This commit is contained in:
27
main.tex
27
main.tex
@@ -14,7 +14,7 @@
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\section{``Cut-free'' Proof}
|
||||
\section{Generalize proof in [Huang \etal{}, IPCO'24]}
|
||||
\begin{problem}[b-free knapsack]\label{bfreeknap}
|
||||
Consider a set of elements $E$ with weights $w:E\to \Z_+$ and cost $c:E\to \Z_+$ and a budget $b\in \Z_+$. Given a feasible set $\mathcal F\subset 2^E$, find $\min_{X\in \mathcal F, F\subset E} w(X\setminus F)$ such that $c(F)\leq b$.
|
||||
\end{problem}
|
||||
@@ -23,7 +23,7 @@ Note that $\mathcal F$ is usually not explicitly given.
|
||||
\begin{problem}[Normalized knapsack]\label{nknap}
|
||||
Given the same input as \autoref{bfreeknap}, find $\min \limits_{X\in \mathcal F, F\subset E} \frac{w(X\setminus F)}{B-c(F)}\; s.t.\; c(F)\leq b$.
|
||||
\end{problem}
|
||||
In \cite{vygen_fptas_2024} the normalized min-cut problem use $B=b+1$. Here we use any integer $B>b$ and see how their method works.
|
||||
In \cite{huang_fptas_2024} the normalized min-cut problem use $B=b+1$. Here we use any integer $B>b$ and see how their method works.
|
||||
|
||||
Let $\tau$ be the optimum of \autoref{nknap}. Define a new weight $w_\tau:\E\to \R$,
|
||||
|
||||
@@ -38,7 +38,7 @@ w_\tau(e)=\begin{cases}
|
||||
Let $(X^N,F^N)$ be the optimal solution to \autoref{nknap}.
|
||||
Every element in $F^N$ is heavy.
|
||||
\end{lemma}
|
||||
The proof is the same as \cite[Lemma 1]{vygen_fptas_2024}.
|
||||
The proof is the same as \cite[Lemma 1]{huang_fptas_2024}.
|
||||
|
||||
The following two lemmas show (a general version of) that the optimal cut $C^N$ to normalized min-cut is exactly the minimum cut under weights $w_\tau$.
|
||||
|
||||
@@ -60,7 +60,7 @@ The following two lemmas show (a general version of) that the optimal cut $C^N$
|
||||
Thus by \autoref{lem:lb}, $X^N$ gets the minimum.
|
||||
\end{proof}
|
||||
|
||||
% Now we show the counter part of \cite[Theorem 5]{vygen_fptas_2024}, which states the optimal solution to \autoref{bfreeknap} is a $\alpha$-approximate solution to $\min_{F\in \mathcal{F}} w_\tau(F)$.
|
||||
% Now we show the counter part of \cite[Theorem 5]{huang_fptas_2024}, which states the optimal solution to \autoref{bfreeknap} is a $\alpha$-approximate solution to $\min_{F\in \mathcal{F}} w_\tau(F)$.
|
||||
|
||||
\begin{lemma}\label{lem:conditionalLB}
|
||||
Let $(X^*,F^*)$ be the optimal solution to \autoref{bfreeknap}.
|
||||
@@ -70,8 +70,8 @@ The following two lemmas show (a general version of) that the optimal cut $C^N$
|
||||
\end{lemma}
|
||||
|
||||
% In fact, corollary 1 and theorem 5 are also the same as those in
|
||||
% \cite{vygen_fptas_2024}.
|
||||
Then following arguments in \cite[Corollary 1]{vygen_fptas_2024}, assume that $X^*$ is not an $\alpha$-approximate solution to $\min_{X\in\mathcal F}
|
||||
% \cite{huang_fptas_2024}.
|
||||
Then following arguments in \cite[Corollary 1]{huang_fptas_2024}, assume that $X^*$ is not an $\alpha$-approximate solution to $\min_{X\in\mathcal F}
|
||||
w_\tau(X)$ for some $\alpha>1$. We have
|
||||
\[
|
||||
\frac{w(C^N\setminus F^N)}{w(C^*\setminus F^*)}\leq \frac{\tau(B-c(F^N))}{\tau(\alpha B-b)}\leq \frac{B}{\alpha B-b},
|
||||
@@ -79,7 +79,7 @@ Then following arguments in \cite[Corollary 1]{vygen_fptas_2024}, assume that $X
|
||||
where the second inequality uses \autoref{lem:conditionalLB}.
|
||||
One can see that if $\alpha>2$, $\frac{w(C^N\setminus F^N)}{w(C^*\setminus F^*)}\leq \frac{B}{\alpha B-b} <1$ which implies $(C^*,F^*)$ is not optimal. Thus for $\alpha >2$, $X^*$ must be a $2$-approximate solution to $\min_{X\in\mathcal F} w_\tau(X)$.
|
||||
|
||||
Finally we get a general version of \cite[Theorem 4]{vygen_fptas_2024}:
|
||||
Finally we get a general version of \cite[Theorem 4]{huang_fptas_2024}:
|
||||
\begin{theorem}\label{thm:main}
|
||||
Let $X^{\min}$ be the optimal solution to $\min_{X\in\mathcal F} w_\tau(X)$.
|
||||
The optimal set $X^*$ in \autoref{bfreeknap} is a
|
||||
@@ -88,7 +88,7 @@ Finally we get a general version of \cite[Theorem 4]{vygen_fptas_2024}:
|
||||
|
||||
Thus to obtain a FPTAS for \autoref{bfreeknap}, one need to design a FPTAS for \autoref{nknap} and a polynomial time algorithm for finding all 2-approximations to $\min_{X\in\mathcal F} w_\tau(X)$.
|
||||
|
||||
\paragraph{FPTAS for \autoref{nknap} in \cite{vygen_fptas_2024}} (The name
|
||||
\paragraph{FPTAS for \autoref{nknap} in \cite{huang_fptas_2024}} (The name
|
||||
``FPTAS'' here is not precise since we do not have a approximation scheme but
|
||||
an enumeration algorithm. But I will use this term anyway.) In their settings,
|
||||
$\mathcal F$ is the collection of all cuts in some graph.
|
||||
@@ -110,7 +110,7 @@ Let $(C,F)$ be the optimal solution to connectivity interdiction. The optimum
|
||||
cut $C$ can be computed in polynomial time.
|
||||
\end{conjecture}
|
||||
|
||||
Note that there is a FPTAS algorithm for finding $C$ in \cite{vygen_fptas_2024}.
|
||||
Note that there is a FPTAS algorithm for finding $C$ in \cite{huang_fptas_2024}.
|
||||
|
||||
\section{Connections}
|
||||
For unit weight and cost, connectivity interdiction with budget $b=k-1$ is the same
|
||||
@@ -118,7 +118,7 @@ problem as finding the minimum weighted edge set whose removal breaks $k$-edge
|
||||
connectivity.
|
||||
|
||||
\autoref{nknap} may come from an intermediate problem of MWU methods for positive covering LPs.
|
||||
% Authors of \cite{vygen_fptas_2024} $\subset$ authors of
|
||||
% Authors of \cite{huang_fptas_2024} $\subset$ authors of
|
||||
% \cite{chalermsook_approximating_2022}.
|
||||
|
||||
Can we get an FPTAS using LP methods?
|
||||
@@ -340,9 +340,12 @@ Note that everything in blue is non-negative.
|
||||
And we get that upperbound of $\lambda^*b$ by throwing away all blue terms and using $c(F^*)\leq b$.
|
||||
|
||||
Can we show that the gap is 0 or much smaller than 2?
|
||||
\begin{enumerate}
|
||||
\item One cannot do better than $b\lambda^*$ for general costs.
|
||||
There are examples (a 4-vertex path with parallel edges) where the gap is almost $b\lambda^*$.\footnote{see \url{https://gitea.talldoor.uk/sxlxc/edge_conn_interdiction/src/branch/master/gap.py}}
|
||||
\item Unit cost. We can assume WLOG that $|C^*|>b$ and that $F^*$ is the set of $b$ edges in $C^*$ with largest weights. By the complementary slackness condition, $(C^{LD},F^{LD})$ is optimal for connectivity interdiction IP. Thus we can see the gap is $1$.
|
||||
\end{enumerate}
|
||||
|
||||
No. There are examples (a 4-vertex path with parallel edges) where the gap is almost $b\lambda^*$.\footnote{see \url{https://gitea.talldoor.uk/sxlxc/edge_conn_interdiction/src/branch/master/gap.py}}
|
||||
One cannot do better than $b\lambda^*$.
|
||||
\end{remark}
|
||||
|
||||
\subsection{general objective function}
|
||||
|
||||
193
ref.bib
193
ref.bib
@@ -1,113 +1,104 @@
|
||||
|
||||
@inproceedings{vygen_fptas_2024,
|
||||
address = {Cham},
|
||||
title = {An {FPTAS} for {Connectivity} {Interdiction}},
|
||||
volume = {14679},
|
||||
isbn = {978-3-031-59834-0 978-3-031-59835-7},
|
||||
url = {https://link.springer.com/10.1007/978-3-031-59835-7_16},
|
||||
language = {en},
|
||||
urldate = {2024-11-04},
|
||||
booktitle = {Integer {Programming} and {Combinatorial} {Optimization}},
|
||||
publisher = {Springer Nature Switzerland},
|
||||
author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai},
|
||||
editor = {Vygen, Jens and Byrka, Jarosław},
|
||||
year = {2024},
|
||||
doi = {10.1007/978-3-031-59835-7_16},
|
||||
pages = {210--223},
|
||||
@inproceedings{huang_fptas_2024,
|
||||
address = {Cham},
|
||||
title = {An {FPTAS} for {Connectivity} {Interdiction}},
|
||||
volume = {14679},
|
||||
isbn = {978-3-031-59834-0 978-3-031-59835-7},
|
||||
url = {https://link.springer.com/10.1007/978-3-031-59835-7_16},
|
||||
language = {en},
|
||||
urldate = {2024-11-04},
|
||||
booktitle = {Integer {Programming} and {Combinatorial} {Optimization}},
|
||||
publisher = {Springer Nature Switzerland},
|
||||
author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai},
|
||||
editor = {Vygen, Jens and Byrka, Jaros\l{}aw},
|
||||
year = {2024},
|
||||
doi = {10.1007/978-3-031-59835-7_16},
|
||||
pages = {210--223}
|
||||
}
|
||||
|
||||
@article{chalermsook_approximating_2022,
|
||||
title = {Approximating k-{Edge}-{Connected} {Spanning} {Subgraphs} via a {Near}-{Linear} {Time} {LP} {Solver}},
|
||||
volume = {229},
|
||||
copyright = {Creative Commons Attribution 4.0 International license, info:eu-repo/semantics/openAccess},
|
||||
issn = {1868-8969},
|
||||
url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.37},
|
||||
doi = {10.4230/LIPICS.ICALP.2022.37},
|
||||
language = {en},
|
||||
urldate = {2025-03-09},
|
||||
journal = {LIPIcs, Volume 229, ICALP 2022},
|
||||
author = {Chalermsook, Parinya and Huang, Chien-Chung and Nanongkai, Danupon and Saranurak, Thatchaphol and Sukprasert, Pattara and Yingchareonthawornchai, Sorrachai},
|
||||
editor = {Bojańczyk, Mikołaj and Merelli, Emanuela and Woodruff, David P.},
|
||||
year = {2022},
|
||||
keywords = {Approximation Algorithms, Data Structures, Theory of computation → Routing and network design problems},
|
||||
pages = {37:1--37:20},
|
||||
title = {Approximating k-{Edge}-{Connected} {Spanning} {Subgraphs} via a {Near}-{Linear} {Time} {LP} {Solver}},
|
||||
volume = {229},
|
||||
copyright = {Creative Commons Attribution 4.0 International license, info:eu-repo/semantics/openAccess},
|
||||
issn = {1868-8969},
|
||||
url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.37},
|
||||
doi = {10.4230/LIPICS.ICALP.2022.37},
|
||||
language = {en},
|
||||
urldate = {2025-03-09},
|
||||
journal = {LIPIcs, Volume 229, ICALP 2022},
|
||||
author = {Chalermsook, Parinya and Huang, Chien-Chung and Nanongkai, Danupon and Saranurak, Thatchaphol and Sukprasert, Pattara and Yingchareonthawornchai, Sorrachai},
|
||||
editor = {Boja\'{n}czyk, Miko\l{}aj and Merelli, Emanuela and Woodruff, David P.},
|
||||
year = {2022},
|
||||
keywords = {Approximation Algorithms, Data Structures, Theory of computation \rightarrow{} Routing and network design problems},
|
||||
pages = {37:1--37:20}
|
||||
}
|
||||
|
||||
@article{garg_faster_nodate,
|
||||
author = {Garg, Naveen and K\"{o}nemann, Jochen},
|
||||
title = {Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems},
|
||||
journal = {SIAM Journal on Computing},
|
||||
volume = {37},
|
||||
number = {2},
|
||||
pages = {630-652},
|
||||
year = {2007},
|
||||
doi = {10.1137/S0097539704446232},
|
||||
URL = { https://doi.org/10.1137/S0097539704446232},
|
||||
eprint = {https://doi.org/10.1137/S0097539704446232},
|
||||
abstract = { This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new, faster, and much simpler algorithms for these problems. }
|
||||
author = {Garg, Naveen and K\"{o}nemann, Jochen},
|
||||
title = {Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems},
|
||||
journal = {SIAM Journal on Computing},
|
||||
volume = {37},
|
||||
number = {2},
|
||||
pages = {630--652},
|
||||
year = {2007},
|
||||
doi = {10.1137/S0097539704446232},
|
||||
url = {https://doi.org/10.1137/S0097539704446232},
|
||||
eprint = {https://doi.org/10.1137/S0097539704446232},
|
||||
abstract = {This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new, faster, and much simpler algorithms for these problems.}
|
||||
}
|
||||
|
||||
@article{cunningham_optimal_1985,
|
||||
title = {Optimal attack and reinforcement of a network},
|
||||
volume = {32},
|
||||
issn = {0004-5411, 1557-735X},
|
||||
url = {https://dl.acm.org/doi/10.1145/3828.3829},
|
||||
doi = {10.1145/3828.3829},
|
||||
language = {en},
|
||||
number = {3},
|
||||
urldate = {2025-04-11},
|
||||
journal = {Journal of the ACM},
|
||||
author = {Cunningham, William H.},
|
||||
month = jul,
|
||||
year = {1985},
|
||||
pages = {549--561},
|
||||
title = {Optimal attack and reinforcement of a network},
|
||||
volume = {32},
|
||||
issn = {0004-5411, 1557-735X},
|
||||
url = {https://dl.acm.org/doi/10.1145/3828.3829},
|
||||
doi = {10.1145/3828.3829},
|
||||
language = {en},
|
||||
number = {3},
|
||||
urldate = {2025-04-11},
|
||||
journal = {Journal of the ACM},
|
||||
author = {Cunningham, William H.},
|
||||
month = jul,
|
||||
year = {1985},
|
||||
pages = {549--561}
|
||||
}
|
||||
|
||||
|
||||
@article{chekuri_lp_2020,
|
||||
title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}},
|
||||
volume = {34},
|
||||
issn = {0895-4801, 1095-7146},
|
||||
url = {https://epubs.siam.org/doi/10.1137/19M1299359},
|
||||
doi = {10.1137/19M1299359},
|
||||
language = {en},
|
||||
number = {2},
|
||||
urldate = {2022-04-10},
|
||||
journal = {SIAM Journal on Discrete Mathematics},
|
||||
author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
|
||||
month = jan,
|
||||
year = {2020},
|
||||
keywords = {Approximation, K-cut, Minimum cut, Tree packing},
|
||||
pages = {1334--1353},
|
||||
title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}},
|
||||
volume = {34},
|
||||
issn = {0895-4801, 1095-7146},
|
||||
url = {https://epubs.siam.org/doi/10.1137/19M1299359},
|
||||
doi = {10.1137/19M1299359},
|
||||
language = {en},
|
||||
number = {2},
|
||||
urldate = {2022-04-10},
|
||||
journal = {SIAM Journal on Discrete Mathematics},
|
||||
author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
|
||||
month = jan,
|
||||
year = {2020},
|
||||
keywords = {Approximation, K-cut, Minimum cut, Tree packing},
|
||||
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}
|
||||
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-\ensuremath{\delta})-time FPTAS for any constant \ensuremath{\delta} > 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}
|
||||
}
|
||||
|
||||
|
||||
@incollection{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}
|
||||
}
|
||||
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}
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user