From 8edb8f8a45f7693dae495bbf39090bb4cea8a679 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Wed, 17 Sep 2025 13:42:59 +0800 Subject: [PATCH] fix bug --- main.tex | 77 +++++++++++++++++++++++++------------------------------- 1 file changed, 34 insertions(+), 43 deletions(-) diff --git a/main.tex b/main.tex index b0363df..a721402 100644 --- a/main.tex +++ b/main.tex @@ -35,22 +35,16 @@ \section{``Cut-free'' Proof} \begin{problem}[b-free knapsack]\label{bfreeknap} - Consider a set of elements $E$ and two weights $w:E\to \Z_+$ and - $c:E\to \Z_+$ 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 capacity $c:E\to \Z_+$ 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} Note that $\mathcal F$ is usually not explicitly given. \begin{problem}[Normalized knapsack]\label{nknap} - Given the same input as \autoref{bfreeknap}, find $\min \limits_{X - \in \mathcal F, F\subset E} \frac{w(X\setminus F)}{B-c(F)}$ such that - $c(F)\leq b$. + Given the same input as \autoref{bfreeknap}, find $\min \limits_{X\in \mathcal F, F\subset E} \frac{w(X\setminus F)}{B-c(F)}$ such that $c(F)\leq b$. \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. -Denote by $\tau$ 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 $w_\tau:\E\to \R$, \[ w_\tau(e)=\begin{cases} @@ -63,7 +57,7 @@ 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 exactly the same as \cite[Lemma 1]{vygen_fptas_2024}. +The proof is the same as \cite[Lemma 1]{vygen_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 $w_\tau$. @@ -85,9 +79,9 @@ 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]{vygen_fptas_2024}, which states the optimal solution to \autoref{bfreeknap} is a $\alpha$-approximate solution to $\min_{F\in \mathcal{F}} w_\tau(F)$. -\begin{lemma}[Lemma 4 in \cite{vygen_fptas_2024}]\label{lem:conditionalLB} +\begin{lemma}\label{lem:conditionalLB} Let $(X^*,F^*)$ be the optimal solution to \autoref{bfreeknap}. $X^*$ is either an $\alpha$-approximate solution to $\min_{X\in\mathcal F} w_\tau(X)$ for some $\alpha>1$, or $w(X^*\setminus F^*)\geq @@ -96,7 +90,7 @@ Now we show the counter part of \cite[Theorem 5]{vygen_fptas_2024}, which states % In fact, corollary 1 and theorem 5 are also the same as those in % \cite{vygen_fptas_2024}. -Then following the argument of Corollary 1 in \cite{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]{vygen_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}, @@ -104,16 +98,14 @@ Then following the argument of Corollary 1 in \cite{vygen_fptas_2024}, assume th 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 knapsack version of Theorem 4: -\begin{theorem}[Theorem 4 in \cite{vygen_fptas_2024}]\label{thm:main} +Finally we get a general version of \cite[Theorem 4]{vygen_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 2-approximation to $X^{\min}$. \end{theorem} -Thus to obtain a FPTAS for \autoref{bfreeknap}, one need to design a FPTAS for -\autoref{nknap} and a polynomial time alg 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 ``FPTAS'' here is not precise since we do not have a approximation scheme but @@ -134,20 +126,19 @@ have a non-zero lowerbound on $\opt^N$. \begin{conjecture} Let $(C,F)$ be the optimal solution to connectivity interdiction. The optimum -cut $C$ can be computed in polynomial time. In other words, connectivity -interdiction is almost as easy as knapsack. +cut $C$ can be computed in polynomial time. \end{conjecture} \section{Connections} -For unit costs, connectivity interdiction with budget $b=k-1$ is the same +For unit weight and capacity, connectivity interdiction with budget $b=k-1$ is the same problem as finding the minimum weighted edge set whose removal breaks $k$-edge connectivity. -It turns out that \autoref{nknap} is just a necessary ingredient for MWU. -Authors of \cite{vygen_fptas_2024} $\subset$ authors of -\cite{chalermsook_approximating_2022}. +\autoref{nknap} may come from an intermediate problem of MWU methods for positive covering LPs. +% Authors of \cite{vygen_fptas_2024} $\subset$ authors of +% \cite{chalermsook_approximating_2022}. -How to derive normalized min cut for connectivity interdiction? +Can we get an FPTAS using LP methods? \begin{equation*} @@ -175,9 +166,9 @@ s.t.& & \sum_{e\in T} x_e&\geq 1 & &\forall T\quad \text{($x$ is a c Now this LP looks similar to the normalized min-cut problem. -A further reformulation (the new $x$ is $x-y$) gives us the following, +A further reformulation ($x\from x-y$) gives us the following integer program, -\begin{equation}\label{lp:conninterdict} +\begin{equation}\label{ip:interdiction} \begin{aligned} \min& & \sum_{e} x_e w(e) & & \\ s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\quad \text{($x+y$ is a cut)}\\ @@ -189,23 +180,21 @@ s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\quad \text{($x+y$ Note that now this is almost a positive covering LP. Let $L(\lambda)= \min \{ w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut $C$}\;\forall F\subset C % \land c(F)\leq b -\}$. -Consider the Lagrangian dual, -\begin{equation*} +\}$ +and consider the Lagrangian dual, +\[ \max_{\lambda\geq 0} L(\lambda)= \max_{\lambda\geq 0} \min \left\{ w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut $C$}\;\forall F\subset C % \land c(F)\leq b -\right\} -\end{equation*} +\right\}. +\] - -% At this point, it becomes clear how the normalized min-cut is implicated in \cite{vygen_fptas_2024}. The optimum of normalized min-cut is exactly the value of $\lambda$ when $L(\lambda)$ is 0. We have shown that the budget $B$ in normalized min-cut does not really matter as long as $B>b$. Note that $L(\lambda)$ and the normalized min-cut look similar to the principal sequence of partitions of a graph and the graph strength problem. \subsection{graph strength} % Assume that the graph $G$ is connected. -Given a graph $G=(V,E)$ and a cost function $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. Note that an alternative formulation of strength (using graphic matroid rank) is $\sigma(G)=\min_{F\subset E} \frac{|E-F|}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing. +For a graph $G=(V,E)$ with edge capacity $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. Note that an alternative formulation of strength (using graphic matroid rank function) is $\sigma(G)=\min_{F\subset E} \frac{|E-F|}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing. -The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. ($L(\lambda)=\min_{F\in E}c(E\setminus F)-\lambda(r(E)-r(F)+1)$) Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}(implicated stated in \cite{cunningham_optimal_1985}). +The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. (alternatively, $L(\lambda)=\min_{F\in E}c(E\setminus F)-\lambda(r(E)-r(F)+1)$) Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}(implicated stated in \cite{cunningham_optimal_1985}). \begin{enumerate} \item We can assume $G$ is connected and deal with the smallest strength component. One can see this by fractional base packing on the direct sum of matroids. Note that on disconnected graphs we should use the edge set definition instead of partitions. \item $L(\lambda)$ is piecewise linear concave since it is the lower envelope of some line arrangement. @@ -237,7 +226,7 @@ Consider $L(\lambda)$ for cut problem. One can see that the optimal $\lambda$ is \subsection{integrality gap} I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem. -First consider the dual of linear relaxation of \autoref{lp:conninterdict}. +First consider the dual of linear relaxation of \autoref{ip:interdiction}. \begin{equation}\label{lp:dualcutint} \begin{aligned} @@ -265,7 +254,7 @@ $\lambda_i=\min \frac{w(C\setminus F)-w(C_{i-1}\setminus F_{i-1})}{c(F_{i-1})-c( The proof is using the argument for showing $\lambda_1=\min \frac{w(C\setminus F)}{B-c(F)}$ and induction. $\lambda_i$ looks similar to normalized mincut but is related to the slope and vertical intercept of a previous segment. \begin{conjecture}\label{conj:intgap} - \autoref{lp:conninterdict} has an integrality gap of 4. + \autoref{ip:interdiction} has an integrality gap of 4. \end{conjecture} However, \autoref{conj:intgap} is wrong. The integrality gap is unbounded. Consider a cycle $C_n$ of $n$ vertices with two special edges $e_1,e_2$. Let $L$ be a large number. @@ -286,9 +275,12 @@ For LP, we assign $x=0$ and $y_e=\frac{1}{n-2}$ for every edge except $e_1$. The What is the gap if we only relax $\lambda$ in the Lagrangian dual? -\section{Random Stuff} +\bibliographystyle{plain} +\bibliography{ref} -\subsection{remove box constraints} +\appendix + +\section{remove box constraints} Given a positive covering LP, \begin{equation*} @@ -317,11 +309,10 @@ suppose that $c(f)k$ since we assume $x_f>c(f)$, which means we can decrease $x_f$ without violating any constraint. Thus it contradicts the assumption that $x^*$ is optimal. Then we can add redundant constraints $x_e\leq c(e) \;\forall e$ to LP2 and see that LP1 and LP2 have the same optimum. -This applies to \cite{chalermsook_approximating_2022} but cannot get an improvement on their algorithm.(MWU does not care the number of constraints.) So does this trick apply to connectivity interdiction? +This applies to \cite{chalermsook_approximating_2022} but cannot get an improvement on their algorithm.(MWU does not care the number of constraints.) \[ \min_{\text{cut C}, f\in C}\frac{\sum_{e\in C\setminus\set{f}}w(e)x_e}{k-c(f)} \] -\bibliographystyle{plain} -\bibliography{ref} + \end{document}