22 Search Results for "Krauthgamer, Robert"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.30
Lower Bounds for Pseudo-Deterministic Counting in a Stream

Authors: Vladimir Braverman, Robert Krauthgamer, Aditya Krishnan, and Shay Sapir

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
Many streaming algorithms provide only a high-probability relative approximation. These two relaxations, of allowing approximation and randomization, seem necessary - for many streaming problems, both relaxations must be employed simultaneously, to avoid an exponentially larger (and often trivial) space complexity. A common drawback of these randomized approximate algorithms is that independent executions on the same input have different outputs, that depend on their random coins. Pseudo-deterministic algorithms combat this issue, and for every input, they output with high probability the same "canonical" solution. We consider perhaps the most basic problem in data streams, of counting the number of items in a stream of length at most n. Morris’s counter [CACM, 1978] is a randomized approximation algorithm for this problem that uses O(log log n) bits of space, for every fixed approximation factor (greater than 1). Goldwasser, Grossman, Mohanty and Woodruff [ITCS 2020] asked whether pseudo-deterministic approximation algorithms can match this space complexity. Our main result answers their question negatively, and shows that such algorithms must use Ω(√{log n / log log n}) bits of space. Our approach is based on a problem that we call Shift Finding, and may be of independent interest. In this problem, one has query access to a shifted version of a known string F ∈ {0,1}^{3n}, which is guaranteed to start with n zeros and end with n ones, and the goal is to find the unknown shift using a small number of queries. We provide for this problem an algorithm that uses O(√n) queries. It remains open whether poly(log n) queries suffice; if true, then our techniques immediately imply a nearly-tight Ω(log n/log log n) space bound for pseudo-deterministic approximate counting.
Cite as

Vladimir Braverman, Robert Krauthgamer, Aditya Krishnan, and Shay Sapir. Lower Bounds for Pseudo-Deterministic Counting in a Stream. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{braverman_et_al:LIPIcs.ICALP.2023.30,
  author =	{Braverman, Vladimir and Krauthgamer, Robert and Krishnan, Aditya and Sapir, Shay},
  title =	{{Lower Bounds for Pseudo-Deterministic Counting in a Stream}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{30:1--30:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.30},
  URN =		{urn:nbn:de:0030-drops-180827},
  doi =		{10.4230/LIPIcs.ICALP.2023.30},
  annote =	{Keywords: streaming algorithms, pseudo-deterministic, approximate counting}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.31
Clustering Permutations: New Techniques with Streaming Applications

Authors: Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
We study the classical metric k-median clustering problem over a set of input rankings (i.e., permutations), which has myriad applications, from social-choice theory to web search and databases. A folklore algorithm provides a 2-approximate solution in polynomial time for all k = O(1), and works irrespective of the underlying distance measure, so long it is a metric; however, going below the 2-factor is a notorious challenge. We consider the Ulam distance, a variant of the well-known edit-distance metric, where strings are restricted to be permutations. For this metric, Chakraborty, Das, and Krauthgamer [SODA, 2021] provided a (2-δ)-approximation algorithm for k = 1, where δ≈ 2^{-40}. Our primary contribution is a new algorithmic framework for clustering a set of permutations. Our first result is a 1.999-approximation algorithm for the metric k-median problem under the Ulam metric, that runs in time (k log (nd))^{O(k)} nd³ for an input consisting of n permutations over [d]. In fact, our framework is powerful enough to extend this result to the streaming model (where the n input permutations arrive one by one) using only polylogarithmic (in n) space. Additionally, we show that similar results can be obtained even in the presence of outliers, which is presumably a more difficult problem.
Cite as

Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Clustering Permutations: New Techniques with Streaming Applications. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 31:1-31:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chakraborty_et_al:LIPIcs.ITCS.2023.31,
  author =	{Chakraborty, Diptarka and Das, Debarati and Krauthgamer, Robert},
  title =	{{Clustering Permutations: New Techniques with Streaming Applications}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{31:1--31:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.31},
  URN =		{urn:nbn:de:0030-drops-175340},
  doi =		{10.4230/LIPIcs.ITCS.2023.31},
  annote =	{Keywords: Clustering, Approximation Algorithms, Ulam Distance, Rank Aggregation, Streaming}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.58
An Algorithmic Bridge Between Hamming and Levenshtein Distances

Authors: Elazar Goldenberg, Tomasz Kociumaka, Robert Krauthgamer, and Barna Saha

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
The edit distance between strings classically assigns unit cost to every character insertion, deletion, and substitution, whereas the Hamming distance only allows substitutions. In many real-life scenarios, insertions and deletions (abbreviated indels) appear frequently but significantly less so than substitutions. To model this, we consider substitutions being cheaper than indels, with cost 1/a for a parameter a ≥ 1. This basic variant, denoted ED_a, bridges classical edit distance (a = 1) with Hamming distance (a → ∞), leading to interesting algorithmic challenges: Does the time complexity of computing ED_a interpolate between that of Hamming distance (linear time) and edit distance (quadratic time)? What about approximating ED_a? We first present a simple deterministic exact algorithm for ED_a and further prove that it is near-optimal assuming the Orthogonal Vectors Conjecture. Our main result is a randomized algorithm computing a (1+ε)-approximation of ED_a(X,Y), given strings X,Y of total length n and a bound k ≥ ED_a(X,Y). For simplicity, let us focus on k ≥ 1 and a constant ε > 0; then, our algorithm takes Õ(n/a + ak³) time. Unless a = Õ(1), in which case ED_a resembles the standard edit distance, and for the most interesting regime of small enough k, this running time is sublinear in n. We also consider a very natural version that asks to find a (k_I, k_S)-alignment, i.e., an alignment with at most k_I indels and k_S substitutions. In this setting, we give an exact algorithm and, more importantly, an Õ((nk_I)/k_S + k_S k_I³)-time (1,1+ε)-bicriteria approximation algorithm. The latter solution is based on the techniques we develop for ED_a for a = Θ(k_S/k_I), and its running time is again sublinear in n whenever k_I ≪ k_S and the overall distance is small enough. These bounds are in stark contrast to unit-cost edit distance, where state-of-the-art algorithms are far from achieving (1+ε)-approximation in sublinear time, even for a favorable choice of k.
Cite as

Elazar Goldenberg, Tomasz Kociumaka, Robert Krauthgamer, and Barna Saha. An Algorithmic Bridge Between Hamming and Levenshtein Distances. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 58:1-58:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{goldenberg_et_al:LIPIcs.ITCS.2023.58,
  author =	{Goldenberg, Elazar and Kociumaka, Tomasz and Krauthgamer, Robert and Saha, Barna},
  title =	{{An Algorithmic Bridge Between Hamming and Levenshtein Distances}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{58:1--58:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.58},
  URN =		{urn:nbn:de:0030-drops-175615},
  doi =		{10.4230/LIPIcs.ITCS.2023.58},
  annote =	{Keywords: edit distance, Hamming distance, Longest Common Extension queries}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.47
Streaming Algorithms for Geometric Steiner Forest

Authors: Artur Czumaj, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý

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

Abstract
We consider an important generalization of the Steiner tree problem, the Steiner forest problem, in the Euclidean plane: the input is a multiset X ⊆ ℝ², partitioned into k color classes C₁, C₂, …, Cₖ ⊆ X. The goal is to find a minimum-cost Euclidean graph G such that every color class Cᵢ is connected in G. We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to X. Each input point x ∈ X arrives with its color color(x) ∈ [k], and as usual for dynamic geometric streams, the input is restricted to the discrete grid {0, …, Δ}². We design a single-pass streaming algorithm that uses poly(k ⋅ log Δ) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio α₂ (currently 1.1547 ≤ α₂ ≤ 1.214). This approximation guarantee matches the state of the art bound for streaming Steiner tree, i.e., when k = 1. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and has so far not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite approximation requires Ω(k) bits of space.
Cite as

Artur Czumaj, Shaofeng H.-C. Jiang, Robert Krauthgamer, and Pavel Veselý. Streaming Algorithms for Geometric Steiner Forest. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 47:1-47:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{czumaj_et_al:LIPIcs.ICALP.2022.47,
  author =	{Czumaj, Artur and Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Vesel\'{y}, Pavel},
  title =	{{Streaming Algorithms for Geometric Steiner Forest}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{47:1--47:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.47},
  URN =		{urn:nbn:de:0030-drops-163880},
  doi =		{10.4230/LIPIcs.ICALP.2022.47},
  annote =	{Keywords: Steiner forest, streaming, sublinear algorithms, dynamic programming}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2021.73
Maximum-Weight Matching in Sliding Windows and Beyond

Authors: Leyla Biabani, Mark de Berg, and Morteza Monemizadeh

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Abstract
We study the maximum-weight matching problem in the sliding-window model. In this model, we are given an adversarially ordered stream of edges of an underlying edge-weighted graph G(V,E), and a parameter L specifying the window size, and we want to maintain an approximation of the maximum-weight matching of the current graph G(t); here G(t) is defined as the subgraph of G consisting of the edges that arrived during the time interval [max(t-L,1),t], where t is the current time. The goal is to do this with Õ(n) space, where n is the number of vertices of G. We present a deterministic (3.5+ε)-approximation algorithm for this problem, thus significantly improving the (6+ε)-approximation algorithm due to Crouch and Stubbs [Michael S. Crouch and Daniel M. Stubbs, 2014]. We also present a generic machinery for approximating subadditve functions in the sliding-window model. A function f is called subadditive if for every disjoint substreams A, B of a stream S it holds that f(AB) ⩽ f(A) + f(B), where AB denotes the concatenation of A and B. We show that given an α-approximation algorithm for a subadditive function f in the insertion-only model we can maintain a (2α+ε)-approximation of f in the sliding-window model. This improves upon recent result Krauthgamer and Reitblat [Robert Krauthgamer and David Reitblat, 2019], who obtained a (2α²+ε)-approximation.
Cite as

Leyla Biabani, Mark de Berg, and Morteza Monemizadeh. Maximum-Weight Matching in Sliding Windows and Beyond. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{biabani_et_al:LIPIcs.ISAAC.2021.73,
  author =	{Biabani, Leyla and de Berg, Mark and Monemizadeh, Morteza},
  title =	{{Maximum-Weight Matching in Sliding Windows and Beyond}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{73:1--73:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.73},
  URN =		{urn:nbn:de:0030-drops-155061},
  doi =		{10.4230/LIPIcs.ISAAC.2021.73},
  annote =	{Keywords: maximum-weight matching, sliding-window model, approximation algorithm, and subadditve functions}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2021.11
Approximate Trace Reconstruction via Median String (In Average-Case)

Authors: Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
We consider an approximate version of the trace reconstruction problem, where the goal is to recover an unknown string s ∈ {0,1}ⁿ from m traces (each trace is generated independently by passing s through a probabilistic insertion-deletion channel with rate p). We present a deterministic near-linear time algorithm for the average-case model, where s is random, that uses only three traces. It runs in near-linear time Õ(n) and with high probability reports a string within edit distance Õ(p² n) from s, which significantly improves over the straightforward bound of O(pn). Technically, our algorithm computes a (1+ε)-approximate median of the three input traces. To prove its correctness, our probabilistic analysis shows that an approximate median is indeed close to the unknown s. To achieve a near-linear time bound, we have to bypass the well-known dynamic programming algorithm that computes an optimal median in time O(n³).
Cite as

Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Approximate Trace Reconstruction via Median String (In Average-Case). In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2021.11,
  author =	{Chakraborty, Diptarka and Das, Debarati and Krauthgamer, Robert},
  title =	{{Approximate Trace Reconstruction via Median String (In Average-Case)}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{11:1--11:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.11},
  URN =		{urn:nbn:de:0030-drops-155228},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.11},
  annote =	{Keywords: Trace Reconstruction, Approximation Algorithms, Edit Distance, String Median}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.MFCS.2021.5
Sublinear Algorithms for Edit Distance (Invited Talk)

Authors: Barna Saha

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
The edit distance is a way of quantifying how similar two strings are to one another by counting the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. A simple dynamic programming computes the edit distance between two strings of length n in O(n²) time, and a more sophisticated algorithm runs in time O(n+t²) where t is the distance (Landau, Myers and Schmidt, SICOMP 1998). In pursuit of obtaining faster running time, the last couple of decades have seen a flurry of research on approximating edit distance, including polylogarithmic approximation in near-linear time (Andoni, Krauthgamer and Onak, FOCS 2010), and a constant-factor approximation in subquadratic time (Chakrabarty, Das, Goldenberg, Koucký and Saks, FOCS 2018). In this talk, we will discuss recent progress that goes beyond linear time, and studies sublinear time algorithms for edit distance. We will also discuss the role preprocessing might play in designing fast algorithms. This is a joint work with Elazar Goldenberg, Tomasz Kociumaka, Robert Krauthgamer, and Aviad Rubinstein.
Cite as

Barna Saha. Sublinear Algorithms for Edit Distance (Invited Talk). In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, p. 5:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{saha:LIPIcs.MFCS.2021.5,
  author =	{Saha, Barna},
  title =	{{Sublinear Algorithms for Edit Distance}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{5:1--5:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.5},
  URN =		{urn:nbn:de:0030-drops-144452},
  doi =		{10.4230/LIPIcs.MFCS.2021.5},
  annote =	{Keywords: Edit distance, sublinear algorithms, string processing}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.12
Mincut Sensitivity Data Structures for the Insertion of an Edge

Authors: Surender Baswana, Shiv Gupta, and Till Knollmann

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Let G = (V,E) be an undirected graph on n vertices with non-negative capacities on its edges. The mincut sensitivity problem for the insertion of an edge is defined as follows. Build a compact data structure for G and a given set S ⊆ V of vertices that, on receiving any edge (x,y) ∈ S×S of positive capacity as query input, can efficiently report the set of all pairs from S× S whose mincut value increases upon insertion of the edge (x,y) to G. The only result that exists for this problem is for a single pair of vertices (Picard and Queyranne, Mathematical Programming Study, 13 (1980), 8-16). We present the following results for the single source and the all-pairs versions of this problem. 1) Single source: Given any designated source vertex s, there exists a data structure of size 𝒪(|S|) that can output all those vertices from S whose mincut value to s increases upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(|S|). 2) All-pairs: There exists an 𝒪(|S|²) size data structure that can output all those pairs of vertices from S× S whose mincut value gets increased upon insertion of any given edge. The time taken by the data structure to answer any query is 𝒪(k), where k is the number of pairs of vertices whose mincut increases. For both these versions, we also address the problem of reporting the values of the mincuts upon insertion of any given edge. To derive our results, we use interesting insights into the nearest and the farthest mincuts for a pair of vertices. In addition, a crucial result, that we establish and use in our data structures, is that there exists a directed acyclic graph of 𝒪(n) size that compactly stores the farthest mincuts from all vertices of V to a designated vertex s in the graph. We believe that this result is of independent interest, especially, because it also complements a previously existing result by Hariharan et al. (STOC 2007) that the nearest mincuts from all vertices of V to s is a laminar family, and hence, can be stored compactly in a tree of 𝒪(n) size.
Cite as

Surender Baswana, Shiv Gupta, and Till Knollmann. Mincut Sensitivity Data Structures for the Insertion of an Edge. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{baswana_et_al:LIPIcs.ESA.2020.12,
  author =	{Baswana, Surender and Gupta, Shiv and Knollmann, Till},
  title =	{{Mincut Sensitivity Data Structures for the Insertion of an Edge}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{12:1--12:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.12},
  URN =		{urn:nbn:de:0030-drops-128781},
  doi =		{10.4230/LIPIcs.ESA.2020.12},
  annote =	{Keywords: Mincut, Sensitivity, Data Structure}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.ICALP.2020.2
Sketching Graphs and Combinatorial Optimization (Invited Talk)

Authors: Robert Krauthgamer

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
Graph-sketching algorithms summarize an input graph G in a manner that suffices to later answer (perhaps approximately) one or more optimization problems on G, like distances, cuts, and matchings. Two famous examples are the Gomory-Hu tree, which represents all the minimum st-cuts in a graph G using a tree on the same vertex set V(G); and the cut-sparsifier of Benczúr and Karger, which is a sparse graph (often a reweighted subgraph) that approximates every cut in G within factor 1±ε. Another genre of these problems limits the queries to designated terminal vertices, denoted T ⊆ V(G), and the sketch size depends on |T| instead of |V(G)|. The talk will survey this topic, particularly cut and flow problems such as the three examples above. Currently, most known sketches are based on a graph representation, often called edge and vertex sparsification, which leaves room for potential improvements like smaller storage by using another representation, and faster running time to answer a query. These algorithms employ a host of techniques, ranging from combinatorial methods, like graph partitioning and edge or vertex sampling, to standard tools in data-stream algorithms and in sparse recovery. There are also several lower bounds known, either combinatorial (for the graph representation) or based on communication complexity and information theory. Many of the recent efforts focus on characterizing the tradeoff between accuracy and sketch size, yet many intriguing and very accessible problems are still open, and I will describe them in the talk.
Cite as

Robert Krauthgamer. Sketching Graphs and Combinatorial Optimization (Invited Talk). In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{krauthgamer:LIPIcs.ICALP.2020.2,
  author =	{Krauthgamer, Robert},
  title =	{{Sketching Graphs and Combinatorial Optimization}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.2},
  URN =		{urn:nbn:de:0030-drops-124090},
  doi =		{10.4230/LIPIcs.ICALP.2020.2},
  annote =	{Keywords: Sketching, edge sparsification, vertex sparsification, Gomory-Hu tree, mimicking networks, graph sampling, succinct data structures}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.25
New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures

Authors: Diptarka Chakraborty and Keerti Choudhary

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
In this paper, we consider the question of computing sparse subgraphs for any input directed graph G = (V,E) on n vertices and m edges, that preserves reachability and/or strong connectivity structures. - We show O(n+min{|P|√n, n√|P|}) bound on a subgraph that is an 1-fault-tolerant reachability preserver for a given vertex-pair set P ⊆ V× V, i.e., it preserves reachability between any pair of vertices in P under single edge (or vertex) failure. Our result is a significant improvement over the previous best O(n |P|) bound obtained as a corollary of single-source reachability preserver construction. We prove our upper bound by exploiting the special structure of single fault-tolerant reachability preserver for any pair, and then considering the interaction among such structures for different pairs. - In the lower bound side, we show that a 2-fault-tolerant reachability preserver for a vertex-pair set P ⊆ V×V of size Ω(n^ε), for even any arbitrarily small ε, requires at least Ω(n^(1+ε/8)) edges. This refutes the existence of linear-sized dual fault-tolerant preservers for reachability for any polynomial sized vertex-pair set. - We also present the first sub-quadratic bound of at most Õ(k 2^k n^(2-1/k)) size, for strong-connectivity preservers of directed graphs under k failures. To the best of our knowledge no non-trivial bound for this problem was known before, for a general k. We get our result by adopting the color-coding technique of Alon, Yuster, and Zwick [JACM'95].
Cite as

Diptarka Chakraborty and Keerti Choudhary. New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chakraborty_et_al:LIPIcs.ICALP.2020.25,
  author =	{Chakraborty, Diptarka and Choudhary, Keerti},
  title =	{{New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.25},
  URN =		{urn:nbn:de:0030-drops-124327},
  doi =		{10.4230/LIPIcs.ICALP.2020.25},
  annote =	{Keywords: Preservers, Strong-connectivity, Reachability, Fault-tolerant, Graph sparsification}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.47
Scattering and Sparse Partitions, and Their Applications

Authors: Arnold Filtser

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
A partition 𝒫 of a weighted graph G is (σ,τ,Δ)-sparse if every cluster has diameter at most Δ, and every ball of radius Δ/σ intersects at most τ clusters. Similarly, 𝒫 is (σ,τ,Δ)-scattering if instead for balls we require that every shortest path of length at most Δ/σ intersects at most τ clusters. Given a graph G that admits a (σ,τ,Δ)-sparse partition for all Δ > 0, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch O(τσ²log_τ n). Given a graph G that admits a (σ,τ,Δ)-scattering partition for all Δ > 0, we construct a solution for the Steiner Point Removal problem with stretch O(τ³σ³). We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.
Cite as

Arnold Filtser. Scattering and Sparse Partitions, and Their Applications. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 47:1-47:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{filtser:LIPIcs.ICALP.2020.47,
  author =	{Filtser, Arnold},
  title =	{{Scattering and Sparse Partitions, and Their Applications}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{47:1--47:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.47},
  URN =		{urn:nbn:de:0030-drops-124547},
  doi =		{10.4230/LIPIcs.ICALP.2020.47},
  annote =	{Keywords: Scattering partitions, sparse partitions, sparse covers, Steiner point removal, Universal Steiner tree, Universal TSP}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2019.7
Towards a Theory of Parameterized Streaming Algorithms

Authors: Rajesh Chitnis and Graham Cormode

Published in: LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Abstract
Parameterized complexity attempts to give a more fine-grained analysis of the complexity of problems: instead of measuring the running time as a function of only the input size, we analyze the running time with respect to additional parameters. This approach has proven to be highly successful in delineating our understanding of NP-hard problems. Given this success with the TIME resource, it seems but natural to use this approach for dealing with the SPACE resource. First attempts in this direction have considered a few individual problems, with some success: Fafianie and Kratsch [MFCS'14] and Chitnis et al. [SODA'15] introduced the notions of streaming kernels and parameterized streaming algorithms respectively. For example, the latter shows how to refine the Omega(n^2) bit lower bound for finding a minimum Vertex Cover (VC) in the streaming setting by designing an algorithm for the parameterized k-VC problem which uses O(k^{2}log n) bits. In this paper, we initiate a systematic study of graph problems from the paradigm of parameterized streaming algorithms. We first define a natural hierarchy of space complexity classes of FPS, SubPS, SemiPS, SupPS and BrutePS, and then obtain tight classifications for several well-studied graph problems such as Longest Path, Feedback Vertex Set, Dominating Set, Girth, Treewidth, etc. into this hierarchy (see Figure 1 and Table 1). On the algorithmic side, our parameterized streaming algorithms use techniques from the FPT world such as bidimensionality, iterative compression and bounded-depth search trees. On the hardness side, we obtain lower bounds for the parameterized streaming complexity of various problems via novel reductions from problems in communication complexity. We also show a general (unconditional) lower bound for space complexity of parameterized streaming algorithms for a large class of problems inspired by the recently developed frameworks for showing (conditional) kernelization lower bounds. Parameterized algorithms and streaming algorithms are approaches to cope with TIME and SPACE intractability respectively. It is our hope that this work on parameterized streaming algorithms leads to two-way flow of ideas between these two previously separated areas of theoretical computer science.
Cite as

Rajesh Chitnis and Graham Cormode. Towards a Theory of Parameterized Streaming Algorithms. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chitnis_et_al:LIPIcs.IPEC.2019.7,
  author =	{Chitnis, Rajesh and Cormode, Graham},
  title =	{{Towards a Theory of Parameterized Streaming Algorithms}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Jansen, Bart M. P. and Telle, Jan Arne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.7},
  URN =		{urn:nbn:de:0030-drops-114682},
  doi =		{10.4230/LIPIcs.IPEC.2019.7},
  annote =	{Keywords: Parameterized Algorithms, Streaming Algorithms, Kernels}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.FSTTCS.2019.2
Sketching Graphs and Combinatorial Optimization (Invited Talk)

Authors: Robert Krauthgamer

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Abstract
Graph-sketching algorithms summarize an input graph G in a manner that suffices to later answer (perhaps approximately) one or more optimization problems on G, like distances, cuts, and matchings. Two famous examples are the Gomory-Hu tree, which represents all the minimum st-cuts in a graph G using a tree on the same vertex set V(G); and the cut-sparsifier of Benczúr and Karger, which is a sparse graph (often a reweighted subgraph) that approximates every cut in G within factor 1 +/- epsilon. Another genre of these problems limits the queries to designated terminal vertices, denoted T subseteq V(G), and the sketch size depends on |T| instead of |V(G)|. The talk will survey this topic, particularly cut and flow problems such as the three examples above. Currently, most known sketches are based on a graphical representation, often called edge and vertex sparsification, which leaves room for potential improvements like smaller storage by using another representation, and faster running time to answer a query. These algorithms employ a host of techniques, ranging from combinatorial methods, like graph partitioning and edge or vertex sampling, to standard tools in data-stream algorithms and in sparse recovery. There are also several lower bounds known, either combinatorial (for the graphical representation) or based on communication complexity and information theory. Many of the recent efforts focus on characterizing the tradeoff between accuracy and sketch size, yet many intriguing and very accessible problems are still open, and I will describe them in the talk.
Cite as

Robert Krauthgamer. Sketching Graphs and Combinatorial Optimization (Invited Talk). In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{krauthgamer:LIPIcs.FSTTCS.2019.2,
  author =	{Krauthgamer, Robert},
  title =	{{Sketching Graphs and Combinatorial Optimization}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.2},
  URN =		{urn:nbn:de:0030-drops-115646},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.2},
  annote =	{Keywords: Sketching, edge sparsification, vertex sparsification, Gomory-Hu tree, mimicking networks, graph sampling, succinct data structures}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.47
Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1

Authors: Ioannis Z. Emiris, Vasilis Margonis, and Ioannis Psarros

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

Abstract
Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (l_2) metric, but much less for the Manhattan (l_1) metric. Our primary motivation is the approximate nearest neighbor problem in l_1. We exploit its reduction to the decision-with-witness version, called approximate near neighbor, which incurs a roughly logarithmic overhead. In 2007, Indyk and Naor, in the context of approximate nearest neighbors, introduced the notion of nearest neighbor-preserving embeddings. These are randomized embeddings between two metric spaces with guaranteed bounded distortion only for the distances between a query point and a point set. Such embeddings are known to exist for both l_2 and l_1 metrics, as well as for doubling subsets of l_2. The case that remained open were doubling subsets of l_1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of l_1. Our approach is to represent the pointset with a carefully chosen covering set, then randomly project the latter. We study two types of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension. We employ Cauchy variables: certain concentration bounds derived should be of independent interest.
Cite as

Ioannis Z. Emiris, Vasilis Margonis, and Ioannis Psarros. Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{emiris_et_al:LIPIcs.APPROX-RANDOM.2019.47,
  author =	{Emiris, Ioannis Z. and Margonis, Vasilis and Psarros, Ioannis},
  title =	{{Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l\underline1}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{47:1--47:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.47},
  URN =		{urn:nbn:de:0030-drops-112628},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.47},
  annote =	{Keywords: Approximate nearest neighbor, Manhattan metric, randomized embedding}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.7
Faster Algorithms for All-Pairs Bounded Min-Cuts

Authors: Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf

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

Abstract
The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum s-t cut (or just its value) for all pairs of vertices s,t. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the k-bounded case, where the algorithm has to find all pairs with min-cut value less than k, and report only those. The most basic case k=1 is the Transitive Closure (TC) problem, which can be solved in graphs with n vertices and m edges in time O(mn) combinatorially, and in time O(n^{omega}) where omega<2.38 is the matrix-multiplication exponent. These time bounds are conjectured to be optimal. We present new algorithms and conditional lower bounds that advance the frontier for larger k, as follows: - A randomized algorithm for vertex capacities that runs in time {O}((nk)^{omega}). This is only a factor k^omega away from the TC bound, and nearly matches it for all k=n^{o(1)}. - Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time {O}(2^{{O}(k^2)}* mn). The second algorithm can be faster on dense DAGs and runs in time {O}((k log n)^{4^{k+o(k)}}* n^{omega}). Previously, Georgiadis et al. [ICALP 2017], could match the TC bound (up to n^{o(1)} factors) only when k=2, and now our two algorithms match it for all k=o(sqrt{log n}) and k=o(log log n). - The first super-cubic lower bound of n^{omega-1-o(1)} k^2 time under the 4-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting k=n. For combinatorial algorithms, our reduction implies an n^{2-o(1)} k^2 conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC. Our three sets of results are obtained via different techniques. The first one adapts the network coding method of Cheung, Lau, and Leung [SICOMP 2013] to vertex-capacitated digraphs. The second set exploits new insights on the structure of latest cuts together with suitable algebraic tools. The lower bounds arise from a novel reduction of a different structure than the SETH-based constructions.
Cite as

Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf. Faster Algorithms for All-Pairs Bounded Min-Cuts. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{abboud_et_al:LIPIcs.ICALP.2019.7,
  author =	{Abboud, Amir and Georgiadis, Loukas and Italiano, Giuseppe F. and Krauthgamer, Robert and Parotsidis, Nikos and Trabelsi, Ohad and Uzna\'{n}ski, Przemys{\l}aw and Wolleb-Graf, Daniel},
  title =	{{Faster Algorithms for All-Pairs Bounded Min-Cuts}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.7},
  URN =		{urn:nbn:de:0030-drops-105833},
  doi =		{10.4230/LIPIcs.ICALP.2019.7},
  annote =	{Keywords: All-pairs min-cut, k-reachability, network coding, Directed graphs, fine-grained complexity}
}
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