31 Search Results for "Assadi, Sepehr"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.44
On the Power of Regular and Permutation Branching Programs

Authors: Chin Ho Lee, Edward Pyne, and Salil Vadhan

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

Abstract
We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation. - Regular SOBPs of length n and width ⌊w(n+1)/2⌋ can exactly simulate general SOBPs of length n and width w, and moreover an n/2-o(n) blow-up in width is necessary for such a simulation. Our result extends and simplifies prior average-case simulations (Reingold, Trevisan, and Vadhan (STOC 2006), Bogdanov, Hoza, Prakriya, and Pyne (CCC 2022)), in particular implying that weighted pseudorandom generators (Braverman, Cohen, and Garg (SICOMP 2020)) for regular SOBPs of width poly(n) or larger automatically extend to general SOBPs. Furthermore, our simulation also extends to general (even read-many) oblivious branching programs. - There exist natural functions computable by regular SOBPs of constant width that are average-case hard for permutation SOBPs of exponential width. Indeed, we show that Inner-Product mod 2 is average-case hard for arbitrary-order permutation ROBPs of exponential width. - There exist functions computable by constant-width arbitrary-order permutation ROBPs that are worst-case hard for exponential-width SOBPs. - Read-twice permutation branching programs of subexponential width can simulate polynomial-width arbitrary-order ROBPs.
Cite as

Chin Ho Lee, Edward Pyne, and Salil Vadhan. On the Power of Regular and Permutation Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 44:1-44:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lee_et_al:LIPIcs.APPROX/RANDOM.2023.44,
  author =	{Lee, Chin Ho and Pyne, Edward and Vadhan, Salil},
  title =	{{On the Power of Regular and Permutation Branching Programs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{44:1--44:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.44},
  URN =		{urn:nbn:de:0030-drops-188698},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.44},
  annote =	{Keywords: Pseudorandomness, Branching Programs}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.57
On Constructing Spanners from Random Gaussian Projections

Authors: Sepehr Assadi, Michael Kapralov, and Huacheng Yu

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

Abstract
Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA'12) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA'21), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work.
Cite as

Sepehr Assadi, Michael Kapralov, and Huacheng Yu. On Constructing Spanners from Random Gaussian Projections. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 57:1-57:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{assadi_et_al:LIPIcs.APPROX/RANDOM.2023.57,
  author =	{Assadi, Sepehr and Kapralov, Michael and Yu, Huacheng},
  title =	{{On Constructing Spanners from Random Gaussian Projections}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{57:1--57:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.57},
  URN =		{urn:nbn:de:0030-drops-188821},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.57},
  annote =	{Keywords: sketching algorithm, lower bound, graph spanner}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.58
Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance

Authors: Vikrant Ashvinkumar, Sepehr Assadi, Chengyuan Deng, Jie Gao, and Chen Wang

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

Abstract
Structural balance theory studies stability in networks. Given a n-vertex complete graph G = (V,E) whose edges are labeled positive or negative, the graph is considered balanced if every triangle either consists of three positive edges (three mutual "friends"), or one positive edge and two negative edges (two "friends" with a common "enemy"). From a computational perspective, structural balance turns out to be a special case of correlation clustering with the number of clusters at most two. The two main algorithmic problems of interest are: (i) detecting whether a given graph is balanced, or (ii) finding a partition that approximates the frustration index, i.e., the minimum number of edge flips that turn the graph balanced. We study these problems in the streaming model where edges are given one by one and focus on memory efficiency. We provide randomized single-pass algorithms for: (i) determining whether an input graph is balanced with O(log n) memory, and (ii) finding a partition that induces a (1 + ε)-approximation to the frustration index with O(n ⋅ polylog(n)) memory. We further provide several new lower bounds, complementing different aspects of our algorithms such as the need for randomization or approximation. To obtain our main results, we develop a method using pseudorandom generators (PRGs) to sample edges between independently-chosen vertices in graph streaming. Furthermore, our algorithm that approximates the frustration index improves the running time of the state-of-the-art correlation clustering with two clusters (Giotis-Guruswami algorithm [SODA 2006]) from n^O(1/ε²) to O(n²log³n/ε² + n log n ⋅ (1/ε)^O(1/ε⁴)) time for (1+ε)-approximation. These results may be of independent interest.
Cite as

Vikrant Ashvinkumar, Sepehr Assadi, Chengyuan Deng, Jie Gao, and Chen Wang. Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 58:1-58:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ashvinkumar_et_al:LIPIcs.APPROX/RANDOM.2023.58,
  author =	{Ashvinkumar, Vikrant and Assadi, Sepehr and Deng, Chengyuan and Gao, Jie and Wang, Chen},
  title =	{{Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{58:1--58:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.58},
  URN =		{urn:nbn:de:0030-drops-188830},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.58},
  annote =	{Keywords: Streaming algorithms, structural balance, pseudo-randomness generator}
}
Document
DOI: 10.4230/LIPIcs.SEA.2023.21
Maximum Coverage in Sublinear Space, Faster

Authors: Stephen Jaud, Anthony Wirth, and Farhana Choudhury

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
Given a collection of m sets from a universe 𝒰, the Maximum Set Coverage problem consists of finding k sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial time algorithm up to a factor 1-1/e. However, this algorithm does not scale well with the input size. In a streaming context, practical high-quality solutions are found, but with space complexity that scales linearly with respect to the size of the universe n = |𝒰|. However, one randomized streaming algorithm has been shown to produce a 1-1/e-ε approximation of the optimal solution with a space complexity that scales only poly-logarithmically with respect to m and n. In order to achieve such a low space complexity, the authors used two techniques in their multi-pass approach: - F₀-sketching, allows to determine with great accuracy the number of distinct elements in a set using less space than the set itself. - Subsampling, consists of only solving the problem on a subspace of the universe. It is implemented using γ-independent hash functions. This article focuses on the sublinear-space algorithm and highlights the time cost of these two techniques, especially subsampling. We present optimizations that significantly reduce the time complexity of the algorithm. Firstly, we give some optimizations that do not alter the space complexity, number of passes and approximation quality of the original algorithm. In particular, we reanalyze the error bounds to show that the original independence factor of Ω(ε^{-2} k log m) can be fine-tuned to Ω(k log m); we also show how F₀-sketching can be removed. Secondly, we derive a new lower bound for the probability of producing a 1-1/e-ε approximation using only pairwise independence: 1- (4/(c k log m)) compared to 1-(2e/(m^{ck/6})) with Ω(k log m)-independence. Although the theoretical guarantees are weaker, suggesting the approximation quality would suffer, for large streams, our algorithms perform well in practice. Finally, our experimental results show that even a pairwise-independent hash-function sampler does not produce worse solution than the original algorithm, while running significantly faster by several orders of magnitude.
Cite as

Stephen Jaud, Anthony Wirth, and Farhana Choudhury. Maximum Coverage in Sublinear Space, Faster. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 21:1-21:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jaud_et_al:LIPIcs.SEA.2023.21,
  author =	{Jaud, Stephen and Wirth, Anthony and Choudhury, Farhana},
  title =	{{Maximum Coverage in Sublinear Space, Faster}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{21:1--21:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.21},
  URN =		{urn:nbn:de:0030-drops-183715},
  doi =		{10.4230/LIPIcs.SEA.2023.21},
  annote =	{Keywords: streaming algorithms, subsampling, maximum set cover, k-wise independent hash functions}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2023.19
Generalizing Greenwald-Khanna Streaming Quantile Summaries for Weighted Inputs

Authors: Sepehr Assadi, Nirmit Joshi, Milind Prabhu, and Vihan Shah

Published in: LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

Abstract
Estimating quantiles, like the median or percentiles, is a fundamental task in data mining and data science. A (streaming) quantile summary is a data structure that can process a set S of n elements in a streaming fashion and at the end, for any ϕ ∈ (0,1], return a ϕ-quantile of S up to an ε error, i.e., return a ϕ'-quantile with ϕ' = ϕ ± ε. We are particularly interested in comparison-based summaries that only compare elements of the universe under a total ordering and are otherwise completely oblivious of the universe. The best known deterministic quantile summary is the 20-year old Greenwald-Khanna (GK) summary that uses O((1/ε) log{(ε n)}) space [SIGMOD'01]. This bound was recently proved to be optimal for all deterministic comparison-based summaries by Cormode and Vesleý [PODS'20]. In this paper, we study weighted quantiles, a generalization of the quantiles problem, where each element arrives with a positive integer weight which denotes the number of copies of that element being inserted. The only known method of handling weighted inputs via GK summaries is the naive approach of breaking each weighted element into multiple unweighted items, and feeding them one by one to the summary, which results in a prohibitively large update time (proportional to the maximum weight of input elements). We give the first non-trivial extension of GK summaries for weighted inputs and show that it takes O((1/ε) log(εn)) space and O(log(1/ε)+log log(εn)) update time per element to process a stream of length n (under some quite mild assumptions on the range of weights and ε). En route to this, we also simplify the original GK summaries for unweighted quantiles.
Cite as

Sepehr Assadi, Nirmit Joshi, Milind Prabhu, and Vihan Shah. Generalizing Greenwald-Khanna Streaming Quantile Summaries for Weighted Inputs. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 19:1-19:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{assadi_et_al:LIPIcs.ICDT.2023.19,
  author =	{Assadi, Sepehr and Joshi, Nirmit and Prabhu, Milind and Shah, Vihan},
  title =	{{Generalizing Greenwald-Khanna Streaming Quantile Summaries for Weighted Inputs}},
  booktitle =	{26th International Conference on Database Theory (ICDT 2023)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-270-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{255},
  editor =	{Geerts, Floris and Vandevoort, Brecht},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2023.19},
  URN =		{urn:nbn:de:0030-drops-177618},
  doi =		{10.4230/LIPIcs.ICDT.2023.19},
  annote =	{Keywords: Streaming algorithms, Quantile summaries, Rank estimation}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.7
All-Norm Load Balancing in Graph Streams via the Multiplicative Weights Update Method

Authors: Sepehr Assadi, Aaron Bernstein, and Zachary Langley

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

Abstract
In the weighted load balancing problem, the input is an n-vertex bipartite graph between a set of clients and a set of servers, and each client comes with some nonnegative real weight. The output is an assignment that maps each client to one of its adjacent servers, and the load of a server is then the sum of the weights of the clients assigned to it. The goal is to find an assignment that is well-balanced, typically captured by (approximately) minimizing either the 𝓁_∞- or 𝓁₂-norm of the server loads. Generalizing both of these objectives, the all-norm load balancing problem asks for an assignment that approximately minimizes all 𝓁_p-norm objectives for p ≥ 1, including p = ∞, simultaneously. Our main result is a deterministic O(log n)-pass O(1)-approximation semi-streaming algorithm for the all-norm load balancing problem. Prior to our work, only an O(log n)-pass O(log n)-approximation algorithm for the 𝓁_∞-norm objective was known in the semi-streaming setting. Our algorithm uses a novel application of the multiplicative weights update method to a mixed covering/packing convex program for the all-norm load balancing problem involving an infinite number of constraints.
Cite as

Sepehr Assadi, Aaron Bernstein, and Zachary Langley. All-Norm Load Balancing in Graph Streams via the Multiplicative Weights Update Method. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 7:1-7:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{assadi_et_al:LIPIcs.ITCS.2023.7,
  author =	{Assadi, Sepehr and Bernstein, Aaron and Langley, Zachary},
  title =	{{All-Norm Load Balancing in Graph Streams via the Multiplicative Weights Update Method}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{7:1--7:24},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.7},
  URN =		{urn:nbn:de:0030-drops-175106},
  doi =		{10.4230/LIPIcs.ITCS.2023.7},
  annote =	{Keywords: Load Balancing, Semi-Streaming Algorithms, Semi-Matching}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.DISC.2022.1
Graph Coloring, Palette Sparsification, and Beyond (Invited Talk)

Authors: Sepehr Assadi

Published in: LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

Abstract
Graph coloring is a central problem in graph theory and has numerous applications in diverse areas of computer science. An important and well-studied case of graph coloring problems is the (Δ+1) (vertex) coloring problem where Δ is the maximum degree of the graph. Not only does every graph admit a (Δ + 1) coloring, but in fact we can find one quite easily in linear time and space via a greedy algorithm. But are there more efficient algorithms for (Δ+1) coloring that can process massive graphs that even this algorithm cannot handle? This talk overviews recent results that answer this question in affirmative across a variety of models dedicated to processing massive graphs - streaming, sublinear-time, massively parallel computation, distributed communication, etc. - via a single unified approach: Palette Sparsification. We survey the ideas behind these results and techniques, their generalizations to various other coloring problems and even beyond (e.g., to clustering problems), as well as their natural limitations. The talk is based on a series of joint works with Noga Alon, Andrew Chen, Yu Chen, Sanjeev Khanna, Pankaj Kumar, Parth Mittal, Glenn Sun, and Chen Wang.
Cite as

Sepehr Assadi. Graph Coloring, Palette Sparsification, and Beyond (Invited Talk). In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, p. 1:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi:LIPIcs.DISC.2022.1,
  author =	{Assadi, Sepehr},
  title =	{{Graph Coloring, Palette Sparsification, and Beyond}},
  booktitle =	{36th International Symposium on Distributed Computing (DISC 2022)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-255-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{246},
  editor =	{Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.1},
  URN =		{urn:nbn:de:0030-drops-171920},
  doi =		{10.4230/LIPIcs.DISC.2022.1},
  annote =	{Keywords: Graph coloring, Palette Sparsification, Sublinear Algorithms}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.48
Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting

Authors: Sepehr Assadi and Hoai-An Nguyen

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
The h-index is a metric used to measure the impact of a user in a publication setting, such as a member of a social network with many highly liked posts or a researcher in an academic domain with many highly cited publications. Specifically, the h-index of a user is the largest integer h such that at least h publications of the user have at least h units of positive feedback. We design an algorithm that, given query access to the n publications of a user and each publication’s corresponding positive feedback number, outputs a (1± ε)-approximation of the h-index of this user with probability at least 1-δ in time O(n⋅ln(1/δ) / (ε²⋅h)), where h is the actual h-index which is unknown to the algorithm a-priori. We then design a novel lower bound technique that allows us to prove that this bound is in fact asymptotically optimal for this problem in all parameters n,h,ε, and δ. Our work is one of the first in sublinear time algorithms that addresses obtaining asymptotically optimal bounds, especially in terms of the error and confidence parameters. As such, we focus on designing novel techniques for this task. In particular, our lower bound technique seems quite general - to showcase this, we also use our approach to prove an asymptotically optimal lower bound for the problem of estimating the number of triangles in a graph in sublinear time, which now is also optimal in the error and confidence parameters. This latter result improves upon prior lower bounds of Eden, Levi, Ron, and Seshadhri (FOCS'15) for this problem, as well as multiple follow-up works that extended this lower bound to other subgraph counting problems.
Cite as

Sepehr Assadi and Hoai-An Nguyen. Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 48:1-48:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi_et_al:LIPIcs.APPROX/RANDOM.2022.48,
  author =	{Assadi, Sepehr and Nguyen, Hoai-An},
  title =	{{Asymptotically Optimal Bounds for Estimating H-Index in Sublinear Time with Applications to Subgraph Counting}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{48:1--48:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.48},
  URN =		{urn:nbn:de:0030-drops-171708},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.48},
  annote =	{Keywords: Sublinear time algorithms, h-index, asymptotically optimal bounds, lower bounds, subgraph counting}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.53
Space Optimal Vertex Cover in Dynamic Streams

Authors: Kheeran K. Naidu and Vihan Shah

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
We optimally resolve the space complexity for the problem of finding an α-approximate minimum vertex cover (αMVC) in dynamic graph streams. We give a randomised algorithm for αMVC which uses O(n²/α²) bits of space matching Dark and Konrad’s lower bound [CCC 2020] up to constant factors. By computing a random greedy matching, we identify "easy" instances of the problem which can trivially be solved by returning the entire vertex set. The remaining "hard" instances, then have sparse induced subgraphs which we exploit to get our space savings and solve αMVC. Achieving this type of optimality result is crucial for providing a complete understanding of a problem, and it has been gaining interest within the dynamic graph streaming community. For connectivity, Nelson and Yu [SODA 2019] improved the lower bound showing that Ω(n log³ n) bits of space is necessary while Ahn, Guha, and McGregor [SODA 2012] have shown that O(n log³ n) bits is sufficient. For finding an α-approximate maximum matching, the upper bound was improved by Assadi and Shah [ITCS 2022] showing that O(n²/α³) bits is sufficient while Dark and Konrad [CCC 2020] have shown that Ω(n²/α³) bits is necessary. The space complexity, however, remains unresolved for many other dynamic graph streaming problems where further improvements can still be made.
Cite as

Kheeran K. Naidu and Vihan Shah. Space Optimal Vertex Cover in Dynamic Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 53:1-53:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{naidu_et_al:LIPIcs.APPROX/RANDOM.2022.53,
  author =	{Naidu, Kheeran K. and Shah, Vihan},
  title =	{{Space Optimal Vertex Cover in Dynamic Streams}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{53:1--53:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.53},
  URN =		{urn:nbn:de:0030-drops-171753},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.53},
  annote =	{Keywords: Graph Streaming Algorithms, Vertex Cover, Dynamic Streams, Approximation Algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.11
Decremental Matching in General Graphs

Authors: Sepehr Assadi, Aaron Bernstein, and Aditi Dudeja

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

Abstract
We consider the problem of maintaining an approximate maximum integral matching in a dynamic graph G, while the adversary makes changes to the edges of the graph. The goal is to maintain a (1+ε)-approximate maximum matching for constant ε > 0, while minimizing the update time. In the fully dynamic setting, where both edge insertion and deletions are allowed, Gupta and Peng (see [Manoj Gupta and Richard Peng, 2013]) gave an algorithm for this problem with an update time of O(√m/ε²). Motivated by the fact that the O_ε(√m) barrier is hard to overcome (see Henzinger, Krinninger, Nanongkai, and Saranurak [Henzinger et al., 2015]; Kopelowitz, Pettie, and Porat [Kopelowitz et al., 2016]), we study this problem in the decremental model, where the adversary is only allowed to delete edges. Recently, Bernstein, Probst-Gutenberg, and Saranurak (see [Bernstein et al., 2020]) gave an O(poly({log n}/ε)) update time decremental algorithm for this problem in bipartite graphs. However, beating O(√m) update time remained an open problem for general graphs. In this paper, we bridge the gap between bipartite and general graphs, by giving an O_ε(poly(log n)) update time algorithm that maintains a (1+ε)-approximate maximum integral matching under adversarial deletions. Our algorithm is randomized, but works against an adaptive adversary. Together with the work of Grandoni, Leonardi, Sankowski, Schwiegelshohn, and Solomon [Fabrizio Grandoni et al., 2019] who give an O_ε(1) update time algorithm for general graphs in the incremental (insertion-only) model, our result essentially completes the picture for partially dynamic matching.
Cite as

Sepehr Assadi, Aaron Bernstein, and Aditi Dudeja. Decremental Matching in General Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi_et_al:LIPIcs.ICALP.2022.11,
  author =	{Assadi, Sepehr and Bernstein, Aaron and Dudeja, Aditi},
  title =	{{Decremental Matching in General Graphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.11},
  URN =		{urn:nbn:de:0030-drops-163528},
  doi =		{10.4230/LIPIcs.ICALP.2022.11},
  annote =	{Keywords: Dynamic algorithms, matching, primal-dual algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.77
Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching

Authors: Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian

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

Abstract
Box-simplex games are a family of bilinear minimax objectives which encapsulate graph-structured problems such as maximum flow [Sherman, 2017], optimal transport [Arun Jambulapati et al., 2019], and bipartite matching [Sepehr Assadi et al., 2022]. We develop efficient near-linear time, high-accuracy solvers for regularized variants of these games. Beyond the immediate applications of such solvers for computing Sinkhorn distances, a prominent tool in machine learning, we show that these solvers can be used to obtain improved running times for maintaining a (fractional) ε-approximate maximum matching in a dynamic decremental bipartite graph against an adaptive adversary. We give a generic framework which reduces this dynamic matching problem to solving regularized graph-structured optimization problems to high accuracy. Through our reduction framework, our regularized box-simplex game solver implies a new algorithm for dynamic decremental bipartite matching in total time Õ(m ⋅ ε^{-3}), from an initial graph with m edges and n nodes. We further show how to use recent advances in flow optimization [Chen et al., 2022] to improve our runtime to m^{1 + o(1)} ⋅ ε^{-2}, thereby demonstrating the versatility of our reduction-based approach. These results improve upon the previous best runtime of Õ(m ⋅ ε^{-4}) [Aaron Bernstein et al., 2020] and illustrate the utility of using regularized optimization problem solvers for designing dynamic algorithms.
Cite as

Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian. Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 77:1-77:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jambulapati_et_al:LIPIcs.ICALP.2022.77,
  author =	{Jambulapati, Arun and Jin, Yujia and Sidford, Aaron and Tian, Kevin},
  title =	{{Regularized Box-Simplex Games and Dynamic Decremental Bipartite Matching}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{77:1--77:20},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.77},
  URN =		{urn:nbn:de:0030-drops-164181},
  doi =		{10.4230/LIPIcs.ICALP.2022.77},
  annote =	{Keywords: bipartite matching, decremental matching, dynamic algorithms, continuous optimization, box-simplex games, primal-dual method}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.9
An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams

Authors: Sepehr Assadi and Vihan Shah

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We present an algorithm for the maximum matching problem in dynamic (insertion-deletions) streams with asymptotically optimal space: for any n-vertex graph, our algorithm with high probability outputs an α-approximate matching in a single pass using O(n²/α³) bits of space. A long line of work on the dynamic streaming matching problem has reduced the gap between space upper and lower bounds first to n^{o(1)} factors [Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to polylog factors [Dark-Konrad; CCC 2020]. Our upper bound now matches the Dark-Konrad lower bound up to O(1) factors, thus completing this research direction. Our approach consists of two main steps: we first (provably) identify a family of graphs, similar to the instances used in prior work to establish the lower bounds for this problem, as the only "hard" instances to focus on. These graphs include an induced subgraph which is both sparse and contains a large matching. We then design a dynamic streaming algorithm for this family of graphs which is more efficient than prior work. The key to this efficiency is a novel sketching method, which bypasses the typical loss of polylog(n)-factors in space compared to standard L₀-sampling primitives, and can be of independent interest in designing optimal algorithms for other streaming problems.
Cite as

Sepehr Assadi and Vihan Shah. An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 9:1-9:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi_et_al:LIPIcs.ITCS.2022.9,
  author =	{Assadi, Sepehr and Shah, Vihan},
  title =	{{An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{9:1--9:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.9},
  URN =		{urn:nbn:de:0030-drops-156054},
  doi =		{10.4230/LIPIcs.ITCS.2022.9},
  annote =	{Keywords: Graph streaming algorithms, Sketching, Maximum matching}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.10
Sublinear Time and Space Algorithms for Correlation Clustering via Sparse-Dense Decompositions

Authors: Sepehr Assadi and Chen Wang

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We present a new approach for solving (minimum disagreement) correlation clustering that results in sublinear algorithms with highly efficient time and space complexity for this problem. In particular, we obtain the following algorithms for n-vertex (+/-)-labeled graphs G: - A sublinear-time algorithm that with high probability returns a constant approximation clustering of G in O(nlog²n) time assuming access to the adjacency list of the (+)-labeled edges of G (this is almost quadratically faster than even reading the input once). Previously, no sublinear-time algorithm was known for this problem with any multiplicative approximation guarantee. - A semi-streaming algorithm that with high probability returns a constant approximation clustering of G in O(n log n) space and a single pass over the edges of the graph G (this memory is almost quadratically smaller than input size). Previously, no single-pass algorithm with o(n²) space was known for this problem with any approximation guarantee. The main ingredient of our approach is a novel connection to sparse-dense graph decompositions that are used extensively in the graph coloring literature. To our knowledge, this connection is the first application of these decompositions beyond graph coloring, and in particular for the correlation clustering problem, and can be of independent interest.
Cite as

Sepehr Assadi and Chen Wang. Sublinear Time and Space Algorithms for Correlation Clustering via Sparse-Dense Decompositions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 10:1-10:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{assadi_et_al:LIPIcs.ITCS.2022.10,
  author =	{Assadi, Sepehr and Wang, Chen},
  title =	{{Sublinear Time and Space Algorithms for Correlation Clustering via Sparse-Dense Decompositions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.10},
  URN =		{urn:nbn:de:0030-drops-156067},
  doi =		{10.4230/LIPIcs.ITCS.2022.10},
  annote =	{Keywords: Correlation Clustering, Sublinear Algorithms, Semi-streaming Algorithms, Sublinear time Algorithms}
}
Document
DOI: 10.4230/LIPIcs.DISC.2021.6
Ruling Sets in Random Order and Adversarial Streams

Authors: Sepehr Assadi and Aditi Dudeja

Published in: LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

Abstract
The goal of this paper is to understand the complexity of a key symmetry breaking problem, namely the (α,β)-ruling set problem in the graph streaming model. Given a graph G = (V,E), an (α, β)-ruling set is a subset I ⊆ V such that the distance between any two vertices in I is at least α and the distance between a vertex in V and the closest vertex in I is at most β. This is a fundamental problem in distributed computing where it finds applications as a useful subroutine for other problems such as maximal matching, distributed colouring, or shortest paths. Additionally, it is a generalization of MIS, which is a (2,1)-ruling set. Our main results are two algorithms for (2,2)-ruling sets: 1) In adversarial streams, where the order in which edges arrive is arbitrary, we give an algorithm with Õ(n^{4/3}) space, improving upon the best known algorithm due to Konrad et al. [DISC 2019], with space Õ(n^{3/2}). 2) In random-order streams, where the edges arrive in a random order, we give a semi-streaming algorithm, that is an algorithm that takes Õ(n) space. Finally, we present new algorithms and lower bounds for (α,β)-ruling sets for other values of α and β. Our algorithms improve and generalize the previous work of Konrad et al. [DISC 2019] for (2,β)-ruling sets, while our lower bound establishes the impossibility of obtaining any non-trivial streaming algorithm for (α,α-1)-ruling sets for all even α > 2.
Cite as

Sepehr Assadi and Aditi Dudeja. Ruling Sets in Random Order and Adversarial Streams. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 6:1-6:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{assadi_et_al:LIPIcs.DISC.2021.6,
  author =	{Assadi, Sepehr and Dudeja, Aditi},
  title =	{{Ruling Sets in Random Order and Adversarial Streams}},
  booktitle =	{35th International Symposium on Distributed Computing (DISC 2021)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-210-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{209},
  editor =	{Gilbert, Seth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.6},
  URN =		{urn:nbn:de:0030-drops-148086},
  doi =		{10.4230/LIPIcs.DISC.2021.6},
  annote =	{Keywords: Symmetry breaking, Ruling sets, Lower bounds, Communication Complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.48
On the Robust Communication Complexity of Bipartite Matching

Authors: Sepehr Assadi and Soheil Behnezhad

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
We study the robust - à la Chakrabarti, Cormode, and McGregor [STOC'08] - communication complexity of the maximum bipartite matching problem. The edges of an adversarially chosen n-vertex bipartite graph G are partitioned randomly between Alice and Bob. Alice has to send a single message to Bob, using which Bob has to output an approximate maximum matching of G. We are particularly interested in understanding the best approximation ratio possible by protocols that use a near-optimal message size of n ⋅ polylog(n). The communication complexity of bipartite matching in this setting under an adversarial partitioning is well-understood. In their beautiful paper, Goel, Kapralov, and Khanna [SODA'12] gave a rac{2} {3}-approximate protocol with O(n) communication and showed that this approximation is tight unless we allow more than a near-linear communication. The complexity of the robust version, i.e., with a random partitioning of the edges, however remains wide open. The best known protocol, implied by a very recent random-order streaming algorithm of the authors [ICALP'21], uses O(n log n) communication to obtain a (rac{2} {3} + ε₀)-approximation for a constant ε₀ ∼ 10^{-14}. The best known lower bound, on the other hand, leaves open the possibility of all the way up to even a (1-ε)-approximation using near-linear communication for constant ε > 0. In this work, we give a new protocol with a significantly better approximation. Particularly, our protocol achieves a 0.716 expected approximation using O(n) communication. This protocol is based on a new notion of distribution-dependent sparsifiers which give a natural way of sparsifying graphs sampled from a known distribution. We then show how to lift the assumption on knowing the graph’s distribution via minimax theorems. We believe this is a particularly powerful method of designing communication protocols and might find further applications.
Cite as

Sepehr Assadi and Soheil Behnezhad. On the Robust Communication Complexity of Bipartite Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 48:1-48:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{assadi_et_al:LIPIcs.APPROX/RANDOM.2021.48,
  author =	{Assadi, Sepehr and Behnezhad, Soheil},
  title =	{{On the Robust Communication Complexity of Bipartite Matching}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{48:1--48:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.48},
  URN =		{urn:nbn:de:0030-drops-147411},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.48},
  annote =	{Keywords: Maximum Matching, Communication Complexity, Random-Order Streaming}
}
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