DROPS

Volume

LIPIcs, Volume 245

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

APPROX/RANDOM 2022, September 19-21, 2022, University of Illinois, Urbana-Champaign, USA (Virtual Conference)

Editors: Amit Chakrabarti and Chaitanya Swamy

Document

DOI: 10.4230/LIPIcs.CCC.2024.2

Streaming Zero-Knowledge Proofs

Authors: Graham Cormode, Marcel Dall'Agnol, Tom Gur, and Chris Hickey

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

Abstract

Streaming interactive proofs (SIPs) enable a space-bounded algorithm with one-pass access to a massive stream of data to verify a computation that requires large space, by communicating with a powerful but untrusted prover. This work initiates the study of zero-knowledge proofs for data streams. We define the notion of zero-knowledge in the streaming setting and construct zero-knowledge SIPs for the two main algorithmic building blocks in the streaming interactive proofs literature: the sumcheck and polynomial evaluation protocols. To the best of our knowledge all known streaming interactive proofs are based on either of these tools, and indeed, this allows us to obtain zero-knowledge SIPs for central streaming problems such as index, point and range queries, median, frequency moments, and inner product. Our protocols are efficient in terms of time and space, as well as communication: the verifier algorithm’s space complexity is polylog(n) and, after a non-interactive setup that uses a random string of near-linear length, the remaining parameters are n^o(1). En route, we develop an algorithmic toolkit for designing zero-knowledge data stream protocols, consisting of an algebraic streaming commitment protocol and a temporal commitment protocol. Our analyses rely on delicate algebraic and information-theoretic arguments and reductions from average-case communication complexity.

Cite as

Graham Cormode, Marcel Dall'Agnol, Tom Gur, and Chris Hickey. Streaming Zero-Knowledge Proofs. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 2:1-2:66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cormode_et_al:LIPIcs.CCC.2024.2,
  author =	{Cormode, Graham and Dall'Agnol, Marcel and Gur, Tom and Hickey, Chris},
  title =	{{Streaming Zero-Knowledge Proofs}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{2:1--2:66},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.2},
  URN =		{urn:nbn:de:0030-drops-203988},
  doi =		{10.4230/LIPIcs.CCC.2024.2},
  annote =	{Keywords: Zero-knowledge proofs, streaming algorithms, computational complexity}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.5

Explicit Time and Space Efficient Encoders Exist Only with Random Access

Authors: Joshua Cook and Dana Moshkovitz

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

Abstract

We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time n^{1 + o(1)} and space polylog(n) provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [Divsalar et al., 1998] and the codes described in [Gál et al., 2013]. To construct our codes, we also give explicit, efficiently invertible, lossless condensers with constant entropy gap and polylogarithmic seed length. In contrast to encoders with random access to the message, we show that encoders with sequential access to the message can not run in almost linear time and polylogarithmic space. Our notion of sequential access is much stronger than streaming access.

Cite as

Joshua Cook and Dana Moshkovitz. Explicit Time and Space Efficient Encoders Exist Only with Random Access. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 5:1-5:54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cook_et_al:LIPIcs.CCC.2024.5,
  author =	{Cook, Joshua and Moshkovitz, Dana},
  title =	{{Explicit Time and Space Efficient Encoders Exist Only with Random Access}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{5:1--5:54},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.5},
  URN =		{urn:nbn:de:0030-drops-204015},
  doi =		{10.4230/LIPIcs.CCC.2024.5},
  annote =	{Keywords: Time-Space Trade Offs, Error Correcting Codes, Encoders, Explicit Constructions, Streaming Lower Bounds, Sequential Access, Time-Space Lower Bounds, Lossless Condensers, Invertible Condensers, Condensers}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.7

Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy

Authors: Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay

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

Abstract

The following question arises naturally in the study of graph streaming algorithms: Is there any graph problem which is "not too hard", in that it can be solved efficiently with total communication (nearly) linear in the number n of vertices, and for which, nonetheless, any streaming algorithm with Õ(n) space (i.e., a semi-streaming algorithm) needs a polynomial n^Ω(1) number of passes? Assadi, Chen, and Khanna [STOC 2019] were the first to prove that this is indeed the case. However, the lower bounds that they obtained are for rather non-standard graph problems. Our first main contribution is to present the first polynomial-pass lower bounds for natural "not too hard" graph problems studied previously in the streaming model: k-cores and degeneracy. We devise a novel communication protocol for both problems with near-linear communication, thus showing that k-cores and degeneracy are natural examples of "not too hard" problems. Indeed, previous work have developed single-pass semi-streaming algorithms for approximating these problems. In contrast, we prove that any semi-streaming algorithm for exactly solving these problems requires (almost) Ω(n^{1/3}) passes. The lower bound follows by a reduction from a generalization of the hidden pointer chasing (HPC) problem of Assadi, Chen, and Khanna, which is also the basis of their earlier semi-streaming lower bounds. Our second main contribution is improved round-communication lower bounds for the underlying communication problems at the basis of these reductions: - We improve the previous lower bound of Assadi, Chen, and Khanna for HPC to achieve optimal bounds for this problem. - We further observe that all current reductions from HPC can also work with a generalized version of this problem that we call MultiHPC, and prove an even stronger and optimal lower bound for this generalization. These two results collectively allow us to improve the resulting pass lower bounds for semi-streaming algorithms by a polynomial factor, namely, from n^{1/5} to n^{1/3} passes.

Cite as

Sepehr Assadi, Prantar Ghosh, Bruno Loff, Parth Mittal, and Sagnik Mukhopadhyay. Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{assadi_et_al:LIPIcs.CCC.2024.7,
  author =	{Assadi, Sepehr and Ghosh, Prantar and Loff, Bruno and Mittal, Parth and Mukhopadhyay, Sagnik},
  title =	{{Polynomial Pass Semi-Streaming Lower Bounds for K-Cores and Degeneracy}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.7},
  URN =		{urn:nbn:de:0030-drops-204035},
  doi =		{10.4230/LIPIcs.CCC.2024.7},
  annote =	{Keywords: Graph streaming, Lower bounds, Communication complexity, k-Cores and degeneracy}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.19

Information Dissemination via Broadcasts in the Presence of Adversarial Noise

Authors: Klim Efremenko, Gillat Kol, Dmitry Paramonov, Ran Raz, and Raghuvansh R. Saxena

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

Abstract

We initiate the study of error correcting codes over the multi-party adversarial broadcast channel. Specifically, we consider the classic information dissemination problem where n parties, each holding an input bit, wish to know each other’s input. For this, they communicate in rounds, where, in each round, one designated party sends a bit to all other parties over a channel governed by an adversary that may corrupt a constant fraction of the received communication. We mention that the dissemination problem was studied in the stochastic noise model since the 80’s. While stochastic noise in multi-party channels has received quite a bit of attention, the case of adversarial noise has largely been avoided, as such channels cannot handle more than a 1/n-fraction of errors. Indeed, this many errors allow an adversary to completely corrupt the incoming or outgoing communication for one of the parties and fail the protocol. Curiously, we show that by eliminating these "trivial" attacks, one can get a simple protocol resilient to a constant fraction of errors. Thus, a model that rules out such attacks is both necessary and sufficient to get a resilient protocol. The main shortcoming of our dissemination protocol is its length: it requires Θ(n²) communication rounds whereas n rounds suffice in the absence of noise. Our main result is a matching lower bound of Ω(n²) on the length of any dissemination protocol in our model. Our proof first "gets rid" of the channel noise by converting it to a form of "input noise", showing that a noisy dissemination protocol implies a (noiseless) protocol for a version of the direct sum gap-majority problem. We conclude the proof with a tight lower bound for the latter problem, which may be of independent interest.

Cite as

Klim Efremenko, Gillat Kol, Dmitry Paramonov, Ran Raz, and Raghuvansh R. Saxena. Information Dissemination via Broadcasts in the Presence of Adversarial Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 19:1-19:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{efremenko_et_al:LIPIcs.CCC.2024.19,
  author =	{Efremenko, Klim and Kol, Gillat and Paramonov, Dmitry and Raz, Ran and Saxena, Raghuvansh R.},
  title =	{{Information Dissemination via Broadcasts in the Presence of Adversarial Noise}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{19:1--19:33},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.19},
  URN =		{urn:nbn:de:0030-drops-204159},
  doi =		{10.4230/LIPIcs.CCC.2024.19},
  annote =	{Keywords: Radio Networks, Interactive Coding, Error Correcting Codes}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.28

Finding Missing Items Requires Strong Forms of Randomness

Authors: Amit Chakrabarti and Manuel Stoeckl

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

Abstract

Adversarially robust streaming algorithms are required to process a stream of elements and produce correct outputs, even when each stream element can be chosen as a function of earlier algorithm outputs. As with classic streaming algorithms, which must only be correct for the worst-case fixed stream, adversarially robust algorithms with access to randomness can use significantly less space than deterministic algorithms. We prove that for the Missing Item Finding problem in streaming, the space complexity also significantly depends on how adversarially robust algorithms are permitted to use randomness. (In contrast, the space complexity of classic streaming algorithms does not depend as strongly on the way randomness is used.) For Missing Item Finding on streams of length 𝓁 with elements in {1,…,n}, and ≤ 1/poly(𝓁) error, we show that when 𝓁 = O(2^√{log n}), "random seed" adversarially robust algorithms, which only use randomness at initialization, require 𝓁^Ω(1) bits of space, while "random tape" adversarially robust algorithms, which may make random decisions at any time, may use O(polylog(𝓁)) random bits. When 𝓁 is between n^Ω(1) and O(√n), "random tape" adversarially robust algorithms need 𝓁^Ω(1) space, while "random oracle" adversarially robust algorithms, which can read from a long random string for free, may use O(polylog(𝓁)) space. The space lower bound for the "random seed" case follows, by a reduction given in prior work, from a lower bound for pseudo-deterministic streaming algorithms given in this paper.

Cite as

Amit Chakrabarti and Manuel Stoeckl. Finding Missing Items Requires Strong Forms of Randomness. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chakrabarti_et_al:LIPIcs.CCC.2024.28,
  author =	{Chakrabarti, Amit and Stoeckl, Manuel},
  title =	{{Finding Missing Items Requires Strong Forms of Randomness}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{28:1--28:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.28},
  URN =		{urn:nbn:de:0030-drops-204242},
  doi =		{10.4230/LIPIcs.CCC.2024.28},
  annote =	{Keywords: Data streaming, lower bounds, space complexity, adversarial robustness, derandomization, sketching, sampling}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.15

List Update with Delays or Time Windows

Authors: Yossi Azar, Shahar Lewkowicz, and Danny Vainstein

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

Abstract

We address the problem of List Update, which is considered one of the fundamental problems in online algorithms and competitive analysis. In this context, we are presented with a list of elements and receive requests for these elements over time. Our objective is to fulfill these requests, incurring a cost proportional to their position in the list. Additionally, we can swap any two consecutive elements at a cost of 1. The renowned "Move to Front" algorithm, introduced by Sleator and Tarjan, immediately moves any requested element to the front of the list. They demonstrated that this algorithm achieves a competitive ratio of 2. While this bound is impressive, the actual cost of the algorithm’s solution can be excessively high. For example, if we request the last half of the list, the resulting solution cost becomes quadratic in the list’s length. To address this issue, we consider a more generalized problem called List Update with Time Windows. In this variant, each request arrives with a specific deadline by which it must be served, rather than being served immediately. Moreover, we allow the algorithm to process multiple requests simultaneously, accessing the corresponding elements in a single pass. The cost incurred in this case is determined by the position of the furthest element accessed, leading to a significant reduction in the total solution cost. We introduce this problem to explore lower solution costs, but it necessitates the development of new algorithms. For instance, Move-to-Front fails when handling the simple scenario of requesting the last half of the list with overlapping time windows. In our work, we present a natural O(1) competitive algorithm for this problem. While the algorithm itself is intuitive, its analysis is intricate, requiring the use of a novel potential function. Additionally, we delve into a more general problem called List Update with Delays, where the fixed deadlines are replaced with arbitrary delay functions. In this case, the cost includes not only the access and swapping costs, but also penalties for the delays incurred until the requests are served. This problem encompasses a special case known as the prize collecting version, where a request may go unserved up to a given deadline, resulting in a specified penalty. For this more comprehensive problem, we establish an O(1) competitive algorithm. However, the algorithm for the delay version is more complex, and its analysis involves significantly more intricate considerations.

Cite as

Yossi Azar, Shahar Lewkowicz, and Danny Vainstein. List Update with Delays or Time Windows. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 15:1-15:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{azar_et_al:LIPIcs.ICALP.2024.15,
  author =	{Azar, Yossi and Lewkowicz, Shahar and Vainstein, Danny},
  title =	{{List Update with Delays or Time Windows}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{15:1--15:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.15},
  URN =		{urn:nbn:de:0030-drops-201583},
  doi =		{10.4230/LIPIcs.ICALP.2024.15},
  annote =	{Keywords: Online, List Update, Delay, Time Window, Deadline}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.37

Fast Approximate Counting of Cycles

Authors: Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams

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

Abstract

We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

Cite as

Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams. Fast Approximate Counting of Cycles. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{censorhillel_et_al:LIPIcs.ICALP.2024.37,
  author =	{Censor-Hillel, Keren and Even, Tomer and Vassilevska Williams, Virginia},
  title =	{{Fast Approximate Counting of Cycles}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.37},
  URN =		{urn:nbn:de:0030-drops-201809},
  doi =		{10.4230/LIPIcs.ICALP.2024.37},
  annote =	{Keywords: Approximate triangle counting, Approximate cycle counting Fast matrix multiplication, Fast rectangular matrix multiplication}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.71

Low-Memory Algorithms for Online Edge Coloring

Authors: Prantar Ghosh and Manuel Stoeckl

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

Abstract

For edge coloring, the online and the W-streaming models seem somewhat orthogonal: the former needs edges to be assigned colors immediately after insertion, typically without any space restrictions, while the latter limits memory to be sublinear in the input size but allows an edge’s color to be announced any time after its insertion. We aim for the best of both worlds by designing small-space online algorithms for edge coloring. Our online algorithms significantly improve upon the memory used by prior ones while achieving an O(1)-competitive ratio. We study the problem under both (adversarial) edge arrivals and vertex arrivals. Under vertex arrivals of any n-node graph with maximum vertex-degree Δ, our online O(Δ)-coloring algorithm uses only semi-streaming space (i.e., Õ(n) space, where the Õ(.) notation hides polylog(n) factors). Under edge arrivals, we obtain an online O(Δ)-coloring in Õ(n√Δ) space. We also achieve a smooth color-space tradeoff: for any t = O(Δ), we get an O(Δt(log²Δ))-coloring in Õ(n√{Δ/t}) space, improving upon the state of the art that used Õ(nΔ/t) space for the same number of colors. The improvements stem from extensive use of random permutations that enable us to avoid previously used colors. Most of our algorithms can be derandomized and extended to multigraphs, where edge coloring is known to be considerably harder than for simple graphs.

Cite as

Prantar Ghosh and Manuel Stoeckl. Low-Memory Algorithms for Online Edge Coloring. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 71:1-71:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ghosh_et_al:LIPIcs.ICALP.2024.71,
  author =	{Ghosh, Prantar and Stoeckl, Manuel},
  title =	{{Low-Memory Algorithms for Online Edge Coloring}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{71:1--71:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.71},
  URN =		{urn:nbn:de:0030-drops-202146},
  doi =		{10.4230/LIPIcs.ICALP.2024.71},
  annote =	{Keywords: Edge coloring, streaming model, online algorithms}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.90

Dynamic PageRank: Algorithms and Lower Bounds

Authors: Rajesh Jayaram, Jakub Łącki, Slobodan Mitrović, Krzysztof Onak, and Piotr Sankowski

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

Abstract

We consider the PageRank problem in the dynamic setting, where the goal is to explicitly maintain an approximate PageRank vector π ∈ ℝⁿ for a graph under a sequence of edge insertions and deletions. Our main result is a complete characterization of the complexity of dynamic PageRank maintenance for both multiplicative and additive (L₁) approximations. First, we establish matching lower and upper bounds for maintaining additive approximate PageRank in both incremental and decremental settings. In particular, we demonstrate that in the worst-case (1/α)^{Θ(log log n)} update time is necessary and sufficient for this problem, where α is the desired additive approximation. On the other hand, we demonstrate that the commonly employed ForwardPush approach performs substantially worse than this optimal runtime. Specifically, we show that ForwardPush requires Ω(n^{1-δ}) time per update on average, for any δ > 0, even in the incremental setting. For multiplicative approximations, however, we demonstrate that the situation is significantly more challenging. Specifically, we prove that any algorithm that explicitly maintains a constant factor multiplicative approximation of the PageRank vector of a directed graph must have amortized update time Ω(n^{1-δ}), for any δ > 0, even in the incremental setting, thereby resolving a 13-year old open question of Bahmani et al. (VLDB 2010). This sharply contrasts with the undirected setting, where we show that poly log n update time is feasible, even in the fully dynamic setting under oblivious adversary.

Cite as

Rajesh Jayaram, Jakub Łącki, Slobodan Mitrović, Krzysztof Onak, and Piotr Sankowski. Dynamic PageRank: Algorithms and Lower Bounds. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 90:1-90:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jayaram_et_al:LIPIcs.ICALP.2024.90,
  author =	{Jayaram, Rajesh and {\L}\k{a}cki, Jakub and Mitrovi\'{c}, Slobodan and Onak, Krzysztof and Sankowski, Piotr},
  title =	{{Dynamic PageRank: Algorithms and Lower Bounds}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{90:1--90:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.90},
  URN =		{urn:nbn:de:0030-drops-202336},
  doi =		{10.4230/LIPIcs.ICALP.2024.90},
  annote =	{Keywords: PageRank, dynamic algorithms, graph algorithms}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.121

Streaming Edge Coloring with Asymptotically Optimal Colors

Authors: Mohammad Saneian and Soheil Behnezhad

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

Abstract

Given a graph G, an edge-coloring is an assignment of colors to edges of G such that any two edges sharing an endpoint receive different colors. By Vizing’s celebrated theorem, any graph of maximum degree Δ needs at least Δ and at most (Δ + 1) colors to be properly edge colored. In this paper, we study edge colorings in the streaming setting. The edges arrive one by one in an arbitrary order. The algorithm takes a single pass over the input and must output a solution using a much smaller space than the input size. Since the output of edge coloring is as large as its input, the assigned colors should also be reported in a streaming fashion. The streaming edge coloring problem has been studied in a series of works over the past few years. The main challenge is that the algorithm cannot "remember" all the color assignments that it returns. To ensure the validity of the solution, existing algorithms use many more colors than Vizing’s bound. Namely, in n-vertex graphs, the state-of-the-art algorithm with Õ(n s) space requires O(Δ²/s + Δ) colors. Note, in particular, that for an asymptotically optimal O(Δ) coloring, this algorithm requires Ω(nΔ) space which is as large as the input. Whether such a coloring can be achieved with sublinear space has been left open. In this paper, we answer this question in the affirmative. We present a randomized algorithm that returns an asymptotically optimal O(Δ) edge coloring using Õ(n √{Δ}) space. More generally, our algorithm returns a proper O(Δ^{1.5}/s + Δ) edge coloring with Õ(n s) space, improving prior algorithms for the whole range of s.

Cite as

Mohammad Saneian and Soheil Behnezhad. Streaming Edge Coloring with Asymptotically Optimal Colors. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 121:1-121:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{saneian_et_al:LIPIcs.ICALP.2024.121,
  author =	{Saneian, Mohammad and Behnezhad, Soheil},
  title =	{{Streaming Edge Coloring with Asymptotically Optimal Colors}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{121:1--121:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.121},
  URN =		{urn:nbn:de:0030-drops-202640},
  doi =		{10.4230/LIPIcs.ICALP.2024.121},
  annote =	{Keywords: Streaming, Edge coloring, Adversarial order}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2024.53

New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification

Authors: Prantar Ghosh and Vihan Shah

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

Abstract

We present novel lower bounds in the Merlin-Arthur (MA) communication model and the related annotated streaming or stream verification model. The MA communication model extends the classical communication model by introducing an all-powerful but untrusted player, Merlin, who knows the inputs of the usual players, Alice and Bob, and attempts to convince them about the output. We focus on the online MA (OMA) model where Alice and Merlin each send a single message to Bob, who needs to catch Merlin if he is dishonest and announce the correct output otherwise. Most known functions have OMA protocols with total communication significantly smaller than what would be needed without Merlin. In this work, we introduce the notion of non-trivial-OMA complexity of a function. This is the minimum total communication required when we restrict ourselves to only non-trivial protocols where Alice sends Bob fewer bits than what she would have sent without Merlin. We exhibit the first explicit functions that have this complexity superlinear - even exponential - in their classical one-way complexity: this means the trivial protocol, where Merlin communicates nothing and Alice and Bob compute the function on their own, is exponentially better than any non-trivial protocol in terms of total communication. These OMA lower bounds also translate to the annotated streaming model, the MA analogue of single-pass data streaming. We show large separations between the classical streaming complexity and the non-trivial annotated streaming complexity (for the analogous notion in this setting) of fundamental problems such as counting distinct items, as well as of graph problems such as connectivity and k-connectivity in a certain edge update model called the support graph turnstile model that we introduce here.

Cite as

Prantar Ghosh and Vihan Shah. New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 53:1-53:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ghosh_et_al:LIPIcs.ITCS.2024.53,
  author =	{Ghosh, Prantar and Shah, Vihan},
  title =	{{New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{53:1--53:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.53},
  URN =		{urn:nbn:de:0030-drops-195815},
  doi =		{10.4230/LIPIcs.ITCS.2024.53},
  annote =	{Keywords: Graph Algorithms, Streaming, Communication Complexity, Stream Verification, Merlin-Arthur Communication, Lower Bounds}
}

Document

Complete Volume

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022

LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume

Authors: Amit Chakrabarti and Chaitanya Swamy

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

Abstract

LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume

Cite as

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 1-1064, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{chakrabarti_et_al:LIPIcs.APPROX/RANDOM.2022,
  title =	{{LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{1--1064},
  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},
  URN =		{urn:nbn:de:0030-drops-171211},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022},
  annote =	{Keywords: LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume}
}

Document

Front Matter

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.0

Front Matter, Table of Contents, Preface, Conference Organization

Authors: Amit Chakrabarti and Chaitanya Swamy

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

Abstract

Front Matter, Table of Contents, Preface, Conference Organization

Cite as

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakrabarti_et_al:LIPIcs.APPROX/RANDOM.2022.0,
  author =	{Chakrabarti, Amit and Swamy, Chaitanya},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{0:i--0:xx},
  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.2022.0},
  URN =		{urn:nbn:de:0030-drops-171229},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.1

A Unified Approach to Discrepancy Minimization

Authors: Nikhil Bansal, Aditi Laddha, and Santosh Vempala

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

Abstract

We study a unified approach and algorithm for constructive discrepancy minimization based on a stochastic process. By varying the parameters of the process, one can recover various state-of-the-art results. We demonstrate the flexibility of the method by deriving a discrepancy bound for smoothed instances, which interpolates between known bounds for worst-case and random instances.

Cite as

Nikhil Bansal, Aditi Laddha, and Santosh Vempala. A Unified Approach to Discrepancy Minimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.APPROX/RANDOM.2022.1,
  author =	{Bansal, Nikhil and Laddha, Aditi and Vempala, Santosh},
  title =	{{A Unified Approach to Discrepancy Minimization}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{1:1--1:22},
  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.2022.1},
  URN =		{urn:nbn:de:0030-drops-171238},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.1},
  annote =	{Keywords: Discrepancy theory, smoothed analysis}
}

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