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main.tex
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main.tex
@@ -226,6 +226,11 @@ It would be nice if we can characterize good parts in rigidity matroids with 1-t
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\section{Greedy base packing}
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\section{Principal sequence + KT contraction}
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Recently, Mohit Daga \cite{Daga_2025} combines the principal sequence of partitions and Kawarabayashi-Thorup contractions to get a sub-$n^k$ deterministic algorithm for $k$-cut in simple unweighted graphs.
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Recall that in the ideal tree packing framework \cite{Thorup_2008}, one finds the first partition with $\geq k$ parts in the principal sequence and then merge parts to get a bound. However, the idea in \cite{Daga_2025} is that, instead of finding the first partition with $\geq k$ parts, one finds the last partition $P$ with $<k$ parts and show that the optimal $k$-cut can be expressed as the $E[G/P]$ together with some internal cuts and some singleton isolations inside parts of $P$.
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\bibliographystyle{plain}
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\bibliography{ref}
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ref.bib
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ref.bib
@@ -1,68 +1,72 @@
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@article{cunningham_optimal_1985,
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title = {Optimal attack and reinforcement of a network},
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volume = {32},
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issn = {0004-5411, 1557-735X},
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url = {https://dl.acm.org/doi/10.1145/3828.3829},
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doi = {10.1145/3828.3829},
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language = {en},
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number = {3},
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urldate = {2025-04-11},
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journal = {Journal of the ACM},
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author = {Cunningham, William H.},
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month = jul,
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year = {1985},
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pages = {549--561},
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title = {Optimal attack and reinforcement of a network},
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volume = {32},
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issn = {0004-5411, 1557-735X},
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url = {https://dl.acm.org/doi/10.1145/3828.3829},
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doi = {10.1145/3828.3829},
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language = {en},
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number = {3},
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urldate = {2025-04-11},
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journal = {Journal of the ACM},
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author = {Cunningham, William H.},
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month = jul,
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year = {1985},
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pages = {549--561}
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}
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@article{chekuri_lp_2020,
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title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}},
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volume = {34},
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issn = {0895-4801, 1095-7146},
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url = {https://epubs.siam.org/doi/10.1137/19M1299359},
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doi = {10.1137/19M1299359},
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language = {en},
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number = {2},
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urldate = {2022-04-10},
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journal = {SIAM Journal on Discrete Mathematics},
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author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
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month = jan,
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year = {2020},
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keywords = {Approximation, K-cut, Minimum cut, Tree packing},
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pages = {1334--1353},
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title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}},
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volume = {34},
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issn = {0895-4801, 1095-7146},
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url = {https://epubs.siam.org/doi/10.1137/19M1299359},
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doi = {10.1137/19M1299359},
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language = {en},
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number = {2},
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urldate = {2022-04-10},
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journal = {SIAM Journal on Discrete Mathematics},
|
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author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
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month = jan,
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year = {2020},
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keywords = {Approximation, K-cut, Minimum cut, Tree packing},
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pages = {1334--1353}
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}
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@inproceedings{Thorup_2008,
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address = {Victoria British Columbia Canada},
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title = {Minimum k-way cuts via deterministic greedy tree packing},
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ISBN = {9781605580470},
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url = {https://dl.acm.org/doi/10.1145/1374376.1374402},
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DOI = {10.1145/1374376.1374402},
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booktitle = {Proceedings of the fortieth annual ACM symposium on Theory of
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computing},
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publisher = {ACM},
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author = {Thorup, Mikkel},
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year = {2008},
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month = may,
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pages = {159–166},
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language = {en},
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address = {Victoria British Columbia Canada},
|
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title = {Minimum k-way cuts via deterministic greedy tree packing},
|
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isbn = {9781605580470},
|
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url = {https://dl.acm.org/doi/10.1145/1374376.1374402},
|
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doi = {10.1145/1374376.1374402},
|
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booktitle = {Proceedings of the fortieth annual ACM symposium on Theory of computing},
|
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publisher = {ACM},
|
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author = {Thorup, Mikkel},
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year = {2008},
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month = may,
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pages = {159–166},
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language = {en}
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}
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@article{thorup_fully-dynamic_2007,
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title = {Fully-{Dynamic} {Min}-{Cut}*},
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volume = {27},
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issn = {0209-9683, 1439-6912},
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url = {http://link.springer.com/10.1007/s00493-007-0045-2},
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doi = {10.1007/s00493-007-0045-2},
|
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language = {en},
|
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number = {1},
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urldate = {2023-02-05},
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journal = {Combinatorica},
|
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author = {Thorup, Mikkel},
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month = feb,
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year = {2007},
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pages = {91--127},
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file = {Thorup_2007_Fully-Dynamic
|
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Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic
|
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Min-Cut.pdf:application/pdf},
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title = {Fully-{Dynamic} {Min}-{Cut}\textasteriskcentered},
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volume = {27},
|
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issn = {0209-9683, 1439-6912},
|
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url = {http://link.springer.com/10.1007/s00493-007-0045-2},
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doi = {10.1007/s00493-007-0045-2},
|
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language = {en},
|
||||
number = {1},
|
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urldate = {2023-02-05},
|
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journal = {Combinatorica},
|
||||
author = {Thorup, Mikkel},
|
||||
month = feb,
|
||||
year = {2007},
|
||||
pages = {91--127},
|
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file = {Thorup_2007_Fully-Dynamic Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic Min-Cut.pdf:application/pdf}
|
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}
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@article{Daga_2025,
|
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title = {Sub-$n^k$ Deterministic algorithm for minimum $k$-way cut in simple graphs},
|
||||
url = {http://arxiv.org/abs/2512.12900},
|
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doi = {10.48550/arXiv.2512.12900},
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note = {arXiv:2512.12900},
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number = {arXiv:2512.12900},
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publisher = {arXiv},
|
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author = {Daga, Mohit},
|
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year = {2025},
|
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month = dec
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}
|
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Reference in New Issue
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