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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
 
 \section{Greedy base packing}
 
+\section{Principal sequence + KT contraction}
+
+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.
+
+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 $P$.
 
 \bibliographystyle{plain}
 \bibliography{ref}

ref.bib

@@ -1,68 +1,72 @@
 @article{cunningham_optimal_1985,
-	title = {Optimal attack and reinforcement of a network},
+	title         = {Optimal attack and reinforcement of a network},
-	volume = {32},
+	volume        = {32},
-	issn = {0004-5411, 1557-735X},
+	issn          = {0004-5411, 1557-735X},
-	url = {https://dl.acm.org/doi/10.1145/3828.3829},
+	url           = {https://dl.acm.org/doi/10.1145/3828.3829},
-	doi = {10.1145/3828.3829},
+	doi           = {10.1145/3828.3829},
-	language = {en},
+	language      = {en},
-	number = {3},
+	number        = {3},
-	urldate = {2025-04-11},
+	urldate       = {2025-04-11},
-	journal = {Journal of the ACM},
+	journal       = {Journal of the ACM},
-	author = {Cunningham, William H.},
+	author        = {Cunningham, William H.},
-	month = jul,
+	month         = jul,
-	year = {1985},
+	year          = {1985},
-	pages = {549--561},
+	pages         = {549--561}
 }
-
-
 @article{chekuri_lp_2020,
-	title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}},
+	title         = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}},
-	volume = {34},
+	volume        = {34},
-	issn = {0895-4801, 1095-7146},
+	issn          = {0895-4801, 1095-7146},
-	url = {https://epubs.siam.org/doi/10.1137/19M1299359},
+	url           = {https://epubs.siam.org/doi/10.1137/19M1299359},
-	doi = {10.1137/19M1299359},
+	doi           = {10.1137/19M1299359},
-	language = {en},
+	language      = {en},
-	number = {2},
+	number        = {2},
-	urldate = {2022-04-10},
+	urldate       = {2022-04-10},
-	journal = {SIAM Journal on Discrete Mathematics},
+	journal       = {SIAM Journal on Discrete Mathematics},
-	author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
+	author        = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
-	month = jan,
+	month         = jan,
-	year = {2020},
+	year          = {2020},
-	keywords = {Approximation, K-cut, Minimum cut, Tree packing},
+	keywords      = {Approximation, K-cut, Minimum cut, Tree packing},
-	pages = {1334--1353},
+	pages         = {1334--1353}
 }
-
 @inproceedings{Thorup_2008,
-	address = {Victoria British Columbia Canada},
+	address       = {Victoria British Columbia Canada},
-	title = {Minimum k-way cuts via deterministic greedy tree packing},
+	title         = {Minimum k-way cuts via deterministic greedy tree packing},
-	ISBN = {9781605580470},
+	isbn          = {9781605580470},
-	url = {https://dl.acm.org/doi/10.1145/1374376.1374402},
+	url           = {https://dl.acm.org/doi/10.1145/1374376.1374402},
-	DOI = {10.1145/1374376.1374402},
+	doi           = {10.1145/1374376.1374402},
-	booktitle = {Proceedings of the fortieth annual ACM symposium on Theory of
+	booktitle     = {Proceedings of the fortieth annual ACM symposium on Theory of computing},
-	             computing},
+	publisher     = {ACM},
-	publisher = {ACM},
+	author        = {Thorup, Mikkel},
-	author = {Thorup, Mikkel},
+	year          = {2008},
-	year = {2008},
+	month         = may,
-	month = may,
+	pages         = {159–166},
-	pages = {159–166},
+	language      = {en}
-	language = {en},
 }
-
 @article{thorup_fully-dynamic_2007,
-	title = {Fully-{Dynamic} {Min}-{Cut}*},
+	title         = {Fully-{Dynamic} {Min}-{Cut}\textasteriskcentered},
-	volume = {27},
+	volume        = {27},
-	issn = {0209-9683, 1439-6912},
+	issn          = {0209-9683, 1439-6912},
-	url = {http://link.springer.com/10.1007/s00493-007-0045-2},
+	url           = {http://link.springer.com/10.1007/s00493-007-0045-2},
-	doi = {10.1007/s00493-007-0045-2},
+	doi           = {10.1007/s00493-007-0045-2},
-	language = {en},
+	language      = {en},
-	number = {1},
+	number        = {1},
-	urldate = {2023-02-05},
+	urldate       = {2023-02-05},
-	journal = {Combinatorica},
+	journal       = {Combinatorica},
-	author = {Thorup, Mikkel},
+	author        = {Thorup, Mikkel},
-	month = feb,
+	month         = feb,
-	year = {2007},
+	year          = {2007},
-	pages = {91--127},
+	pages         = {91--127},
-	file = {Thorup_2007_Fully-Dynamic
+	file          = {Thorup_2007_Fully-Dynamic Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic Min-Cut.pdf:application/pdf}
-	        Min-Cut.pdf:/Users/congyu/Zotero/storage/Q329VHH3/Thorup_2007_Fully-Dynamic
+}
-	        Min-Cut.pdf:application/pdf},
+@article{Daga_2025,
+	title         = {Sub-$n^k$ Deterministic algorithm for minimum $k$-way cut in simple graphs},
+	url           = {http://arxiv.org/abs/2512.12900},
+	doi           = {10.48550/arXiv.2512.12900},
+	note          = {arXiv:2512.12900},
+	number        = {arXiv:2512.12900},
+	publisher     = {arXiv},
+	author        = {Daga, Mohit},
+	year          = {2025},
+	month         = dec
 }