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DOI: 10.4230/LIPIcs.FSTTCS.2024.29

Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model

Authors: Christian Konrad, Andrew McGregor, Rik Sengupta, and Cuong Than

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract

We consider the problem of estimating the size of a maximum matching in low-arboricity graphs in the dynamic graph stream model. In this setting, an algorithm with limited memory makes multiple passes over a stream of edge insertions and deletions, resulting in a low-arboricity graph. Let n be the number of vertices of the input graph, and α be its arboricity. We give the following results. 1) As our main result, we give a three-pass streaming algorithm that produces an (α + 2)(1 + ε)-approximation and uses space O(ε^{-2}⋅α²⋅n^{1/2}⋅log n). This result should be contrasted with the Ω(α^{-5/2}⋅n^{1/2}) space lower bound established by [Assadi et al., SODA'17] for one-pass algorithms, showing that, for graphs of constant arboricity, the one-pass space lower bound can be achieved in three passes (up to poly-logarithmic factors). Furthermore, we obtain a two-pass algorithm that uses space O(ε^{-2}⋅α²⋅n^{3/5}⋅log n). 2) We also give a (1+ε)-approximation multi-pass algorithm, where the space used is parameterized by an upper bound on the size of a largest matching. For example, using O(log log n) passes, the space required is O(ε^{-1}⋅α²⋅k⋅log n), where k denotes an upper bound on the size of a largest matching. Finally, we define a notion of arboricity in the context of matrices. This is a natural measure of the sparsity of a matrix that is more nuanced than simply bounding the total number of nonzero entries, but less restrictive than bounding the number of nonzero entries in each row and column. For such matrices, we exploit our results on estimating matching size to present upper bounds for the problem of rank estimation in the dynamic data stream model.

Cite as

Christian Konrad, Andrew McGregor, Rik Sengupta, and Cuong Than. Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{konrad_et_al:LIPIcs.FSTTCS.2024.29,
  author =	{Konrad, Christian and McGregor, Andrew and Sengupta, Rik and Than, Cuong},
  title =	{{Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.29},
  URN =		{urn:nbn:de:0030-drops-222187},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.29},
  annote =	{Keywords: Data Streams, Graph Matching, Graph Arboricity}
}

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DOI: 10.4230/LIPIcs.ESA.2024.8

Interval Selection in Sliding Windows

Authors: Cezar-Mihail Alexandru and Christian Konrad

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract

We initiate the study of the Interval Selection problem in the (streaming) sliding window model of computation. In this problem, an algorithm receives a potentially infinite stream of intervals on the line, and the objective is to maintain at every moment an approximation to a largest possible subset of disjoint intervals among the L most recent intervals, for some integer L. We give the following results: 1) In the unit-length intervals case, we give a 2-approximation sliding window algorithm with space Õ(|OPT|), and we show that any sliding window algorithm that computes a (2-ε)-approximation requires space Ω(L), for any ε > 0. 2) In the arbitrary-length case, we give a (11/3+ε)-approximation sliding window algorithm with space Õ(|OPT|), for any constant ε > 0, which constitutes our main result. We also show that space Ω(L) is needed for algorithms that compute a (2.5-ε)-approximation, for any ε > 0. Our main technical contribution is an improvement over the smooth histogram technique, which consists of running independent copies of a traditional streaming algorithm with different start times. By employing the one-pass 2-approximation streaming algorithm by Cabello and Pérez-Lantero [Theor. Comput. Sci. '17] for Interval Selection on arbitrary-length intervals as the underlying algorithm, the smooth histogram technique immediately yields a (4+ε)-approximation in this setting. Our improvement is obtained by forwarding the structure of the intervals identified in a run to the subsequent run, which constrains the shape of an optimal solution and allows us to target optimal intervals differently.

Cite as

Cezar-Mihail Alexandru and Christian Konrad. Interval Selection in Sliding Windows. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 8:1-8:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alexandru_et_al:LIPIcs.ESA.2024.8,
  author =	{Alexandru, Cezar-Mihail and Konrad, Christian},
  title =	{{Interval Selection in Sliding Windows}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{8:1--8:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.8},
  URN =		{urn:nbn:de:0030-drops-210795},
  doi =		{10.4230/LIPIcs.ESA.2024.8},
  annote =	{Keywords: Sliding window algorithms, Streaming algorithms, Interval selection}
}

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DOI: 10.4230/LIPIcs.ITP.2024.23

A Generalised Union of Rely-Guarantee and Separation Logic Using Permission Algebras

Authors: Vincent Jackson, Toby Murray, and Christine Rizkallah

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

This paper describes GenRGSep, an Isabelle/HOL library for the development of RGSep logics using a general algebraic state model. In particular, we develop an algebraic state models based on resource algebras that assume neither the presence of unit resources or the cancellativity law. If a new resource model is required, its components need only be proven an instance of a permission algebra, and then they can be composed together using tuples and functions. The proof of soundness is performed by Vafeiadis' operational soundness method. This method was originally formulated with respect to a concrete heap model. This paper adapts it to account for the absence of both units as well as the cancellativity law.

Cite as

Vincent Jackson, Toby Murray, and Christine Rizkallah. A Generalised Union of Rely-Guarantee and Separation Logic Using Permission Algebras. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jackson_et_al:LIPIcs.ITP.2024.23,
  author =	{Jackson, Vincent and Murray, Toby and Rizkallah, Christine},
  title =	{{A Generalised Union of Rely-Guarantee and Separation Logic Using Permission Algebras}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.23},
  URN =		{urn:nbn:de:0030-drops-207510},
  doi =		{10.4230/LIPIcs.ITP.2024.23},
  annote =	{Keywords: verification, concurrency, rely-guarantee, separation logic, resource algebras}
}

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DOI: 10.4230/LIPIcs.ITP.2024.18

Mechanized HOL Reasoning in Set Theory

Authors: Simon Guilloud, Sankalp Gambhir, Andrea Gilot, and Viktor Kunčak

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract

We present a mechanized embedding of higher-order logic (HOL) and algebraic data types (ADTs) into first-order logic with ZFC axioms. Our approach interprets types as sets, with function (arrow) types coinciding with set-theoretic function spaces. We assume traditional FOL syntax without notation for term-level binders. To embed λ-terms, we define a notion of context, defining the closure of all abstractions occuring inside a term. We implement the embedding in the Lisa proof assistant for schematic first-order logic and its library based on axiomatic set theory (presented at ITP 2023). We show how to implement type checking and the proof steps of HOL Light as proof-producing tactics in Lisa. The embedded HOL theorems and proofs are interoperable with the existing Lisa library. This yields a form of soft type system supporting top-level polymorphism and ADTs within set theory. The approach offers tools for Lisa users to carry HOL-style proofs within set theory. It also enables the import of HOL Light theorem statements into Lisa, as well as the replay of small HOL Light kernel proofs.

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Simon Guilloud, Sankalp Gambhir, Andrea Gilot, and Viktor Kunčak. Mechanized HOL Reasoning in Set Theory. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{guilloud_et_al:LIPIcs.ITP.2024.18,
  author =	{Guilloud, Simon and Gambhir, Sankalp and Gilot, Andrea and Kun\v{c}ak, Viktor},
  title =	{{Mechanized HOL Reasoning in Set Theory}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{18:1--18:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.18},
  URN =		{urn:nbn:de:0030-drops-207464},
  doi =		{10.4230/LIPIcs.ITP.2024.18},
  annote =	{Keywords: Proof assistant, First Order Logic, Set Theory, Higher Order Logic}
}

Document

DOI: 10.4230/LIPIcs.WABI.2024.5

Memoization on Shared Subtrees Accelerates Computations on Genealogical Forests

Authors: Lukas Hübner and Alexandros Stamatakis

Published in: LIPIcs, Volume 312, 24th International Workshop on Algorithms in Bioinformatics (WABI 2024)

Abstract

The field of population genetics attempts to advance our understanding of evolutionary processes. It has applications, for example, in medical research, wildlife conservation, and - in conjunction with recent advances in ancient DNA sequencing technology - studying human migration patterns over the past few thousand years. The basic toolbox of population genetics includes genealogical trees, which describe the shared evolutionary history among individuals of the same species. They are calculated on the basis of genetic variations. However, in recombining organisms, a single tree is insufficient to describe the evolutionary history of the whole genome. Instead, a collection of correlated trees can be used, where each describes the evolutionary history of a consecutive region of the genome. The current corresponding state of-the-art data structure, tree sequences, compresses these genealogical trees via edit operations when moving from one tree to the next along the genome instead of storing the full, often redundant, description for each tree. We propose a new data structure, genealogical forests, which compresses the set of genealogical trees into a DAG. In this DAG identical subtrees that are shared across the input trees are encoded only once, thereby allowing for straight-forward memoization of intermediate results. Additionally, we provide a C++ implementation of our proposed data structure, called gfkit, which is 2.1 to 11.2 (median 4.0) times faster than the state-of-the-art tool on empirical and simulated datasets at computing important population genetics statistics such as the Allele Frequency Spectrum, Patterson’s f, the Fixation Index, Tajima’s D, pairwise Lowest Common Ancestors, and others. On Lowest Common Ancestor queries with more than two samples as input, gfkit scales asymptotically better than the state-of-the-art, and is thus up to 990 times faster. In conclusion, our proposed data structure compresses genealogical trees by storing shared subtrees only once, thereby enabling straight-forward memoization of intermediate results, yielding a substantial runtime reduction and a potentially more intuitive data representation over the state-of-the-art. Our improvements will boost the development of novel analyses and models in the field of population genetics and increases scalability to ever-growing genomic datasets.

Cite as

Lukas Hübner and Alexandros Stamatakis. Memoization on Shared Subtrees Accelerates Computations on Genealogical Forests. In 24th International Workshop on Algorithms in Bioinformatics (WABI 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 312, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hubner_et_al:LIPIcs.WABI.2024.5,
  author =	{H\"{u}bner, Lukas and Stamatakis, Alexandros},
  title =	{{Memoization on Shared Subtrees Accelerates Computations on Genealogical Forests}},
  booktitle =	{24th International Workshop on Algorithms in Bioinformatics (WABI 2024)},
  pages =	{5:1--5:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-340-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{312},
  editor =	{Pissis, Solon P. and Sung, Wing-Kin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2024.5},
  URN =		{urn:nbn:de:0030-drops-206499},
  doi =		{10.4230/LIPIcs.WABI.2024.5},
  annote =	{Keywords: bioinformatics, population genetics, algorithms}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2023.24

Interval Selection in Data Streams: Weighted Intervals and the Insertion-Deletion Setting

Authors: Jacques Dark, Adithya Diddapur, and Christian Konrad

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract

We study the Interval Selection problem in data streams: Given a stream of n intervals on the line, the objective is to compute a largest possible subset of non-overlapping intervals using O(|OPT|) space, where |OPT| is the size of an optimal solution. Previous work gave a 3/2-approximation for unit-length and a 2-approximation for arbitrary-length intervals [Emek et al., ICALP'12]. We extend this line of work to weighted intervals as well as to insertion-deletion streams. Our results include: 1) When considering weighted intervals, a (3/2+ε)-approximation can be achieved for unit intervals, but any constant factor approximation for arbitrary-length intervals requires space Ω(n). 2) In the insertion-deletion setting where intervals can both be added and deleted, we prove that, even without weights, computing a constant factor approximation for arbitrary-length intervals requires space Ω(n), whereas in the weighted unit-length intervals case a (2+ε)-approximation can be obtained. Our lower bound results are obtained via reductions to the recently introduced Chained-Index communication problem, further demonstrating the strength of this problem in the context of streaming geometric independent set problems.

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Jacques Dark, Adithya Diddapur, and Christian Konrad. Interval Selection in Data Streams: Weighted Intervals and the Insertion-Deletion Setting. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dark_et_al:LIPIcs.FSTTCS.2023.24,
  author =	{Dark, Jacques and Diddapur, Adithya and Konrad, Christian},
  title =	{{Interval Selection in Data Streams: Weighted Intervals and the Insertion-Deletion Setting}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{24:1--24:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.24},
  URN =		{urn:nbn:de:0030-drops-193976},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.24},
  annote =	{Keywords: Streaming Algorithms, Interval Selection, Weighted Intervals, Insertion-deletion Streams}
}

Document

DOI: 10.4230/LIPIcs.STACS.2023.6

Improved Weighted Matching in the Sliding Window Model

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

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

Abstract

We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set V of size n. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the L most recent edges, for some integer L, using as little space as possible. Prior to our work, the state-of-the-art results were a (3.5+ε)-approximation algorithm for MWM by Biabani et al. [ISAAC'21] and a (3+ε)-approximation for (unweighted) Maximum Matching (MM) by Crouch et al. [ESA'13]. Both algorithms use space Õ(n). We give the following results: 1) We give a (2+ε)-approximation algorithm for MWM with space Õ(√{nL}). Under the reasonable assumption that the graphs spanned by the edges in each sliding window are simple, our algorithm uses space Õ(n √n). 2) In the Õ(n) space regime, we give a (3+ε)-approximation algorithm for MWM, thereby closing the gap between the best-known approximation ratio for MWM and MM. Similar to Biabani et al.’s MWM algorithm, both our algorithms execute multiple instances of the (2+ε)-approximation Õ(n)-space streaming algorithm for MWM by Paz and Schwartzman [SODA'17] on different portions of the stream. Our improvements are obtained by selecting these substreams differently. Furthermore, our (2+ε)-approximation algorithm runs the Paz-Schwartzman algorithm in reverse direction over some parts of the stream, and in forward direction over other parts, which allows for an improved approximation guarantee at the cost of increased space requirements.

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Cezar-Mihail Alexandru, Pavel Dvořák, Christian Konrad, and Kheeran K. Naidu. Improved Weighted Matching in the Sliding Window Model. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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

Document

DOI: 10.4230/LIPIcs.STACS.2023.41

Maximum Matching via Maximal Matching Queries

Authors: Christian Konrad, Kheeran K. Naidu, and Arun Steward

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

Abstract

We study approximation algorithms for Maximum Matching that are given access to the input graph solely via an edge-query maximal matching oracle. More specifically, in each round, an algorithm queries a set of potential edges and the oracle returns a maximal matching in the subgraph spanned by the query edges that are also contained in the input graph. This model is more general than the vertex-query model introduced by binti Khalil and Konrad [FSTTCS'20], where each query consists of a subset of vertices and the oracle returns a maximal matching in the subgraph of the input graph induced by the queried vertices. In this paper, we give tight bounds for deterministic edge-query algorithms for up to three rounds. In more detail: 1) As our main result, we give a deterministic 3-round edge-query algorithm with approximation factor 0.625 on bipartite graphs. This result establishes a separation between the edge-query and the vertex-query models since every deterministic 3-round vertex-query algorithm has an approximation factor of at most 0.6 [binti Khalil, Konrad, FSTTCS'20], even on bipartite graphs. Our algorithm can also be implemented in the semi-streaming model of computation in a straightforward manner and improves upon the state-of-the-art 3-pass 0.6111-approximation algorithm by Feldman and Szarf [APPROX'22] for bipartite graphs. 2) We show that the aforementioned algorithm is optimal in that every deterministic 3-round edge-query algorithm has an approximation factor of at most 0.625, even on bipartite graphs. 3) Last, we also give optimal bounds for one and two query rounds, where the best approximation factors achievable are 1/2 and 1/2 + Θ(1/n), respectively, where n is the number of vertices in the input graph.

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Christian Konrad, Kheeran K. Naidu, and Arun Steward. Maximum Matching via Maximal Matching Queries. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{konrad_et_al:LIPIcs.STACS.2023.41,
  author =	{Konrad, Christian and Naidu, Kheeran K. and Steward, Arun},
  title =	{{Maximum Matching via Maximal Matching Queries}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{41:1--41:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.41},
  URN =		{urn:nbn:de:0030-drops-176935},
  doi =		{10.4230/LIPIcs.STACS.2023.41},
  annote =	{Keywords: Maximum Matching, Query Model, Algorithms, Lower Bounds}
}

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APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.33

Maximum Matching Sans Maximal Matching: A New Approach for Finding Maximum Matchings in the Data Stream Model

Authors: Moran Feldman and Ariel Szarf

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

Abstract

The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1/2-approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continues this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work [Christian Konrad and Kheeran K. Naidu, 2021] shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1/2-approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms are currently known (to the best of our knowledge), and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs (the result for three passes improves over the previous state-of-the-art, but is worse than the result of this paper mentioned in the previous paragraph for general graphs). The improvements obtained by these results are, unfortunately, numerically not very impressive, but the main importance (in our opinion) of these results is in demonstrating the potential of non-maximal-matching-first algorithms.

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Moran Feldman and Ariel Szarf. Maximum Matching Sans Maximal Matching: A New Approach for Finding Maximum Matchings in the Data Stream Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 33:1-33:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{feldman_et_al:LIPIcs.APPROX/RANDOM.2022.33,
  author =	{Feldman, Moran and Szarf, Ariel},
  title =	{{Maximum Matching Sans Maximal Matching: A New Approach for Finding Maximum Matchings in the Data Stream Model}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{33:1--33:24},
  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.2022.33},
  URN =		{urn:nbn:de:0030-drops-171559},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.33},
  annote =	{Keywords: Maximum matching, semi-streaming algorithms, multi-pass algorithms}
}

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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

DOI: 10.4230/LIPIcs.ITCS.2022.93

Optimal Bounds for Dominating Set in Graph Streams

Authors: Sanjeev Khanna and Christian Konrad

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

Abstract

We resolve the space complexity of one-pass streaming algorithms for Minimum Dominating Set (MDS) in both insertion-only and insertion-deletion streams (up to poly-logarithmic factors) where an input graph is revealed by a sequence of edge updates. Recently, streaming algorithms for the related Set Cover problem have received significant attention. Even though MDS can be viewed as a special case of Set Cover, it is however harder to solve in the streaming setting since the input stream consists of individual edges rather than entire vertex-neighborhoods, as is the case in Set Cover. We prove the following results (n is the number of vertices of the input graph): 1) In insertion-only streams, we give a one-pass semi-streaming algorithm (meaning Õ(n) space) with approximation factor Õ(√n). We also prove that every one-pass streaming algorithm with space o(n) has an approximation factor of Ω(n/log n). Combined with a result by [Assadi et al., STOC'16] for Set Cover which, translated to MDS, shows that space Θ̃(n² / α) is necessary and sufficient for computing an α-approximation for every α = o(√n), this completely settles the space requirements for MDS in the insertion-only setting. 2) In insertion-deletion streams, we prove that space Ω(n² / (α log n)) is necessary for every approximation factor α ≤ Θ(n / log³ n). Combined with the Set Cover algorithm of [Assadi et al., STOC'16], which can be adapted to MDS even in the insertion-deletion setting to give an α-approximation in Õ(n² / α) space, this completely settles the space requirements for MDS in the insertion-deletion setting.

Cite as

Sanjeev Khanna and Christian Konrad. Optimal Bounds for Dominating Set in Graph Streams. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 93:1-93:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{khanna_et_al:LIPIcs.ITCS.2022.93,
  author =	{Khanna, Sanjeev and Konrad, Christian},
  title =	{{Optimal Bounds for Dominating Set in Graph Streams}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{93:1--93:23},
  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.2022.93},
  URN =		{urn:nbn:de:0030-drops-156894},
  doi =		{10.4230/LIPIcs.ITCS.2022.93},
  annote =	{Keywords: Streaming algorithms, communication complexity, information complexity, dominating set}
}

Document

APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.19

On Two-Pass Streaming Algorithms for Maximum Bipartite Matching

Authors: Christian Konrad and Kheeran K. Naidu

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

Abstract

We study two-pass streaming algorithms for Maximum Bipartite Matching (MBM). All known two-pass streaming algorithms for MBM operate in a similar fashion: They compute a maximal matching in the first pass and find 3-augmenting paths in the second in order to augment the matching found in the first pass. Our aim is to explore the limitations of this approach and to determine whether current techniques can be used to further improve the state-of-the-art algorithms. We give the following results: We show that every two-pass streaming algorithm that solely computes a maximal matching in the first pass and outputs a (2/3+ε)-approximation requires n^{1+Ω(1/(log log n))} space, for every ε > 0, where n is the number of vertices of the input graph. This result is obtained by extending the Ruzsa-Szemerédi graph construction of [Goel et al., SODA'12] so as to ensure that the resulting graph has a close to perfect matching, the key property needed in our construction. This result may be of independent interest. Furthermore, we combine the two main techniques, i.e., subsampling followed by the Greedy matching algorithm [Konrad, MFCS'18] which gives a 2-√2 ≈ 0.5857-approximation, and the computation of degree-bounded semi-matchings [Esfandiari et al., ICDMW'16][Kale and Tirodkar, APPROX'17] which gives a 1/2 + 1/12 ≈ 0.5833-approximation, and obtain a meta-algorithm that yields Konrad’s and Esfandiari et al.’s algorithms as special cases. This unifies two strands of research. By optimizing parameters, we discover that Konrad’s algorithm is optimal for the implied class of algorithms and, perhaps surprisingly, that there is a second optimal algorithm. We show that the analysis of our meta-algorithm is best possible. Our results imply that further improvements, if possible, require new techniques.

Cite as

Christian Konrad and Kheeran K. Naidu. On Two-Pass Streaming Algorithms for Maximum Bipartite Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{konrad_et_al:LIPIcs.APPROX/RANDOM.2021.19,
  author =	{Konrad, Christian and Naidu, Kheeran K.},
  title =	{{On Two-Pass Streaming Algorithms for Maximum Bipartite Matching}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{19:1--19:18},
  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.2021.19},
  URN =		{urn:nbn:de:0030-drops-147128},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.19},
  annote =	{Keywords: Data streaming, matchings, lower bounds}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2020.26

Constructing Large Matchings via Query Access to a Maximal Matching Oracle

Authors: Lidiya Khalidah binti Khalil and Christian Konrad

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

Abstract

Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including 2-pass streaming algorithms that break the 1/2-approximation barrier, and (1-ε)-approximation streaming algorithms that run in O(poly 1/ε) passes in bipartite graphs and in O((1/ε)^(1/ε)) or O(poly (1/ε) ⋅ log n) passes in general graphs, where n is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for example, even the existence of 2-pass small space streaming algorithms with approximation factor 0.999 has not yet been ruled out. The key building block of all multi-pass streaming algorithms for Maximum Matching is the Greedy matching algorithm. Our aim is to understand the limitations of this approach: How many passes are required if the algorithm solely relies on the invocation of the Greedy algorithm? In this paper, we initiate the study of lower bounds for restricted families of multi-pass streaming algorithms for Maximum Matching. We focus on the simple yet powerful class of algorithms that in each pass run Greedy on a vertex-induced subgraph of the input graph. In bipartite graphs, we show that 3 passes are necessary and sufficient to improve on the trivial approximation factor of 1/2: We give a lower bound of 0.6 on the approximation ratio of such algorithms, which is optimal. We further show that Ω(1/ε) passes are required for computing a (1-ε)-approximation, even in bipartite graphs. Last, the considered class of algorithms is not well-suited to general graphs: We show that Ω(n) passes are required in order to improve on the trivial approximation factor of 1/2.

Cite as

Lidiya Khalidah binti Khalil and Christian Konrad. Constructing Large Matchings via Query Access to a Maximal Matching Oracle. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{khalil_et_al:LIPIcs.FSTTCS.2020.26,
  author =	{Khalil, Lidiya Khalidah binti and Konrad, Christian},
  title =	{{Constructing Large Matchings via Query Access to a Maximal Matching Oracle}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.26},
  URN =		{urn:nbn:de:0030-drops-132673},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.26},
  annote =	{Keywords: Maximum matching approximation, Query model, Streaming algorithms}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.30

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Authors: Jacques Dark and Christian Konrad

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract

In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice’s edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al. [SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication. Our lower bound is simpler and stronger in several aspects: The lower bound of Assadi et al. only holds for algorithms that (1) are able to process streams that contain a triple exponential number of deletions in n, the number of vertices of the input graph; (2) are able to process multi-graphs; and (3) never output edges that do not exist in the input graph when the randomized algorithm errs. In contrast, our lower bound even holds for algorithms that (1) rely on short (O(n²)-length) input streams; (2) are only able to process simple graphs; and (3) may output non-existing edges when the algorithm errs.

Cite as

Jacques Dark and Christian Konrad. Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dark_et_al:LIPIcs.CCC.2020.30,
  author =	{Dark, Jacques and Konrad, Christian},
  title =	{{Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{30:1--30:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.30},
  URN =		{urn:nbn:de:0030-drops-125824},
  doi =		{10.4230/LIPIcs.CCC.2020.30},
  annote =	{Keywords: Maximum matching, Minimum vertex cover, Dynamic graph streams, Communication complexity, Lower bounds}
}

Document

DOI: 10.4230/LIPIcs.DISC.2019.26

The Complexity of Symmetry Breaking in Massive Graphs

Authors: Christian Konrad, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson

Published in: LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

Abstract

The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related beta-ruling set problem, in two computational models suited for large-scale graph processing, namely the k-machine model and the graph streaming model. We present a number of results. For MIS in the k-machine model, we improve the O~(m/k^2 + Delta/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an O~(m/k^2)-round algorithm. We also present an Omega~(n/k^2) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For beta-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the k-machine model and also in the graph streaming model. More specifically, we obtain a k-machine algorithm that runs in O~(beta n Delta^{1/beta}/k^2) rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use O(beta * n^{1+1/2^{beta-1}}) space. The latter result establishes a clear separation between MIS, which is known to require Omega(n^2) space (Cormode et al., ICALP 2019), and beta-ruling sets, even for beta = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in O~(n/k^{2-epsilon} + k^{1-epsilon}) rounds for any epsilon, 0 <=epsilon <=1. For a wide range of values of k this round complexity simplifies to O~(n/k^2) rounds, which we conjecture is optimal. Our results use a variety of techniques. For our upper bounds, we prove and use simulation theorems for beeping algorithms, hierarchical sampling, and L_0-sampling, whereas for our lower bounds we use information-theoretic arguments and reductions to 2-party communication complexity problems.

Cite as

Christian Konrad, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson. The Complexity of Symmetry Breaking in Massive Graphs. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{konrad_et_al:LIPIcs.DISC.2019.26,
  author =	{Konrad, Christian and Pemmaraju, Sriram V. and Riaz, Talal and Robinson, Peter},
  title =	{{The Complexity of Symmetry Breaking in Massive Graphs}},
  booktitle =	{33rd International Symposium on Distributed Computing (DISC 2019)},
  pages =	{26:1--26:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-126-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{146},
  editor =	{Suomela, Jukka},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.26},
  URN =		{urn:nbn:de:0030-drops-113337},
  doi =		{10.4230/LIPIcs.DISC.2019.26},
  annote =	{Keywords: communication complexity, information theory, k-machine model, maximal independent set, ruling set, streaming algorithms}
}

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