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Documents authored by Dvořák, Pavel

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Dvorák, Pavel

Document
DOI: 10.4230/LIPIcs.CCC.2024.23
Exponential Separation Between Powers of Regular and General Resolution over Parities

Authors: Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014]. Very recently, Efremenko, Garlík and Itsykson [Klim Efremenko et al., 2023] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exist short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and that of regular ResLin for any natural notion of regularity. Our argument, while building upon the work of Efremenko et al. [Klim Efremenko et al., 2023], uses additional ideas from the literature on lifting theorems.
Cite as

Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 23:1-23:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhattacharya_et_al:LIPIcs.CCC.2024.23,
  author =	{Bhattacharya, Sreejata Kishor and Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel},
  title =	{{Exponential Separation Between Powers of Regular and General Resolution over Parities}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{23:1--23:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.23},
  URN =		{urn:nbn:de:0030-drops-204191},
  doi =		{10.4230/LIPIcs.CCC.2024.23},
  annote =	{Keywords: Proof Complexity, Regular Reslin, Branching Programs, Lifting}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.6
Improved Weighted Matching in the Sliding Window Model

Authors: Cezar-Mihail Alexandru, Pavel Dvořák, Christian Konrad, and Kheeran K. Naidu

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

Abstract
We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set V of size n. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the L most recent edges, for some integer L, using as little space as possible. Prior to our work, the state-of-the-art results were a (3.5+ε)-approximation algorithm for MWM by Biabani et al. [ISAAC'21] and a (3+ε)-approximation for (unweighted) Maximum Matching (MM) by Crouch et al. [ESA'13]. Both algorithms use space Õ(n). We give the following results: 1) We give a (2+ε)-approximation algorithm for MWM with space Õ(√{nL}). Under the reasonable assumption that the graphs spanned by the edges in each sliding window are simple, our algorithm uses space Õ(n √n). 2) In the Õ(n) space regime, we give a (3+ε)-approximation algorithm for MWM, thereby closing the gap between the best-known approximation ratio for MWM and MM. Similar to Biabani et al.’s MWM algorithm, both our algorithms execute multiple instances of the (2+ε)-approximation Õ(n)-space streaming algorithm for MWM by Paz and Schwartzman [SODA'17] on different portions of the stream. Our improvements are obtained by selecting these substreams differently. Furthermore, our (2+ε)-approximation algorithm runs the Paz-Schwartzman algorithm in reverse direction over some parts of the stream, and in forward direction over other parts, which allows for an improved approximation guarantee at the cost of increased space requirements.
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Cezar-Mihail Alexandru, Pavel Dvořák, Christian Konrad, and Kheeran K. Naidu. Improved Weighted Matching in the Sliding Window Model. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alexandru_et_al:LIPIcs.STACS.2023.6,
  author =	{Alexandru, Cezar-Mihail and Dvo\v{r}\'{a}k, Pavel and Konrad, Christian and Naidu, Kheeran K.},
  title =	{{Improved Weighted Matching in the Sliding Window Model}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{6:1--6:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.6},
  URN =		{urn:nbn:de:0030-drops-176585},
  doi =		{10.4230/LIPIcs.STACS.2023.6},
  annote =	{Keywords: Sliding window algorithms, Streaming algorithms, Maximum-weight matching}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.30
List Locally Surjective Homomorphisms in Hereditary Graph Classes

Authors: Pavel Dvořák, Tomáš Masařík, Jana Novotná, Monika Krawczyk, Paweł Rzążewski, and Aneta Żuk

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time. We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family 𝒮 consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P₃,C₄} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ∈ 𝒮 (unless the ETH fails).
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Pavel Dvořák, Tomáš Masařík, Jana Novotná, Monika Krawczyk, Paweł Rzążewski, and Aneta Żuk. List Locally Surjective Homomorphisms in Hereditary Graph Classes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2022.30,
  author =	{Dvo\v{r}\'{a}k, Pavel and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Novotn\'{a}, Jana and Krawczyk, Monika and Rz\k{a}\.{z}ewski, Pawe{\l} and \.{Z}uk, Aneta},
  title =	{{List Locally Surjective Homomorphisms in Hereditary Graph Classes}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{30:1--30:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.30},
  URN =		{urn:nbn:de:0030-drops-173154},
  doi =		{10.4230/LIPIcs.ISAAC.2022.30},
  annote =	{Keywords: Homomorphism, Hereditary graphs, Subexponential-time algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.39
Data Structures Lower Bounds and Popular Conjectures

Authors: Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li [Li and Li, 2004] and several data structure problems such as non-adaptive function inversion of Hellman [M. Hellman, 1980] and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.
Cite as

Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová. Data Structures Lower Bounds and Popular Conjectures. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.ESA.2021.39,
  author =	{Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal and Kr\'{a}l, Karel and Sl{\'\i}vov\'{a}, Veronika},
  title =	{{Data Structures Lower Bounds and Popular Conjectures}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.39},
  URN =		{urn:nbn:de:0030-drops-146207},
  doi =		{10.4230/LIPIcs.ESA.2021.39},
  annote =	{Keywords: Data structures, Circuits, Lower bounds, Network Coding Conjecture}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.26
Barrington Plays Cards: The Complexity of Card-Based Protocols

Authors: Pavel Dvořák and Michal Koucký

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
In this paper we study the computational complexity of functions that have efficient card-based protocols. A study of card-based protocols was initiated by den Boer [den Boer, 1990] as a means for secure two-party computation. Our contribution is two-fold: We classify a large class of protocols with respect to the computational complexity of functions they compute, and we propose other encodings of inputs which require fewer cards than the usual 2-card representation.
Cite as

Pavel Dvořák and Michal Koucký. Barrington Plays Cards: The Complexity of Card-Based Protocols. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2021.26,
  author =	{Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal},
  title =	{{Barrington Plays Cards: The Complexity of Card-Based Protocols}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.26},
  URN =		{urn:nbn:de:0030-drops-136715},
  doi =		{10.4230/LIPIcs.STACS.2021.26},
  annote =	{Keywords: Efficient card-based protocol, Branching program, Turing machine}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.20
Lower Bounds for Semi-adaptive Data Structures via Corruption

Authors: Pavel Dvořák and Bruno Loff

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
In a dynamic data structure problem we wish to maintain an encoding of some data in memory, in such a way that we may efficiently carry out a sequence of queries and updates to the data. A long-standing open problem in this area is to prove an unconditional polynomial lower bound of a trade-off between the update time and the query time of an adaptive dynamic data structure computing some explicit function. Ko and Weinstein provided such lower bound for a restricted class of semi-adaptive data structures, which compute the Disjointness function. There, the data are subsets x₁,… ,x_k and y of {1,… ,n}, the updates can modify y (by inserting and removing elements), and the queries are an index i ∈ {1,… ,k} (query i should answer whether x_i and y are disjoint, i.e., it should compute the Disjointness function applied to (x_i, y)). The semi-adaptiveness places a restriction in how the data structure can be accessed in order to answer a query. We generalize the lower bound of Ko and Weinstein to work not just for the Disjointness, but for any function having high complexity under the smooth corruption bound.
Cite as

Pavel Dvořák and Bruno Loff. Lower Bounds for Semi-adaptive Data Structures via Corruption. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dvorak_et_al:LIPIcs.FSTTCS.2020.20,
  author =	{Dvo\v{r}\'{a}k, Pavel and Loff, Bruno},
  title =	{{Lower Bounds for Semi-adaptive Data Structures via Corruption}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.20},
  URN =		{urn:nbn:de:0030-drops-132617},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.20},
  annote =	{Keywords: semi-adaptive dynamic data structure, polynomial lower bound, corruption bound, information theory}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.18
Target Set Selection in Dense Graph Classes

Authors: Pavel Dvorák, Dusan Knop, and Tomás Toufar

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
In this paper we study the Target Set Selection problem from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex the task is to find a set of vertices (called a target set) to activate at the beginning which activates the whole graph during the following iterative process. A vertex outside the active set becomes active if the number of so far activated vertices in its neighborhood is at least its threshold. We give two parameterized algorithms for a special case where each vertex has the threshold set to the half of its neighbors (the so called Majority Target Set Selection problem) for parameterizations by the neighborhood diversity and the twin cover number of the input graph. We complement these results from the negative side. We give a hardness proof for the Majority Target Set Selection problem when parameterized by (a restriction of) the modular-width - a natural generalization of both previous structural parameters. We show that the Target Set Selection problem parameterized by the neighborhood diversity when there is no restriction on the thresholds is W[1]-hard.
Cite as

Pavel Dvorák, Dusan Knop, and Tomás Toufar. Target Set Selection in Dense Graph Classes. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2018.18,
  author =	{Dvor\'{a}k, Pavel and Knop, Dusan and Toufar, Tom\'{a}s},
  title =	{{Target Set Selection in Dense Graph Classes}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{18:1--18:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.18},
  URN =		{urn:nbn:de:0030-drops-99666},
  doi =		{10.4230/LIPIcs.ISAAC.2018.18},
  annote =	{Keywords: parameterized complexity, target set selection, dense graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.26
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Authors: Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.
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Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý. Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2018.26,
  author =	{Dvor\'{a}k, Pavel and Feldmann, Andreas Emil and Knop, Du\v{s}an and Masar{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s} and Vesel\'{y}, Pavel},
  title =	{{Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.26},
  URN =		{urn:nbn:de:0030-drops-85158},
  doi =		{10.4230/LIPIcs.STACS.2018.26},
  annote =	{Keywords: Steiner Tree, Steiner Forest, Approximation Algorithms, Parameterized Algorithms, Lossy Kernelization}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.21
Lower Bounds for Elimination via Weak Regularity

Authors: Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds. To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.
Cite as

Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Lower Bounds for Elimination via Weak Regularity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2017.21,
  author =	{Chattopadhyay, Arkadev and Dvor\'{a}k, Pavel and Kouck\'{y}, Michal and Loff, Bruno and Mukhopadhyay, Sagnik},
  title =	{{Lower Bounds for Elimination via Weak Regularity}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.21},
  URN =		{urn:nbn:de:0030-drops-70128},
  doi =		{10.4230/LIPIcs.STACS.2017.21},
  annote =	{Keywords: communication complexity, elimination, discrepancy, regularity, greater-than}
}

Dvořák, Pavel

Document
DOI: 10.4230/LIPIcs.CCC.2024.23
Exponential Separation Between Powers of Regular and General Resolution over Parities

Authors: Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014]. Very recently, Efremenko, Garlík and Itsykson [Klim Efremenko et al., 2023] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exist short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and that of regular ResLin for any natural notion of regularity. Our argument, while building upon the work of Efremenko et al. [Klim Efremenko et al., 2023], uses additional ideas from the literature on lifting theorems.
Cite as

Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 23:1-23:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhattacharya_et_al:LIPIcs.CCC.2024.23,
  author =	{Bhattacharya, Sreejata Kishor and Chattopadhyay, Arkadev and Dvo\v{r}\'{a}k, Pavel},
  title =	{{Exponential Separation Between Powers of Regular and General Resolution over Parities}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{23:1--23:32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.23},
  URN =		{urn:nbn:de:0030-drops-204191},
  doi =		{10.4230/LIPIcs.CCC.2024.23},
  annote =	{Keywords: Proof Complexity, Regular Reslin, Branching Programs, Lifting}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.6
Improved Weighted Matching in the Sliding Window Model

Authors: Cezar-Mihail Alexandru, Pavel Dvořák, Christian Konrad, and Kheeran K. Naidu

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

Abstract
We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set V of size n. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the L most recent edges, for some integer L, using as little space as possible. Prior to our work, the state-of-the-art results were a (3.5+ε)-approximation algorithm for MWM by Biabani et al. [ISAAC'21] and a (3+ε)-approximation for (unweighted) Maximum Matching (MM) by Crouch et al. [ESA'13]. Both algorithms use space Õ(n). We give the following results: 1) We give a (2+ε)-approximation algorithm for MWM with space Õ(√{nL}). Under the reasonable assumption that the graphs spanned by the edges in each sliding window are simple, our algorithm uses space Õ(n √n). 2) In the Õ(n) space regime, we give a (3+ε)-approximation algorithm for MWM, thereby closing the gap between the best-known approximation ratio for MWM and MM. Similar to Biabani et al.’s MWM algorithm, both our algorithms execute multiple instances of the (2+ε)-approximation Õ(n)-space streaming algorithm for MWM by Paz and Schwartzman [SODA'17] on different portions of the stream. Our improvements are obtained by selecting these substreams differently. Furthermore, our (2+ε)-approximation algorithm runs the Paz-Schwartzman algorithm in reverse direction over some parts of the stream, and in forward direction over other parts, which allows for an improved approximation guarantee at the cost of increased space requirements.
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Cezar-Mihail Alexandru, Pavel Dvořák, Christian Konrad, and Kheeran K. Naidu. Improved Weighted Matching in the Sliding Window Model. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alexandru_et_al:LIPIcs.STACS.2023.6,
  author =	{Alexandru, Cezar-Mihail and Dvo\v{r}\'{a}k, Pavel and Konrad, Christian and Naidu, Kheeran K.},
  title =	{{Improved Weighted Matching in the Sliding Window Model}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{6:1--6:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.6},
  URN =		{urn:nbn:de:0030-drops-176585},
  doi =		{10.4230/LIPIcs.STACS.2023.6},
  annote =	{Keywords: Sliding window algorithms, Streaming algorithms, Maximum-weight matching}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.30
List Locally Surjective Homomorphisms in Hereditary Graph Classes

Authors: Pavel Dvořák, Tomáš Masařík, Jana Novotná, Monika Krawczyk, Paweł Rzążewski, and Aneta Żuk

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time. We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family 𝒮 consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P₃,C₄} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ∈ 𝒮 (unless the ETH fails).
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Pavel Dvořák, Tomáš Masařík, Jana Novotná, Monika Krawczyk, Paweł Rzążewski, and Aneta Żuk. List Locally Surjective Homomorphisms in Hereditary Graph Classes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2022.30,
  author =	{Dvo\v{r}\'{a}k, Pavel and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Novotn\'{a}, Jana and Krawczyk, Monika and Rz\k{a}\.{z}ewski, Pawe{\l} and \.{Z}uk, Aneta},
  title =	{{List Locally Surjective Homomorphisms in Hereditary Graph Classes}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{30:1--30:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.30},
  URN =		{urn:nbn:de:0030-drops-173154},
  doi =		{10.4230/LIPIcs.ISAAC.2022.30},
  annote =	{Keywords: Homomorphism, Hereditary graphs, Subexponential-time algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.39
Data Structures Lower Bounds and Popular Conjectures

Authors: Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li [Li and Li, 2004] and several data structure problems such as non-adaptive function inversion of Hellman [M. Hellman, 1980] and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.
Cite as

Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová. Data Structures Lower Bounds and Popular Conjectures. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.ESA.2021.39,
  author =	{Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal and Kr\'{a}l, Karel and Sl{\'\i}vov\'{a}, Veronika},
  title =	{{Data Structures Lower Bounds and Popular Conjectures}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.39},
  URN =		{urn:nbn:de:0030-drops-146207},
  doi =		{10.4230/LIPIcs.ESA.2021.39},
  annote =	{Keywords: Data structures, Circuits, Lower bounds, Network Coding Conjecture}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.26
Barrington Plays Cards: The Complexity of Card-Based Protocols

Authors: Pavel Dvořák and Michal Koucký

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
In this paper we study the computational complexity of functions that have efficient card-based protocols. A study of card-based protocols was initiated by den Boer [den Boer, 1990] as a means for secure two-party computation. Our contribution is two-fold: We classify a large class of protocols with respect to the computational complexity of functions they compute, and we propose other encodings of inputs which require fewer cards than the usual 2-card representation.
Cite as

Pavel Dvořák and Michal Koucký. Barrington Plays Cards: The Complexity of Card-Based Protocols. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2021.26,
  author =	{Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal},
  title =	{{Barrington Plays Cards: The Complexity of Card-Based Protocols}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.26},
  URN =		{urn:nbn:de:0030-drops-136715},
  doi =		{10.4230/LIPIcs.STACS.2021.26},
  annote =	{Keywords: Efficient card-based protocol, Branching program, Turing machine}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.20
Lower Bounds for Semi-adaptive Data Structures via Corruption

Authors: Pavel Dvořák and Bruno Loff

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
In a dynamic data structure problem we wish to maintain an encoding of some data in memory, in such a way that we may efficiently carry out a sequence of queries and updates to the data. A long-standing open problem in this area is to prove an unconditional polynomial lower bound of a trade-off between the update time and the query time of an adaptive dynamic data structure computing some explicit function. Ko and Weinstein provided such lower bound for a restricted class of semi-adaptive data structures, which compute the Disjointness function. There, the data are subsets x₁,… ,x_k and y of {1,… ,n}, the updates can modify y (by inserting and removing elements), and the queries are an index i ∈ {1,… ,k} (query i should answer whether x_i and y are disjoint, i.e., it should compute the Disjointness function applied to (x_i, y)). The semi-adaptiveness places a restriction in how the data structure can be accessed in order to answer a query. We generalize the lower bound of Ko and Weinstein to work not just for the Disjointness, but for any function having high complexity under the smooth corruption bound.
Cite as

Pavel Dvořák and Bruno Loff. Lower Bounds for Semi-adaptive Data Structures via Corruption. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dvorak_et_al:LIPIcs.FSTTCS.2020.20,
  author =	{Dvo\v{r}\'{a}k, Pavel and Loff, Bruno},
  title =	{{Lower Bounds for Semi-adaptive Data Structures via Corruption}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.20},
  URN =		{urn:nbn:de:0030-drops-132617},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.20},
  annote =	{Keywords: semi-adaptive dynamic data structure, polynomial lower bound, corruption bound, information theory}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.18
Target Set Selection in Dense Graph Classes

Authors: Pavel Dvorák, Dusan Knop, and Tomás Toufar

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
In this paper we study the Target Set Selection problem from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex the task is to find a set of vertices (called a target set) to activate at the beginning which activates the whole graph during the following iterative process. A vertex outside the active set becomes active if the number of so far activated vertices in its neighborhood is at least its threshold. We give two parameterized algorithms for a special case where each vertex has the threshold set to the half of its neighbors (the so called Majority Target Set Selection problem) for parameterizations by the neighborhood diversity and the twin cover number of the input graph. We complement these results from the negative side. We give a hardness proof for the Majority Target Set Selection problem when parameterized by (a restriction of) the modular-width - a natural generalization of both previous structural parameters. We show that the Target Set Selection problem parameterized by the neighborhood diversity when there is no restriction on the thresholds is W[1]-hard.
Cite as

Pavel Dvorák, Dusan Knop, and Tomás Toufar. Target Set Selection in Dense Graph Classes. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.ISAAC.2018.18,
  author =	{Dvor\'{a}k, Pavel and Knop, Dusan and Toufar, Tom\'{a}s},
  title =	{{Target Set Selection in Dense Graph Classes}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{18:1--18:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.18},
  URN =		{urn:nbn:de:0030-drops-99666},
  doi =		{10.4230/LIPIcs.ISAAC.2018.18},
  annote =	{Keywords: parameterized complexity, target set selection, dense graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.26
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

Authors: Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.
Cite as

Pavel Dvorák, Andreas Emil Feldmann, Dušan Knop, Tomáš Masarík, Tomáš Toufar, and Pavel Veselý. Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dvorak_et_al:LIPIcs.STACS.2018.26,
  author =	{Dvor\'{a}k, Pavel and Feldmann, Andreas Emil and Knop, Du\v{s}an and Masar{\'\i}k, Tom\'{a}\v{s} and Toufar, Tom\'{a}\v{s} and Vesel\'{y}, Pavel},
  title =	{{Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.26},
  URN =		{urn:nbn:de:0030-drops-85158},
  doi =		{10.4230/LIPIcs.STACS.2018.26},
  annote =	{Keywords: Steiner Tree, Steiner Forest, Approximation Algorithms, Parameterized Algorithms, Lossy Kernelization}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.21
Lower Bounds for Elimination via Weak Regularity

Authors: Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let f: {0,1}^2n -> {0,1} be any boolean function. Alice and Bob get k inputs x_1, ..., x_k and y_1, ..., y_k respectively, with x_i,y_i in {0,1}^n. They want to output a k-bit vector v, such that there exists one index i for which v_i is not equal f(x_i,y_i). We prove a general result lower bounding the randomized communication complexity of the elimination problem for f using its discrepancy. Consequently, we obtain strong lower bounds for the functions Inner-Product and Greater-Than, that work for exponentially larger values of k than the best previous bounds. To prove our result, we use a pseudo-random notion called regularity that was first used by Raz and Wigderson. We show that functions with small discrepancy are regular. We also observe that a weaker notion, that we call weak-regularity, already implies hardness of elimination. Finally, we give a different proof, borrowing ideas from Viola, to show that Greater-Than is weakly regular.
Cite as

Arkadev Chattopadhyay, Pavel Dvorák, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Lower Bounds for Elimination via Weak Regularity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chattopadhyay_et_al:LIPIcs.STACS.2017.21,
  author =	{Chattopadhyay, Arkadev and Dvor\'{a}k, Pavel and Kouck\'{y}, Michal and Loff, Bruno and Mukhopadhyay, Sagnik},
  title =	{{Lower Bounds for Elimination via Weak Regularity}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{21:1--21:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.21},
  URN =		{urn:nbn:de:0030-drops-70128},
  doi =		{10.4230/LIPIcs.STACS.2017.21},
  annote =	{Keywords: communication complexity, elimination, discrepancy, regularity, greater-than}
}
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