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

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.

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

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

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.

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

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

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.

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} }

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APPROX

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

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.

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} }

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