z

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

@@ -11,10 +11,8 @@
 	author        = {Cunningham, William H.},
 	month         = jul,
 	year          = {1985},
-	pages = {549--561},
+	pages         = {549--561}
 }
-
-
 @article{chekuri_lp_2020,
 	title         = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}},
 	volume        = {34},
@@ -29,27 +27,24 @@
 	month         = jan,
 	year          = {2020},
 	keywords      = {Approximation, K-cut, Minimum cut, Tree packing},
-	pages = {1334--1353},
+	pages         = {1334--1353}
 }
-
 @inproceedings{Thorup_2008,
 	address       = {Victoria British Columbia Canada},
 	title         = {Minimum k-way cuts via deterministic greedy tree packing},
-	ISBN = {9781605580470},
+	isbn          = {9781605580470},
 	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},
 	author        = {Thorup, Mikkel},
 	year          = {2008},
 	month         = may,
 	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},
 	issn          = {0209-9683, 1439-6912},
 	url           = {http://link.springer.com/10.1007/s00493-007-0045-2},
@@ -62,7 +57,16 @@
 	month         = feb,
 	year          = {2007},
 	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
 }