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Documents authored by Blikstad, Joakim

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DOI: 10.4230/LIPIcs.ESA.2023.22
Incremental (1-ε)-Approximate Dynamic Matching in O(poly(1/ε)) Update Time

Authors: Joakim Blikstad and Peter Kiss

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
In the dynamic approximate maximum bipartite matching problem we are given bipartite graph G undergoing updates and our goal is to maintain a matching of G which is large compared the maximum matching size μ(G). We define a dynamic matching algorithm to be α (respectively (α, β))-approximate if it maintains matching M such that at all times |M | ≥ μ(G) ⋅ α (respectively |M| ≥ μ(G) ⋅ α - β). We present the first deterministic (1-ε)-approximate dynamic matching algorithm with O(poly(ε^{-1})) amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or exponential in 1/ε [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive (1, ε⋅n)-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for (1-ε)-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of G in a fully dynamic manner. Our algorithm relies on the weighted variant of the celebrated Edge-Degree-Constrained-Subgraph (EDCS) datastructure introduced by [Bernstein-Stein ICALP'15]. As far as we are aware we introduce the first application of the weighted-EDCS for arbitrarily dense graphs. We also present a significantly simplified proof for the approximation ratio of weighed-EDCS as a matching sparsifier compared to [Bernstein-Stein], as well as simple descriptions of a fractional matching and fractional vertex cover defined on top of the EDCS. Considering the wide range of applications EDCS has found in settings such as streaming, sub-linear, stochastic and more we hope our techniques will be of independent research interest outside of the dynamic setting.
Cite as

Joakim Blikstad and Peter Kiss. Incremental (1-ε)-Approximate Dynamic Matching in O(poly(1/ε)) Update Time. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blikstad_et_al:LIPIcs.ESA.2023.22,
  author =	{Blikstad, Joakim and Kiss, Peter},
  title =	{{Incremental (1-\epsilon)-Approximate Dynamic Matching in O(poly(1/\epsilon)) Update Time}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{22:1--22:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.22},
  URN =		{urn:nbn:de:0030-drops-186756},
  doi =		{10.4230/LIPIcs.ESA.2023.22},
  annote =	{Keywords: Bipartite Matching, Incremental Matching, Dynamic Algorithms, Approximation Algorithms, EDCS}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.25
Sublinear-Round Parallel Matroid Intersection

Authors: Joakim Blikstad

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

Abstract
Despite a lot of recent progress in obtaining faster sequential matroid intersection algorithms, the fastest parallel poly(n)-query algorithm was still the straightforward O(n)-round parallel implementation of Edmonds' augmenting paths algorithm from the 1960s. Very recently, Chakrabarty-Chen-Khanna [FOCS'21] showed the lower bound that any, possibly randomized, parallel matroid intersection algorithm making poly(n) rank-queries requires Ω̃(n^{1/3}) rounds of adaptivity. They ask, as an open question, if the lower bound can be improved to Ω̃(n), or if there can be sublinear-round, poly(n)-query algorithms for matroid intersection. We resolve this open problem by presenting the first sublinear-round parallel matroid intersection algorithms. Perhaps surprisingly, we do not only break the Õ(n)-barrier in the rank-oracle model, but also in the weaker independence-oracle model. Our rank-query algorithm guarantees O(n^{3/4}) rounds of adaptivity, while the independence-query algorithm uses O(n^{7/8}) rounds of adaptivity, both making a total of poly(n) queries.
Cite as

Joakim Blikstad. Sublinear-Round Parallel Matroid Intersection. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blikstad:LIPIcs.ICALP.2022.25,
  author =	{Blikstad, Joakim},
  title =	{{Sublinear-Round Parallel Matroid Intersection}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.25},
  URN =		{urn:nbn:de:0030-drops-163662},
  doi =		{10.4230/LIPIcs.ICALP.2022.25},
  annote =	{Keywords: Matroid Intersection, Combinatorial Optimization, Parallel Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.31
Breaking O(nr) for Matroid Intersection

Authors: Joakim Blikstad

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
We present algorithms that break the Õ(nr)-independence-query bound for the Matroid Intersection problem for the full range of r; where n is the size of the ground set and r ≤ n is the size of the largest common independent set. The Õ(nr) bound was due to the efficient implementations [CLSSW FOCS'19; Nguyên 2019] of the classic algorithm of Cunningham [SICOMP'86]. It was recently broken for large r (r = ω(√n)), first by the Õ(n^{1.5}/ε^{1.5})-query (1-ε)-approximation algorithm of CLSSW [FOCS'19], and subsequently by the Õ(n^{6/5}r^{3/5})-query exact algorithm of BvdBMN [STOC'21]. No algorithm - even an approximation one - was known to break the Õ(nr) bound for the full range of r. We present an Õ(n√r/ε)-query (1-ε)-approximation algorithm and an Õ(nr^{3/4})-query exact algorithm. Our algorithms improve the Õ(nr) bound and also the bounds by CLSSW and BvdBMN for the full range of r.
Cite as

Joakim Blikstad. Breaking O(nr) for Matroid Intersection. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blikstad:LIPIcs.ICALP.2021.31,
  author =	{Blikstad, Joakim},
  title =	{{Breaking O(nr) for Matroid Intersection}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{31:1--31:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela 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.ICALP.2021.31},
  URN =		{urn:nbn:de:0030-drops-141004},
  doi =		{10.4230/LIPIcs.ICALP.2021.31},
  annote =	{Keywords: Matroid Intersection, Combinatorial Optimization, Approximation Algorithms}
}
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