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main.tex
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@@ -14,7 +14,7 @@
\begin{document} \begin{document}
\maketitle \maketitle
\section{``Cut-free'' Proof} \section{Generalize proof in [Huang \etal{}, IPCO'24]}
\begin{problem}[b-free knapsack]\label{bfreeknap} \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 $c(F)\leq b$. 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} \end{problem}
@@ -23,7 +23,7 @@ Note that $\mathcal F$ is usually not explicitly given.
\begin{problem}[Normalized knapsack]\label{nknap} \begin{problem}[Normalized knapsack]\label{nknap}
Given the same input as \autoref{bfreeknap}, find $\min \limits_{X\in \mathcal F, F\subset E} \frac{w(X\setminus F)}{B-c(F)}\; s.t.\; c(F)\leq b$. 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} \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 $w_\tau:\E\to \R$, 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}. Let $(X^N,F^N)$ be the optimal solution to \autoref{nknap}.
Every element in $F^N$ is heavy. Every element in $F^N$ is heavy.
\end{lemma} \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 $w_\tau$. 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. Thus by \autoref{lem:lb}, $X^N$ gets the minimum.
\end{proof} \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} \begin{lemma}\label{lem:conditionalLB}
Let $(X^*,F^*)$ be the optimal solution to \autoref{bfreeknap}. 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} \end{lemma}
% In fact, corollary 1 and theorem 5 are also the same as those in % In fact, corollary 1 and theorem 5 are also the same as those in
% \cite{vygen_fptas_2024}. % \cite{huang_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} 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 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}, \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}. 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 $\min_{X\in\mathcal F} w_\tau(X)$. 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} \begin{theorem}\label{thm:main}
Let $X^{\min}$ be the optimal solution to $\min_{X\in\mathcal F} w_\tau(X)$. 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 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 $\min_{X\in\mathcal F} w_\tau(X)$. 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 ``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, an enumeration algorithm. But I will use this term anyway.) In their settings,
$\mathcal F$ is the collection of all cuts in some graph. $\mathcal F$ 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. cut $C$ can be computed in polynomial time.
\end{conjecture} \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} \section{Connections}
For unit weight and cost, connectivity interdiction with budget $b=k-1$ is the same 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. connectivity.
\autoref{nknap} may come from an intermediate problem of MWU methods for positive covering LPs. \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}. % \cite{chalermsook_approximating_2022}.
Can we get an FPTAS using LP methods? 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 $c(F^*)\leq b$. 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? 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} \end{remark}
\subsection{general objective function} \subsection{general objective function}

95
ref.bib
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@@ -1,5 +1,4 @@
@inproceedings{huang_fptas_2024,
@inproceedings{vygen_fptas_2024,
address = {Cham}, address = {Cham},
title = {An {FPTAS} for {Connectivity} {Interdiction}}, title = {An {FPTAS} for {Connectivity} {Interdiction}},
volume = {14679}, volume = {14679},
@@ -10,12 +9,11 @@
booktitle = {Integer {Programming} and {Combinatorial} {Optimization}}, booktitle = {Integer {Programming} and {Combinatorial} {Optimization}},
publisher = {Springer Nature Switzerland}, publisher = {Springer Nature Switzerland},
author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai}, author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai},
editor = {Vygen, Jens and Byrka, Jarosław}, editor = {Vygen, Jens and Byrka, Jaros\l{}aw},
year = {2024}, year = {2024},
doi = {10.1007/978-3-031-59835-7_16}, doi = {10.1007/978-3-031-59835-7_16},
pages = {210--223}, pages = {210--223}
} }
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volume = {229}, volume = {229},
@@ -27,26 +25,24 @@
urldate = {2025-03-09}, urldate = {2025-03-09},
journal = {LIPIcs, Volume 229, ICALP 2022}, journal = {LIPIcs, Volume 229, ICALP 2022},
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editor = {Bojańczyk, Mikołaj and Merelli, Emanuela and Woodruff, David P.}, editor = {Boja\'{n}czyk, Miko\l{}aj and Merelli, Emanuela and Woodruff, David P.},
year = {2022}, year = {2022},
keywords = {Approximation Algorithms, Data Structures, Theory of computation → Routing and network design problems}, keywords = {Approximation Algorithms, Data Structures, Theory of computation \rightarrow{} Routing and network design problems},
pages = {37:1--37:20}, pages = {37:1--37:20}
} }
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@@ -60,10 +56,8 @@ abstract = { This paper considers the problem of designing fast, approximate, co
author = {Cunningham, William H.}, author = {Cunningham, William H.},
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volume = {34}, volume = {34},
@@ -78,36 +72,33 @@ abstract = { This paper considers the problem of designing fast, approximate, co
month = jan, month = jan,
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author = {Chen, Lin and Lian, Jiayi and Mao, Yuchen and Zhang, Guochuan}, author = {Chen, Lin and Lian, Jiayi and Mao, Yuchen and Zhang, Guochuan},
title = {A Nearly Quadratic-Time FPTAS for Knapsack}, title = {A Nearly Quadratic-Time FPTAS for Knapsack},
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doi = {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.}, 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.},
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numpages = {12}, numpages = {12},
keywords = {Approximation scheme, Knapsack}, keywords = {Approximation scheme, Knapsack},
location = {Vancouver, BC, Canada}, location = {Vancouver, BC, Canada},
series = {STOC 2024} series = {STOC 2024}
} }
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