Our goal is to make decisions about unknown probability distributions, using a sample that has been collected by a confused collector, such as a machine-learning classifier that has not learned to distinguish all elements of the domain. The confused collector holds an unknown clustering of the domain and an input distribution μ, and provides two oracles: a sample oracle which produces a sample from μ that has been labeled according to the clustering; and a label-query oracle which returns the label of a query point x according to the clustering. Our first set of results shows that identity, uniformity, and equivalence of distributions can be tested efficiently, under the earth-mover distance, with remarkably weak conditions on the confused collector, even when the unknown clustering is adversarial. This requires defining a variant of the distribution testing task (inspired by the recent testable learning framework of Rubinfeld & Vasilyan), where the algorithm should test a joint property of the distribution and its clustering. As an example, we get efficient testers when the distribution tester is allowed to reject if it detects that the confused collector clustering is "far" from being a decision tree. The second set of results shows that we can sometimes do significantly better when the clustering is random instead of adversarial. For certain one-dimensional random clusterings, we show that uniformity can be tested under the TV distance using Õ((√n)/(ρ^{3/2} ε²)) samples and zero queries, where ρ ∈ (0,1] controls the "resolution" of the clustering. We improve this to O((√n)/(ρ ε²)) when queries are allowed.

Cite as

Renato Ferreira Pinto Jr. and Nathaniel Harms. Distribution Testing with a Confused Collector. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{ferreirapintojr._et_al:LIPIcs.ITCS.2024.47,
  author =	{Ferreira Pinto Jr., Renato and Harms, Nathaniel},
  title =	{{Distribution Testing with a Confused Collector}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.47},
  URN =		{urn:nbn:de:0030-drops-195755},
  doi =		{10.4230/LIPIcs.ITCS.2024.47},
  annote =	{Keywords: Distribution testing, property testing, uniformity testing, identity testing, earth-mover distance, sublinear algorithms}
}
Document
RANDOM
Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions

Authors: Renato Ferreira Pinto Jr.

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)


Abstract
We study the connection between directed isoperimetric inequalities and monotonicity testing. In recent years, this connection has unlocked breakthroughs for testing monotonicity of functions defined on discrete domains. Inspired the rich history of isoperimetric inequalities in continuous settings, we propose that studying the relationship between directed isoperimetry and monotonicity in such settings is essential for understanding the full scope of this connection. Hence, we ask whether directed isoperimetric inequalities hold for functions f:[0,1]ⁿ → R, and whether this question has implications for monotonicity testing. We answer both questions affirmatively. For Lipschitz functions f:[0,1]ⁿ → ℝ, we show the inequality d^mono₁(f) ≲ 𝔼 [‖∇^- f‖₁], which upper bounds the L¹ distance to monotonicity of f by a measure of its "directed gradient". A key ingredient in our proof is the monotone rearrangement of f, which generalizes the classical "sorting operator" to continuous settings. We use this inequality to give an L¹ monotonicity tester for Lipschitz functions f:[0,1]ⁿ → ℝ, and this framework also implies similar results for testing real-valued functions on the hypergrid.

Cite as

Renato Ferreira Pinto Jr.. Directed Poincaré Inequalities and L¹ Monotonicity Testing of Lipschitz Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{ferreirapintojr.:LIPIcs.APPROX/RANDOM.2023.61,
  author =	{Ferreira Pinto Jr., Renato},
  title =	{{Directed Poincar\'{e} Inequalities and L¹ Monotonicity Testing of Lipschitz Functions}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{61:1--61:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.61},
  URN =		{urn:nbn:de:0030-drops-188867},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.61},
  annote =	{Keywords: Monotonicity testing, property testing, isoperimetric inequalities, Poincar\'{e} inequalities}
}
Document
Box Covers and Domain Orderings for Beyond Worst-Case Join Processing

Authors: Kaleb Alway, Eric Blais, and Semih Salihoglu

Published in: LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)


Abstract
Recent beyond worst-case optimal join algorithms Minesweeper and its generalization Tetris have brought the theory of indexing and join processing together by developing a geometric framework for joins. These algorithms take as input an index ℬ, referred to as a box cover, that stores output gaps that can be inferred from traditional indexes, such as B+ trees or tries, on the input relations. The performances of these algorithms highly depend on the certificate of ℬ, which is the smallest subset of gaps in ℬ whose union covers all of the gaps in the output space of a query Q. Different box covers can have different size certificates and the sizes of both the box covers and certificates highly depend on the ordering of the domain values of the attributes in Q. We study how to generate box covers that contain small size certificates to guarantee efficient runtimes for these algorithms. First, given a query Q over a set of relations of size N and a fixed set of domain orderings for the attributes, we give a Õ(N)-time algorithm called GAMB which generates a box cover for Q that is guaranteed to contain the smallest size certificate across any box cover for Q. Second, we show that finding a domain ordering to minimize the box cover size and certificate is NP-hard through a reduction from the 2 consecutive block minimization problem on boolean matrices. Our third contribution is a Õ(N)-time approximation algorithm called ADORA to compute domain orderings, under which one can compute a box cover of size Õ(K^r), where K is the minimum box cover for Q under any domain ordering and r is the maximum arity of any relation. This guarantees certificates of size Õ(K^r). We combine ADORA and GAMB with Tetris to form a new algorithm we call TetrisReordered, which provides several new beyond worst-case bounds. On infinite families of queries, TetrisReordered’s runtimes are unboundedly better than the bounds stated in prior work.

Cite as

Kaleb Alway, Eric Blais, and Semih Salihoglu. Box Covers and Domain Orderings for Beyond Worst-Case Join Processing. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{alway_et_al:LIPIcs.ICDT.2021.3,
  author =	{Alway, Kaleb and Blais, Eric and Salihoglu, Semih},
  title =	{{Box Covers and Domain Orderings for Beyond Worst-Case Join Processing}},
  booktitle =	{24th International Conference on Database Theory (ICDT 2021)},
  pages =	{3:1--3:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-179-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{186},
  editor =	{Yi, Ke and Wei, Zhewei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.3},
  URN =		{urn:nbn:de:0030-drops-137114},
  doi =		{10.4230/LIPIcs.ICDT.2021.3},
  annote =	{Keywords: Beyond worst-case join algorithms, Tetris, Box covers, Domain orderings}
}
Document
RANDOM
On Testing and Robust Characterizations of Convexity

Authors: Eric Blais and Abhinav Bommireddi

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)


Abstract
A body K ⊂ ℝⁿ is convex if and only if the line segment between any two points in K is completely contained within K or, equivalently, if and only if the convex hull of a set of points in K is contained within K. We show that neither of those characterizations of convexity are robust: there are bodies in ℝⁿ that are far from convex - in the sense that the volume of the symmetric difference between the set K and any convex set C is a constant fraction of the volume of K - for which a line segment between two randomly chosen points x,y ∈ K or the convex hull of a random set X of points in K is completely contained within K except with exponentially small probability. These results show that any algorithms for testing convexity based on the natural line segment and convex hull tests have exponential query complexity.

Cite as

Eric Blais and Abhinav Bommireddi. On Testing and Robust Characterizations of Convexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{blais_et_al:LIPIcs.APPROX/RANDOM.2020.18,
  author =	{Blais, Eric and Bommireddi, Abhinav},
  title =	{{On Testing and Robust Characterizations of Convexity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.18},
  URN =		{urn:nbn:de:0030-drops-126214},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.18},
  annote =	{Keywords: Convexity, Line segment test, Convex hull test, Intersecting cones}
}
Document
Universal Communication, Universal Graphs, and Graph Labeling

Authors: Nathaniel Harms

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family ℱ, and inputs x and y. Alice and Bob use shared randomness to send a message to a third party who cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ℓ(x), ℓ(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ≤ k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ≤ 2 in modular lattices (a superset of distributive lattices) has super-constant Ω(n^{1/4}) communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an O(k) protocol for deciding dist(x,y) ≤ k and planar graphs have an O(1) protocol for dist(x,y) ≤ 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs.

Cite as

Nathaniel Harms. Universal Communication, Universal Graphs, and Graph Labeling. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 33:1-33:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{harms:LIPIcs.ITCS.2020.33,
  author =	{Harms, Nathaniel},
  title =	{{Universal Communication, Universal Graphs, and Graph Labeling}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{33:1--33:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.33},
  URN =		{urn:nbn:de:0030-drops-117182},
  doi =		{10.4230/LIPIcs.ITCS.2020.33},
  annote =	{Keywords: Universal graphs, graph labeling, distance labeling, planar graphs, lattices, hamming distance}
}
Document
Optimal Separation and Strong Direct Sum for Randomized Query Complexity

Authors: Eric Blais and Joshua Brody

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We establish two results regarding the query complexity of bounded-error randomized algorithms. Bounded-error separation theorem. There exists a total function f : {0,1}^n -> {0,1} whose epsilon-error randomized query complexity satisfies overline{R}_epsilon(f) = Omega(R(f) * log 1/epsilon). Strong direct sum theorem. For every function f and every k >= 2, the randomized query complexity of computing k instances of f simultaneously satisfies overline{R}_epsilon(f^k) = Theta(k * overline{R}_{epsilon/k}(f)). As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function f for which R(f^k) = Theta(k log k * R(f)). This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of Göös, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies R^{cc}(f^k) = Theta(k log k * R^{cc}(f)), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).

Cite as

Eric Blais and Joshua Brody. Optimal Separation and Strong Direct Sum for Randomized Query Complexity. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{blais_et_al:LIPIcs.CCC.2019.29,
  author =	{Blais, Eric and Brody, Joshua},
  title =	{{Optimal Separation and Strong Direct Sum for Randomized Query Complexity}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.29},
  URN =		{urn:nbn:de:0030-drops-108511},
  doi =		{10.4230/LIPIcs.CCC.2019.29},
  annote =	{Keywords: Decision trees, query complexity, communication complexity}
}
Document
Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs

Authors: Amit Levi and Erik Waingarten

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)


Abstract
We introduce a new model for testing graph properties which we call the rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Omega~(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f : {0,1}^n -> {0,1}: - Tolerant k-junta testing with non-adaptive queries requires Omega~(k^2) queries. - Tolerant unateness testing requires Omega~(n) queries. - Tolerant unateness testing with non-adaptive queries requires Omega~(n^{3/2}) queries. Given the O~(k^{3/2})-query non-adaptive junta tester of Blais [Eric Blais, 2008], we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O~(n^{3/4})-query unateness tester of Chen, Waingarten, and Xie [Xi Chen et al., 2017] and the O~(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri [Roksana Baleshzar et al., 2017], we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions.

Cite as

Amit Levi and Erik Waingarten. Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 52:1-52:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{levi_et_al:LIPIcs.ITCS.2019.52,
  author =	{Levi, Amit and Waingarten, Erik},
  title =	{{Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{52:1--52:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.52},
  URN =		{urn:nbn:de:0030-drops-101452},
  doi =		{10.4230/LIPIcs.ITCS.2019.52},
  annote =	{Keywords: Property Testing, Juntas, Tolerant Testing, Boolean functions}
}
Document
Complete Volume
LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume

Authors: Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer

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


Abstract
LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume

Cite as

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@Proceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2018,
  title =	{{LIPIcs, Volume 116, APPROX/RANDOM'18, Complete Volume}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018},
  URN =		{urn:nbn:de:0030-drops-97254},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018},
  annote =	{Keywords: Mathematics of computing, Theory of computation}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer

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


Abstract
Front Matter, Table of Contents, Preface, Conference Organization

Cite as

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{blais_et_al:LIPIcs.APPROX-RANDOM.2018.0,
  author =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{0:i--0:xvi},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.0},
  URN =		{urn:nbn:de:0030-drops-94043},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems

Authors: Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi

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


Abstract
Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w: V(G) - >R^+, find a minimum weight subset S subseteq V(G) such that G-S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log^{O(1)} n)-approximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log^{O(1)} n)-approximation algorithms for the following vertex deletion problems. - Let {F} be a finite set of graphs containing a planar graph, and F=G(F) be the family of graphs such that every graph H in G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F=G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log^{1.5} n) and O(log^2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012]. - We give an O(log^2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. - We give an O(log^3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one. We believe that our recursive scheme can be applied to obtain O(log^{O(1)} n)-approximation algorithms for many other problems as well.

Cite as

Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{agrawal_et_al:LIPIcs.APPROX-RANDOM.2018.1,
  author =	{Agrawal, Akanksha and Lokshtanov, Daniel and Misra, Pranabendu and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{1:1--1:15},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.1},
  URN =		{urn:nbn:de:0030-drops-94058},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.1},
  annote =	{Keywords: Approximation Algorithms, Planar- F-Deletion, Separator}
}
Document
Improved Approximation Bounds for the Minimum Constraint Removal Problem

Authors: Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi Varadarajan

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


Abstract
In the minimum constraint removal problem, we are given a set of geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable, and (perhaps surprisingly) no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is a new approximation technique that gives O(sqrt{n})-approximation for rectangles, disks as well as rectilinear polygons. The technique also gives O(sqrt{n})-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.

Cite as

Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi Varadarajan. Improved Approximation Bounds for the Minimum Constraint Removal Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{bandyapadhyay_et_al:LIPIcs.APPROX-RANDOM.2018.2,
  author =	{Bandyapadhyay, Sayan and Kumar, Neeraj and Suri, Subhash and Varadarajan, Kasturi},
  title =	{{Improved Approximation Bounds for the Minimum Constraint Removal Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{2:1--2:19},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.2},
  URN =		{urn:nbn:de:0030-drops-94066},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.2},
  annote =	{Keywords: Minimum Constraint Removal, Minimum Color Path, Barrier Resilience, Obstacle Removal, Obstacle Free Path, Approximation}
}
Document
A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees

Authors: Amariah Becker

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


Abstract
Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded and such that the sum of the tour lengths is minimized. In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of (sqrt{41}-1)/4 by Asano et al.[Asano et al., 2001], while using the same lower bound. Asano et al. show that there exist instances whose optimal cost is 4/3 times this lower bound. Notably, our 4/3 approximation ratio is therefore tight for this lower bound, achieving the best-possible performance.

Cite as

Amariah Becker. A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{becker:LIPIcs.APPROX-RANDOM.2018.3,
  author =	{Becker, Amariah},
  title =	{{A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{3:1--3:15},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.3},
  URN =		{urn:nbn:de:0030-drops-94075},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.3},
  annote =	{Keywords: Approximation algorithms, Graph algorithms, Capacitated vehicle routing}
}
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