complexity

main.tex

@@ -356,11 +356,16 @@ for each 2-approx mincut $C$ in $(G,w_\lambda)$:\\
 return the optimal $(C,F)$
 \end{algo}
 
-\paragraph{time for $\lambda^*$} $L(\lambda)-b\lambda$ is pwl concave. The number of segments is at most $3^m$. We need almost linear time to find the solution to a fixed $\lambda$. So parametric seach gives complexity $m^{1+o(1)} O(\log 3^m)$.
-\note{need to check this}
+To compute $\lambda^*$ we need to use parametric search.
+\begin{lemma}[\cite{salowe_parametric}]
+Let $S(n)$ be the complexity of solving the Lagrangian dual problem for fixed $\lambda$ (where $n$ is the size of the input), then one can compute $\lambda^*$ using parametric search in $O(S(n)^2)$ time.
+\end{lemma}
+It follows directly from the preceding lemma that $\lambda^*$ can be computed in $\tilde O(m^2)$ time.
 
-\paragraph{time for the rest parts} Reweighting takes linear time.
-Finding $<2$-approx mincut takes $\tilde O(n^3)$. FPTAS for knapsack takes $O(\frac{1}{\e}m^2)$. The total complexity is $O(\frac{1}{\e}m^2n^3)$.
+Reweighting the graph takes linear time.
+Finding $<2$-approx mincut takes $\tilde O(n^3)$. 
+An $1+\e$ approximate solution to knapsack can be found in time $\tilde O(m+\frac{1}{\e^2})$ \cite{10.1145/3618260.3649730}.
+The total complexity is $\tilde O(mn^3+\frac{n^3}{\e^2})$.
 
 \bibliographystyle{plain}
 \bibliography{ref}

ref.bib

@@ -81,5 +81,33 @@ abstract = { This paper considers the problem of designing fast, approximate, co
 	pages = {1334--1353},
 }
 
+@inproceedings{10.1145/3618260.3649730,
+author = {Chen, Lin and Lian, Jiayi and Mao, Yuchen and Zhang, Guochuan},
+title = {A Nearly Quadratic-Time FPTAS for Knapsack},
+year = {2024},
+isbn = {9798400703836},
+publisher = {Association for Computing Machinery},
+address = {New York, NY, USA},
+url = {https://doi.org/10.1145/3618260.3649730},
+doi = {10.1145/3618260.3649730},
+abstract = {We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in O(n + (1/)2) time. Prior to our work, the best running time is O(n + (1/)11/5) [Deng, Jin, and Mao’23]. Our algorithm is the best possible (up to a polylogarithmic factor), as Knapsack has no O((n + 1/)2−δ)-time FPTAS for any constant δ > 0, conditioned on the conjecture that (min, +)-convolution has no truly subquadratic-time algorithm.},
+booktitle = {Proceedings of the 56th Annual ACM Symposium on Theory of Computing},
+pages = {283–294},
+numpages = {12},
+keywords = {Approximation scheme, Knapsack},
+location = {Vancouver, BC, Canada},
+series = {STOC 2024}
+}
 
 
+@inbook{salowe_parametric,
+author = {Salowe, Jeffrey S.},
+title = {Parametric search},
+year = {1997},
+isbn = {0849385245},
+publisher = {CRC Press, Inc.},
+address = {USA},
+booktitle = {Handbook of Discrete and Computational Geometry},
+pages = {683–695},
+numpages = {13}
+}