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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.

Cite as

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}
}

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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.

Cite as

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.

Cite as

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.

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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}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2019.21

Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs

Authors: Christian Konrad and Viktor Zamaraev

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract

We give deterministic distributed (1+epsilon)-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in O( (1 / epsilon) log n) rounds, and our independent set algorithm has a runtime of O( (1/epsilon) log(1/epsilon)log^* n) rounds. For coloring, existing lower bounds imply that the dependencies on 1/epsilon and log n are best possible. For independent set, we prove that Omega(1/epsilon) rounds are necessary. Both our algorithms make use of the tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O(log n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first O(log (1/epsilon)) layers are required as they already contain a large enough independent set. We develop a (1+epsilon)-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers. This work raises the question as to how useful tree decompositions are for distributed computing.

Cite as

Christian Konrad and Viktor Zamaraev. Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 21:1-21:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{konrad_et_al:LIPIcs.MFCS.2019.21,
  author =	{Konrad, Christian and Zamaraev, Viktor},
  title =	{{Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.21},
  URN =		{urn:nbn:de:0030-drops-109651},
  doi =		{10.4230/LIPIcs.MFCS.2019.21},
  annote =	{Keywords: local model, approximation algorithms, minimum vertex coloring, maximum independent set, chordal graphs}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2019.45

Independent Sets in Vertex-Arrival Streams

Authors: Graham Cormode, Jacques Dark, and Christian Konrad

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

Abstract

We consider the maximal and maximum independent set problems in three models of graph streams: - In the edge model we see a stream of edges which collectively define a graph; this model is well-studied for a variety of problems. We show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that it is not much easier if we only require approximate maximality. This contrasts strongly with the other two vertex-based models, where one can greedily find an exact solution in only the space needed to store the independent set. - In the "explicit" vertex model, the input stream is a sequence of vertices making up the graph. Every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than in edge-arrival streams. We show that every one-pass c-approximation streaming algorithm for maximum independent set (MIS) on explicit vertex streams requires Omega({n^2}/{c^6}) bits of space, where n is the number of vertices of the input graph. It is already known that Theta~({n^2}/{c^2}) bits of space are necessary and sufficient in the edge arrival model (Halldórsson et al. 2012), thus the MIS problem is not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction from a new multi-party communication problem closely related to pointer jumping. - In the "implicit" vertex model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that maps pairs of objects to the presence or absence of edges, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for interval and square intersection graphs, in both explicit and implicit streams. In particular, we show a gap between the hardness of the explicit and implicit vertex models for interval graphs.

Cite as

Graham Cormode, Jacques Dark, and Christian Konrad. Independent Sets in Vertex-Arrival Streams. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cormode_et_al:LIPIcs.ICALP.2019.45,
  author =	{Cormode, Graham and Dark, Jacques and Konrad, Christian},
  title =	{{Independent Sets in Vertex-Arrival Streams}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{45:1--45:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.45},
  URN =		{urn:nbn:de:0030-drops-106212},
  doi =		{10.4230/LIPIcs.ICALP.2019.45},
  annote =	{Keywords: streaming algorithms, independent set size, lower bounds}
}

Document

DOI: 10.4230/LIPIcs.DISC.2018.16

Detecting Cliques in CONGEST Networks

Authors: Artur Czumaj and Christian Konrad

Published in: LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

Abstract

The problem of detecting network structures plays a central role in distributed computing. One of the fundamental problems studied in this area is to determine whether for a given graph H, the input network contains a subgraph isomorphic to H or not. We investigate this problem for H being a clique K_l in the classical distributed CONGEST model, where the communication topology is the same as the topology of the underlying network, and with limited communication bandwidth on the links. Our first and main result is a lower bound, showing that detecting K_l requires Omega(sqrt{n} / b) communication rounds, for every 4 <=l <=sqrt{n}, and Omega(n / (l b)) rounds for every l >= sqrt{n}, where b is the bandwidth of the communication links. This result is obtained by using a reduction to the set disjointness problem in the framework of two-party communication complexity. We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph, which up to constant factors communicates the same number of bits as our lower bound for K_4 detection. This demonstrates that our lower bound cannot be improved using the two-party communication framework.

Cite as

Artur Czumaj and Christian Konrad. Detecting Cliques in CONGEST Networks. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 16:1-16:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{czumaj_et_al:LIPIcs.DISC.2018.16,
  author =	{Czumaj, Artur and Konrad, Christian},
  title =	{{Detecting Cliques in CONGEST Networks}},
  booktitle =	{32nd International Symposium on Distributed Computing (DISC 2018)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-092-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{121},
  editor =	{Schmid, Ulrich and Widder, Josef},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.16},
  URN =		{urn:nbn:de:0030-drops-98057},
  doi =		{10.4230/LIPIcs.DISC.2018.16},
  annote =	{Keywords: Lower bounds, CONGEST, subgraph detection, two-party communication}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2018.74

A Simple Augmentation Method for Matchings with Applications to Streaming Algorithms

Authors: Christian Konrad

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract

Given a graph G, it is well known that any maximal matching M in G is at least half the size of a maximum matching M^*. In this paper, we show that if G is bipartite, then running the Greedy matching algorithm on a sampled subgraph of G produces enough additional edges that can be used to augment M such that the resulting matching is of size at least (2 - sqrt{2})|M^*| ~~ 0.5857 |M^*| (ignoring lower order terms) with high probability. The main applications of our method lie in the area of data streaming algorithms, where an algorithm performs few passes over the edges of an n-vertex graph while maintaining a memory of size O(n polylog n). Our method immediately yields a very simple two-pass algorithm for Maximum Bipartite Matching (MBM) with approximation factor 0.5857, which only runs the Greedy matching algorithm in each pass. This slightly improves on the much more involved 0.583-approximation algorithm of Esfandiari et al. [ICDMW 2016]. To obtain our main result, we combine our method with a residual sparsity property of the random order Greedy algorithm and give a one-pass random order streaming algorithm for MBM with approximation factor 0.5395. This substantially improves upon the one-pass random order 0.505-approximation algorithm of Konrad et al. [APPROX 2012].

Cite as

Christian Konrad. A Simple Augmentation Method for Matchings with Applications to Streaming Algorithms. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 74:1-74:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{konrad:LIPIcs.MFCS.2018.74,
  author =	{Konrad, Christian},
  title =	{{A Simple Augmentation Method for Matchings with Applications to Streaming Algorithms}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{74:1--74:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.74},
  URN =		{urn:nbn:de:0030-drops-96560},
  doi =		{10.4230/LIPIcs.MFCS.2018.74},
  annote =	{Keywords: Matchings, augmenting paths, streaming algorithms, random order}
}

Document

DOI: 10.4230/LITES-v004-i002-a002

EMSBench: Benchmark and Testbed for Reactive Real-Time Systems

Authors: Florian Kluge, Christine Rochange, and Theo Ungerer

Published in: LITES, Volume 4, Issue 2 (2017). Leibniz Transactions on Embedded Systems, Volume 4, Issue 2

Abstract

Benchmark suites for real-time embedded systems (RTES) usually contain only pure computations that are often used in this domain. They allow to evaluate computing performance, but do not reproduce the complexity and behaviour that is typical for such systems. Actual RTES have to interact with the physical environment, which is often reflected by code that is executed concurrently. In this article, we present the software package EMSBench that mimics such complex behaviour, and highlight some of its use cases. The benchmark code ems of EMSBench is based on the open-source engine management system (EMS) FreeEMS. Additionally, EMSBench contains a trace generator (tg) that provides input signals for ems and enables to execute ems close to reality. We provide detailed descriptions of the ems's execution behaviour and of trace generation. EMSBench can be used as test or benchmark program to compare different hardware platforms, e.g. in terms of schedulability. Also, we use EMSBench as a benchmark for static worst-case execution time (WCET) analysis and compare these results to measurements performed on existing hardware. Our results based on the OTAWA WCET estimation tool show WCET overestimations by the static analysis from 11.9% to 41.1% depending on the complexity of the analysed functions.

Cite as

Florian Kluge, Christine Rochange, and Theo Ungerer. EMSBench: Benchmark and Testbed for Reactive Real-Time Systems. In LITES, Volume 4, Issue 2 (2017). Leibniz Transactions on Embedded Systems, Volume 4, Issue 2, pp. 02:1-02:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@Article{kluge_et_al:LITES-v004-i002-a002,
  author =	{Kluge, Florian and Rochange, Christine and Ungerer, Theo},
  title =	{{EMSBench: Benchmark and Testbed for Reactive Real-Time Systems}},
  booktitle =	{LITES, Volume 4, Issue 2 (2017)},
  pages =	{02:1--02:23},
  journal =	{Leibniz Transactions on Embedded Systems},
  ISSN =	{2199-2002},
  year =	{2017},
  volume =	{4},
  number =	{2},
  editor =	{Kluge, Florian and Rochange, Christine and Ungerer, Theo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LITES-v004-i002-a002},
  doi =		{10.4230/LITES-v004-i002-a002},
  annote =	{Keywords: Real-time benchmark, WCET Analysis, Engine Management System}
}

Document

DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.6

The Densest k-Subhypergraph Problem

Authors: Eden Chlamtac, Michael Dinitz, Christian Konrad, Guy Kortsarz, and George Rabanca

Published in: LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Abstract

The Densest k-Subgraph (DkS) problem, and its corresponding minimization problem Smallest p-Edge Subgraph (SpES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both DkS and SpES from graphs to hypergraphs. We consider the Densest k-Subhypergraph problem (given a hypergraph (V, E), find a subset W subseteq V of k vertices so as to maximize the number of hyperedges contained in W) and define the Minimum p-Union problem (given a hypergraph, choose p of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n^{4(4-sqrt{3})/13 + epsilon}) <= O(n^{0.697831+epsilon})-approximation (for arbitrary constant epsilon > 0) for Densest k-Subhypergraph and an ~O(n^{2/5})-approximation for Minimum p-Union. We also give an O(sqrt{m})-approximation for Minimum p-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.

Cite as

Eden Chlamtac, Michael Dinitz, Christian Konrad, Guy Kortsarz, and George Rabanca. The Densest k-Subhypergraph Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 6:1-6:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chlamtac_et_al:LIPIcs.APPROX-RANDOM.2016.6,
  author =	{Chlamtac, Eden and Dinitz, Michael and Konrad, Christian and Kortsarz, Guy and Rabanca, George},
  title =	{{The Densest k-Subhypergraph Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.6},
  URN =		{urn:nbn:de:0030-drops-66298},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.6},
  annote =	{Keywords: Hypergraphs, Approximation algorithms}
}

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