737 Search Results for "Mihal'�k, Mat�s"


Document
Contributions to the Domino Problem: Seeding, Recurrence and Satisfiability

Authors: Nicolás Bitar

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
We study the seeded domino problem, the recurring domino problem and the k-SAT problem on finitely generated groups. These problems are generalization of their original versions on ℤ² that were shown to be undecidable using the domino problem. We show that the seeded and recurring domino problems on a group are invariant under changes in the generating set, are many-one reduced from the respective problems on subgroups, and are positive equivalent to the problems on finite index subgroups. This leads to showing that the recurring domino problem is decidable for free groups. Coupled with the invariance properties, we conjecture that the only groups in which the seeded and recurring domino problems are decidable are virtually free groups. In the case of the k-SAT problem, we introduce a new generalization that is compatible with decision problems on finitely generated groups. We show that the subgroup membership problem many-one reduces to the 2-SAT problem, that in certain cases the k-SAT problem many one reduces to the domino problem, and finally that the domino problem reduces to 3-SAT for the class of scalable groups.

Cite as

Nicolás Bitar. Contributions to the Domino Problem: Seeding, Recurrence and Satisfiability. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bitar:LIPIcs.STACS.2024.17,
  author =	{Bitar, Nicol\'{a}s},
  title =	{{Contributions to the Domino Problem: Seeding, Recurrence and Satisfiability}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{17:1--17:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.17},
  URN =		{urn:nbn:de:0030-drops-197275},
  doi =		{10.4230/LIPIcs.STACS.2024.17},
  annote =	{Keywords: Tilings, Domino problem, SAT, Computability, Finitely generated groups}
}
Document
Local Certification of Local Properties: Tight Bounds, Trade-Offs and New Parameters

Authors: Nicolas Bousquet, Laurent Feuilloley, and Sébastien Zeitoun

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
Local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. For many classic global properties, like checking the acyclicity of the network, the optimal size of the certificates depends on the size of the network, n. In this paper, we focus on properties for which the size of the certificates does not depend on n but on other parameters. We focus on three such important properties and prove tight bounds for all of them. Namely, we prove that the optimal certification size is: Θ(log k) for k-colorability (and even exactly ⌈ log k ⌉ bits in the anonymous model while previous works had only proved a 2-bit lower bound); (1/2)log t+o(log t) for dominating sets at distance t (an unexpected and tighter-than-usual bound) ; and Θ(log Δ) for perfect matching in graphs of maximum degree Δ (the first non-trivial bound parameterized by Δ). We also prove some surprising upper bounds, for example, certifying the existence of a perfect matching in a planar graph can be done with only two bits. In addition, we explore various specific cases for these properties, in particular improving our understanding of the trade-off between locality of the verification and certificate size.

Cite as

Nicolas Bousquet, Laurent Feuilloley, and Sébastien Zeitoun. Local Certification of Local Properties: Tight Bounds, Trade-Offs and New Parameters. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bousquet_et_al:LIPIcs.STACS.2024.21,
  author =	{Bousquet, Nicolas and Feuilloley, Laurent and Zeitoun, S\'{e}bastien},
  title =	{{Local Certification of Local Properties: Tight Bounds, Trade-Offs and New Parameters}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{21:1--21:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.21},
  URN =		{urn:nbn:de:0030-drops-197317},
  doi =		{10.4230/LIPIcs.STACS.2024.21},
  annote =	{Keywords: Local certification, local properties, proof-labeling schemes, locally checkable proofs, optimal certification size, colorability, dominating set, perfect matching, fault-tolerance, graph structure}
}
Document
Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs

Authors: Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).

Cite as

Marek Filakovský, Tamio-Vesa Nakajima, Jakub Opršal, Gianluca Tasinato, and Uli Wagner. Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{filakovsky_et_al:LIPIcs.STACS.2024.34,
  author =	{Filakovsk\'{y}, Marek and Nakajima, Tamio-Vesa and Opr\v{s}al, Jakub and Tasinato, Gianluca and Wagner, Uli},
  title =	{{Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{34:1--34:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.34},
  URN =		{urn:nbn:de:0030-drops-197445},
  doi =		{10.4230/LIPIcs.STACS.2024.34},
  annote =	{Keywords: constraint satisfaction problem, hypergraph colouring, promise problem, topological methods}
}
Document
The AC⁰-Complexity of Visibly Pushdown Languages

Authors: Stefan Göller and Nathan Grosshans

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
We study the question of which visibly pushdown languages (VPLs) are in the complexity class AC⁰ and how to effectively decide this question. Our contribution is to introduce a particular subclass of one-turn VPLs, called intermediate VPLs, for which the raised question is entirely unclear: to the best of our knowledge our research community is unaware of containment or non-containment in AC⁰ for any language in our newly introduced class. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs exactly one of the following: that its language L is in AC⁰, some m ≥ 2 such that MODₘ (the words over {0,1} having a number of 1’s divisible by m) is constant-depth reducible to L (implying that L is not in AC⁰), or a finite disjoint union of intermediate VPLs that L is constant-depth equivalent to. In the latter of the three cases one can moreover effectively compute k,l ∈ ℕ_{> 0} with k≠l such that the concrete intermediate VPL L(S → ε ∣ ac^{k-1}Sb₁ ∣ ac^{l-1}Sb₂) is constant-depth reducible to the language L. Due to their particular nature we conjecture that either all intermediate VPLs are in AC⁰ or all are not. As a corollary of our main result we obtain that in case the input language is a visibly counter language our algorithm can effectively determine if it is in AC⁰ - hence our main result generalizes a result by Krebs et al. stating that it is decidable if a given visibly counter language is in AC⁰ (when restricted to well-matched words). For our proofs we revisit so-called Ext-algebras (introduced by Czarnetzki et al.), which are closely related to forest algebras (introduced by Bojańczyk and Walukiewicz), and use Green’s relations.

Cite as

Stefan Göller and Nathan Grosshans. The AC⁰-Complexity of Visibly Pushdown Languages. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{goller_et_al:LIPIcs.STACS.2024.38,
  author =	{G\"{o}ller, Stefan and Grosshans, Nathan},
  title =	{{The AC⁰-Complexity of Visibly Pushdown Languages}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{38:1--38:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.38},
  URN =		{urn:nbn:de:0030-drops-197483},
  doi =		{10.4230/LIPIcs.STACS.2024.38},
  annote =	{Keywords: Visibly pushdown languages, Circuit Complexity, AC0}
}
Document
A Faster Algorithm for Vertex Cover Parameterized by Solution Size

Authors: David G. Harris and N. S. Narayanaswamy

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
We describe a new algorithm for vertex cover with runtime O^*(1.25284^k), where k is the size of the desired solution and O^* hides polynomial factors in the input size. This improves over the previous runtime of O^*(1.2738^k) due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a measure which simultaneously tracks k as well as the optimal value λ of the vertex cover LP relaxation. This allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both k and μ = k - λ are decreased at each step. There can be local obstructions in the graph that prevent μ from decreasing in this process; we develop a number of novel branching steps to handle these situations.

Cite as

David G. Harris and N. S. Narayanaswamy. A Faster Algorithm for Vertex Cover Parameterized by Solution Size. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{harris_et_al:LIPIcs.STACS.2024.40,
  author =	{Harris, David G. and Narayanaswamy, N. S.},
  title =	{{A Faster Algorithm for Vertex Cover Parameterized by Solution Size}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{40:1--40:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.40},
  URN =		{urn:nbn:de:0030-drops-197508},
  doi =		{10.4230/LIPIcs.STACS.2024.40},
  annote =	{Keywords: Vertex cover, FPT, Graph algorithm}
}
Document
Decremental Sensitivity Oracles for Covering and Packing Minors

Authors: Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Peter Strulo

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
In this paper, we present the first decremental fixed-parameter sensitivity oracles for a number of basic covering and packing problems on graphs. In particular, we obtain the first decremental sensitivity oracles for Vertex Planarization (delete k vertices to make the graph planar) and Cycle Packing (pack k vertex-disjoint cycles in the given graph). That is, we give a sensitivity oracle that preprocesses the given graph in time f(k,𝓁)n^{{O}(1)} such that, when given a set of 𝓁 edge deletions, the data structure decides in time f(k,𝓁) whether the updated graph is a positive instance of the problem. These results are obtained as a corollary of our central result, which is the first decremental sensitivity oracle for Topological Minor Deletion (cover all topological minors in the input graph that belong to a specified set, using k vertices). Though our methodology closely follows the literature, we are able to produce the first explicit bounds on the preprocessing and query times for several problems. We also initiate the study of fixed-parameter sensitivity oracles with so-called structural parameterizations and give sufficient conditions for the existence of fixed-parameter sensitivity oracles where the parameter is just the treewidth of the graph. In contrast, all existing literature on this topic and the aforementioned results in this paper assume a bound on the solution size (a weaker parameter than treewidth for many problems). As corollaries, we obtain decremental sensitivity oracles for well-studied problems such as Vertex Cover and Dominating Set when only the treewidth of the input graph is bounded. A feature of our methodology behind these results is that we are able to obtain query times independent of treewidth.

Cite as

Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Peter Strulo. Decremental Sensitivity Oracles for Covering and Packing Minors. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{kanesh_et_al:LIPIcs.STACS.2024.44,
  author =	{Kanesh, Lawqueen and Panolan, Fahad and Ramanujan, M. S. and Strulo, Peter},
  title =	{{Decremental Sensitivity Oracles for Covering and Packing Minors}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{44:1--44:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.44},
  URN =		{urn:nbn:de:0030-drops-197544},
  doi =		{10.4230/LIPIcs.STACS.2024.44},
  annote =	{Keywords: Sensitivity oracles, Data Structures, FPT algorithms}
}
Document
The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds

Authors: Karl Bringmann, Allan Grønlund, Marvin Künnemann, and Kasper Green Larsen

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


Abstract
We pose the fine-grained hardness hypothesis that the textbook algorithm for the NFA Acceptance problem is optimal up to subpolynomial factors, even for dense NFAs and fixed alphabets. We show that this barrier appears in many variations throughout the algorithmic literature by introducing a framework of Colored Walk problems. These yield fine-grained equivalent formulations of the NFA Acceptance problem as problems concerning detection of an s-t-walk with a prescribed color sequence in a given edge- or node-colored graph. For NFA Acceptance on sparse NFAs (or equivalently, Colored Walk in sparse graphs), a tight lower bound under the Strong Exponential Time Hypothesis has been rediscovered several times in recent years. We show that our hardness hypothesis, which concerns dense NFAs, has several interesting implications: - It gives a tight lower bound for Context-Free Language Reachability. This proves conditional optimality for the class of 2NPDA-complete problems, explaining the cubic bottleneck of interprocedural program analysis. - It gives a tight (n+nm^{1/3})^{1-o(1)} lower bound for the Word Break problem on strings of length n and dictionaries of total size m. - It implies the popular OMv hypothesis. Since the NFA acceptance problem is a static (i.e., non-dynamic) problem, this provides a static reason for the hardness of many dynamic problems. Thus, a proof of the NFA Acceptance hypothesis would resolve several interesting barriers. Conversely, a refutation of the NFA Acceptance hypothesis may lead the way to attacking the current barriers observed for Context-Free Language Reachability, the Word Break problem and the growing list of dynamic problems proven hard under the OMv hypothesis.

Cite as

Karl Bringmann, Allan Grønlund, Marvin Künnemann, and Kasper Green Larsen. The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 22:1-22:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bringmann_et_al:LIPIcs.ITCS.2024.22,
  author =	{Bringmann, Karl and Gr{\o}nlund, Allan and K\"{u}nnemann, Marvin and Larsen, Kasper Green},
  title =	{{The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{22:1--22:25},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.22},
  URN =		{urn:nbn:de:0030-drops-195500},
  doi =		{10.4230/LIPIcs.ITCS.2024.22},
  annote =	{Keywords: Fine-grained complexity theory, non-deterministic finite automata}
}
Document
Simple and Optimal Online Contention Resolution Schemes for k-Uniform Matroids

Authors: Atanas Dinev and S. Matthew Weinberg

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


Abstract
We provide a simple (1-O(1/(√{k)}))-selectable Online Contention Resolution Scheme for k-uniform matroids against a fixed-order adversary. If A_i and G_i denote the set of selected elements and the set of realized active elements among the first i (respectively), our algorithm selects with probability 1-1/(√{k)} any active element i such that |A_{i-1}| + 1 ≤ (1-1/(√{k)})⋅ 𝔼[|G_i|]+√k. This implies a (1-O(1/(√{k)})) prophet inequality against fixed-order adversaries for k-uniform matroids that is considerably simpler than previous algorithms [Alaei, 2014; Azar et al., 2014; Jiang et al., 2022]. We also prove that no OCRS can be (1-Ω(√{(log k)/k}))-selectable for k-uniform matroids against an almighty adversary. This guarantee is matched by the (known) simple greedy algorithm that selects every active element with probability 1-Θ(√{(log k)/k}) [Hajiaghayi et al., 2007].

Cite as

Atanas Dinev and S. Matthew Weinberg. Simple and Optimal Online Contention Resolution Schemes for k-Uniform Matroids. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{dinev_et_al:LIPIcs.ITCS.2024.39,
  author =	{Dinev, Atanas and Weinberg, S. Matthew},
  title =	{{Simple and Optimal Online Contention Resolution Schemes for k-Uniform Matroids}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{39:1--39:23},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.39},
  URN =		{urn:nbn:de:0030-drops-195677},
  doi =		{10.4230/LIPIcs.ITCS.2024.39},
  annote =	{Keywords: online contention resolutions schemes, prophet inequalities, online algorithms, approximation algorithms}
}
Document
An Improved Protocol for ExactlyN with More Than 3 Players

Authors: Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, and Adi Shraibman

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


Abstract
The ExactlyN problem in the number-on-forehead (NOF) communication setting asks k players, each of whom can see every input but their own, if the k input numbers add up to N. Introduced by Chandra, Furst and Lipton in 1983, ExactlyN is important for its role in understanding the strength of randomness in communication complexity with many players. It is also tightly connected to the field of combinatorics: its k-party NOF communication complexity is related to the size of the largest corner-free subset in [N]^{k-1}. In 2021, Linial and Shraibman gave more efficient protocols for ExactlyN for 3 players. As an immediate consequence, this also gave a new construction of larger corner-free subsets in [N]². Later that year Green gave a further refinement to their argument. These results represent the first improvements to the highest-order term for k = 3 since the famous work of Behrend in 1946. In this paper we give a corresponding improvement to the highest-order term for k > 3, the first since Rankin in 1961. That is, we give a more efficient protocol for ExactlyN as well as larger corner-free sets in higher dimensions. Nearly all previous results in this line of research approached the problem from the combinatorics perspective, implicitly resulting in non-constructive protocols for ExactlyN. Approaching the problem from the communication complexity point of view and constructing explicit protocols for ExactlyN was key to the improvements in the k = 3 setting. As a further contribution we provide explicit protocols for ExactlyN for any number of players which serves as a base for our improvement.

Cite as

Lianna Hambardzumyan, Toniann Pitassi, Suhail Sherif, Morgan Shirley, and Adi Shraibman. An Improved Protocol for ExactlyN with More Than 3 Players. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{hambardzumyan_et_al:LIPIcs.ITCS.2024.58,
  author =	{Hambardzumyan, Lianna and Pitassi, Toniann and Sherif, Suhail and Shirley, Morgan and Shraibman, Adi},
  title =	{{An Improved Protocol for ExactlyN with More Than 3 Players}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{58:1--58:23},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.58},
  URN =		{urn:nbn:de:0030-drops-195868},
  doi =		{10.4230/LIPIcs.ITCS.2024.58},
  annote =	{Keywords: Corner-free sets, number-on-forehead communication}
}
Document
Exponential-Time Approximation Schemes via Compression

Authors: Tanmay Inamdar, Madhumita Kundu, Pekka Parviainen, M. S. Ramanujan, and Saket Saurabh

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


Abstract
In this paper, we give a framework to design exponential-time approximation schemes for basic graph partitioning problems such as k-way cut, Multiway Cut, Steiner k-cut and Multicut, where the goal is to minimize the number of edges going across the parts. Our motivation to focus on approximation schemes for these problems comes from the fact that while it is possible to solve them exactly in 2^nn^{{𝒪}(1)} time (note that this is already faster than brute-forcing over all partitions or edge sets), it is not known whether one can do better. Using our framework, we design the first (1+ε)-approximation algorithms for the above problems that run in time 2^{f(ε)n} (for f(ε) < 1) for all these problems. As part of our framework, we present two compression procedures. The first of these is a "lossless" procedure, which is inspired by the seminal randomized contraction algorithm for Global Min-cut of Karger [SODA '93]. Here, we reduce the graph to an equivalent instance where the total number of edges is linearly bounded in the number of edges in an optimal solution of the original instance. Following this, we show how a careful combination of greedy choices and the best exact algorithm for the respective problems can exploit this structure and lead to our approximation schemes. Our first compression procedure bounds the number of edges linearly in the optimal solution, but this could still leave a dense graph as the solution size could be superlinear in the number of vertices. However, for several problems, it is known that they admit significantly faster algorithms on instances where solution size is linear in the number of vertices, in contrast to general instances. Hence, a natural question arises here. Could one reduce the solution size to linear in the number of vertices, at least in the case where we are willing to settle for a near-optimal solution, so that the aforementioned faster algorithms could be exploited? In the second compression procedure, using cut sparsifiers (this time, inspired by Benczúr and Karger [STOC '96]) we introduce "solution linearization" as a methodology to give an approximation-preserving reduction to the regime where solution size is linear in the number of vertices for certain cut problems. Using this, we obtain the first polynomial-space approximation schemes faster than 2^nn^{{𝒪}(1)} for Minimum bisection and Edge Bipartization. Along the way, we also design the first polynomial-space exact algorithms for these problems that run in time faster than 2^nn^{{𝒪}(1)}, in the regime where solution size is linear in the number of vertices. The use of randomized contraction and cut sparsifiers in the exponential-time setting is novel to the best of our knowledge and forms our conceptual contribution.

Cite as

Tanmay Inamdar, Madhumita Kundu, Pekka Parviainen, M. S. Ramanujan, and Saket Saurabh. Exponential-Time Approximation Schemes via Compression. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 64:1-64:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{inamdar_et_al:LIPIcs.ITCS.2024.64,
  author =	{Inamdar, Tanmay and Kundu, Madhumita and Parviainen, Pekka and Ramanujan, M. S. and Saurabh, Saket},
  title =	{{Exponential-Time Approximation Schemes via Compression}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{64:1--64:22},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.64},
  URN =		{urn:nbn:de:0030-drops-195920},
  doi =		{10.4230/LIPIcs.ITCS.2024.64},
  annote =	{Keywords: Exponential-Time Algorithms, Approximation Algorithms, Graph Algorithms, Cut Problems}
}
Document
A Combinatorial Approach to Robust PCA

Authors: Weihao Kong, Mingda Qiao, and Rajat Sen

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


Abstract
We study the problem of recovering Gaussian data under adversarial corruptions when the noises are low-rank and the corruptions are on the coordinate level. Concretely, we assume that the Gaussian noises lie in an unknown k-dimensional subspace U ⊆ ℝ^d, and s randomly chosen coordinates of each data point fall into the control of an adversary. This setting models the scenario of learning from high-dimensional yet structured data that are transmitted through a highly-noisy channel, so that the data points are unlikely to be entirely clean. Our main result is an efficient algorithm that, when ks² = O(d), recovers every single data point up to a nearly-optimal 𝓁₁ error of Õ(ks/d) in expectation. At the core of our proof is a new analysis of the well-known Basis Pursuit (BP) method for recovering a sparse signal, which is known to succeed under additional assumptions (e.g., incoherence or the restricted isometry property) on the underlying subspace U. In contrast, we present a novel approach via studying a natural combinatorial problem and show that, over the randomness in the support of the sparse signal, a high-probability error bound is possible even if the subspace U is arbitrary.

Cite as

Weihao Kong, Mingda Qiao, and Rajat Sen. A Combinatorial Approach to Robust PCA. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 70:1-70:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{kong_et_al:LIPIcs.ITCS.2024.70,
  author =	{Kong, Weihao and Qiao, Mingda and Sen, Rajat},
  title =	{{A Combinatorial Approach to Robust PCA}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{70:1--70:22},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.70},
  URN =		{urn:nbn:de:0030-drops-195984},
  doi =		{10.4230/LIPIcs.ITCS.2024.70},
  annote =	{Keywords: Robust PCA, Sparse Recovery, Robust Statistics}
}
Document
On the Size Overhead of Pairwise Spanners

Authors: Ofer Neiman and Idan Shabat

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


Abstract
Given an undirected possibly weighted n-vertex graph G = (V,E) and a set 𝒫 ⊆ V² of pairs, a subgraph S = (V,E') is called a P-pairwise α-spanner of G, if for every pair (u,v) ∈ 𝒫 we have d_S(u,v) ≤ α⋅ d_G(u,v). The parameter α is called the stretch of the spanner, and its size overhead is define as |E'|/|P|. A surprising connection was recently discussed between the additive stretch of (1+ε,β)-spanners, to the hopbound of (1+ε,β)-hopsets. A long sequence of works showed that if the spanner/hopset has size ≈ n^{1+1/k} for some parameter k ≥ 1, then β≈(1/ε)^{log k}. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if |P|≈ n^{1+1/k}, then a P-pairwise (1+ε)-spanner must have size at least β⋅ |P| with β≈(1/ε)^{log k} (a near matching upper bound was recently shown in [Michael Elkin and Idan Shabat, 2023]). That is, the size overhead of pairwise spanners has similar bounds to the hopbound of hopsets, and to the additive stretch of spanners. We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for P-pairwise α-spanners. In particular, we show that if |P|≈ n^{1+1/k}, then the size overhead is β≈k/α. A source-wise spanner is a special type of pairwise spanner, for which P = A×V for some A ⊆ V. A prioritized spanner is given also a ranking of the vertices V = (v₁,… ,v_n), and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions: from pairwise spanners to source-wise spanners to prioritized spanners, we improve on the state-of-the-art results for source-wise and prioritized spanners. Since our spanners can be equipped with a path-reporting mechanism, we also substantially improve the known bounds for path-reporting prioritized distance oracles. Specifically, we provide a path-reporting distance oracle, with size O(n⋅(log log n)²), that has a constant stretch for any query that contains a vertex ranked among the first n^{1-δ} vertices (for any constant δ > 0). Such a result was known before only for non-path-reporting distance oracles.

Cite as

Ofer Neiman and Idan Shabat. On the Size Overhead of Pairwise Spanners. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 83:1-83:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{neiman_et_al:LIPIcs.ITCS.2024.83,
  author =	{Neiman, Ofer and Shabat, Idan},
  title =	{{On the Size Overhead of Pairwise Spanners}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{83:1--83:22},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.83},
  URN =		{urn:nbn:de:0030-drops-196110},
  doi =		{10.4230/LIPIcs.ITCS.2024.83},
  annote =	{Keywords: Graph Algorithms, Shortest Paths, Spanners}
}
Document
Differentially Private Approximate Pattern Matching

Authors: Teresa Anna Steiner

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


Abstract
Differential privacy is the de facto privacy standard in data analysis and widely researched in various application areas. On the other hand, analyzing sequences, or strings, is essential to many modern data analysis tasks, and those data often include highly sensitive personal data. While the problem of sanitizing sequential data to protect privacy has received growing attention, there is a surprising lack of theoretical studies of algorithms analyzing sequential data that preserve differential privacy while giving provable guarantees on the accuracy of such an algorithm. The goal of this paper is to initiate such a study. Specifically, in this paper, we consider the k-approximate pattern matching problem under differential privacy, where the goal is to report or count all substrings of a given string S which have a Hamming distance at most k to a pattern P, or decide whether such a substring exists. In our definition of privacy, individual positions of the string S are protected. To be able to answer queries under differential privacy, we allow some slack on k, i.e. we allow reporting or counting substrings of S with a distance at most (1+γ)k+α to P, for a multiplicative error γ and an additive error α. We analyze which values of α and γ are necessary or sufficient to solve the k-approximate pattern matching problem while satisfying ε-differential privacy. Let n denote the length of S. We give - an ε-differentially private algorithm with an additive error of O(ε^{-1}log n) and no multiplicative error for the existence variant; - an ε-differentially private algorithm with an additive error O(ε^{-1}max(k,log n)⋅log n) for the counting variant; - an ε-differentially private algorithm with an additive error of O(ε^{-1}log n) and multiplicative error O(1) for the reporting variant for a special class of patterns. The error bounds hold with high probability. All of these algorithms return a witness, that is, if there exists a substring of S with distance at most k to P, then the algorithm returns a substring of S with distance at most (1+γ)k+α to P. Further, we complement these results by a lower bound, showing that any algorithm for the existence variant which also returns a witness must have an additive error of Ω(ε^{-1}log n) with constant probability.

Cite as

Teresa Anna Steiner. Differentially Private Approximate Pattern Matching. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 94:1-94:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{steiner:LIPIcs.ITCS.2024.94,
  author =	{Steiner, Teresa Anna},
  title =	{{Differentially Private Approximate Pattern Matching}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{94:1--94:18},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.94},
  URN =		{urn:nbn:de:0030-drops-196223},
  doi =		{10.4230/LIPIcs.ITCS.2024.94},
  annote =	{Keywords: Differential privacy, pattern matching}
}
Document
Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness

Authors: Iddo Tzameret and Lu-Ming Zhang

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


Abstract
We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich’s original work [S. Rudich, 1997], and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following: Demi-bit stretch: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are secure against nondeterministic statistical tests [S. Rudich, 1997]. They were introduced to rule out certain approaches to proving strong complexity lower bounds beyond the limitations set out by the Natural Proofs barrier of Razborov and Rudich [A. A. Razborov and S. Rudich, 1997]. Whether demi-bits are stretchable at all had been an open problem since their introduction. We answer this question affirmatively by showing that: every demi-bit b:{0,1}ⁿ → {0,1}^{n+1} can be stretched into sublinear many demi-bits b':{0,1}ⁿ → {0,1}^{n+n^{c}}, for every constant 0 < c < 1. Average-case hardness: Using work by Santhanam [Rahul Santhanam, 2020], we apply our results to obtain new average-case Kolmogorov complexity results: we show that K^{poly}[n-O(1)] is zero-error average-case hard against NP/poly machines iff K^{poly}[n-o(n)] is, where for a function s(n):ℕ → ℕ, K^{poly}[s(n)] denotes the languages of all strings x ∈ {0,1}ⁿ for which there are (fixed) polytime Turing machines of description-length at most s(n) that output x. Characterising super-bits by nondeterministic unpredictability: In the deterministic setting, Yao [Yao, 1982] proved that super-polynomial hardness of pseudorandom generators is equivalent to ("next-bit") unpredictability. Unpredictability roughly means that given any strict prefix of a random string, it is infeasible to predict the next bit. We initiate the study of unpredictability beyond the deterministic setting (in the cryptographic regime), and characterise the nondeterministic hardness of generators from an unpredictability perspective. Specifically, we propose four stronger notions of unpredictability: NP/poly-unpredictability, coNP/poly-unpredictability, ∩-unpredictability and ∪-unpredictability, and show that super-polynomial nondeterministic hardness of generators lies between ∩-unpredictability and ∪-unpredictability. Characterising super-bits by nondeterministic hard-core predicates: We introduce a nondeterministic variant of hard-core predicates, called super-core predicates. We show that the existence of a super-bit is equivalent to the existence of a super-core of some non-shrinking function. This serves as an analogue of the equivalence between the existence of a strong pseudorandom generator and the existence of a hard-core of some one-way function [Goldreich and Levin, 1989; Håstad et al., 1999], and provides a first alternative characterisation of super-bits. We also prove that a certain class of functions, which may have hard-cores, cannot possess any super-core.

Cite as

Iddo Tzameret and Lu-Ming Zhang. Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 95:1-95:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{tzameret_et_al:LIPIcs.ITCS.2024.95,
  author =	{Tzameret, Iddo and Zhang, Lu-Ming},
  title =	{{Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{95:1--95:22},
  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.2024.95},
  URN =		{urn:nbn:de:0030-drops-196234},
  doi =		{10.4230/LIPIcs.ITCS.2024.95},
  annote =	{Keywords: Pseudorandomness, Cryptography, Natural Proofs, Nondeterminism, Lower bounds}
}
Document
Advanced Composition Theorems for Differential Obliviousness

Authors: Mingxun Zhou, Mengshi Zhao, T-H. Hubert Chan, and Elaine Shi

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


Abstract
Differential obliviousness (DO) is a privacy notion which mandates that the access patterns of a program satisfy differential privacy. Earlier works have shown that in numerous applications, differential obliviousness allows us to circumvent fundamental barriers pertaining to fully oblivious algorithms, resulting in asymptotical (and sometimes even polynomial) performance improvements. Although DO has been applied to various contexts, including the design of algorithms, data structures, and protocols, its compositional properties are not explored until the recent work of Zhou et al. (Eurocrypt'23). Specifically, Zhou et al. showed that the original DO notion is not composable. They then proposed a refinement of DO called neighbor-preserving differential obliviousness (NPDO), and proved a basic composition for NPDO. In Zhou et al.’s basic composition theorem for NPDO, the privacy loss is linear in k for k-fold composition. In comparison, for standard differential privacy, we can enjoy roughly √k loss for k-fold composition by applying the well-known advanced composition theorem given an appropriate parameter range. Therefore, a natural question left open by their work is whether we can also prove an analogous advanced composition for NPDO. In this paper, we answer this question affirmatively. As a key step in proving an advanced composition theorem for NPDO, we define a more operational notion called symmetric NPDO which we prove to be equivalent to NPDO. Using symmetric NPDO as a stepping stone, we also show how to generalize NPDO to more general notions of divergence, resulting in Rényi-NPDO, zero-concentrated-NPDO, Gassian-NPDO, and g-NPDO notions. We also prove composition theorems for these generalized notions of NPDO.

Cite as

Mingxun Zhou, Mengshi Zhao, T-H. Hubert Chan, and Elaine Shi. Advanced Composition Theorems for Differential Obliviousness. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 103:1-103:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{zhou_et_al:LIPIcs.ITCS.2024.103,
  author =	{Zhou, Mingxun and Zhao, Mengshi and Chan, T-H. Hubert and Shi, Elaine},
  title =	{{Advanced Composition Theorems for Differential Obliviousness}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{103:1--103:24},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.103},
  URN =		{urn:nbn:de:0030-drops-196315},
  doi =		{10.4230/LIPIcs.ITCS.2024.103},
  annote =	{Keywords: Differential Privacy, Oblivious Algorithms}
}
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