From 5c6f64542fc1ab4ab8b57a7de3b0d7b9bf1a02f7 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sat, 12 Apr 2025 15:47:32 +0800 Subject: [PATCH] refine normalized knapsack problem formulation and update related lemmas --- main.pdf | Bin 236555 -> 238214 bytes main.tex | 36 ++++++++++++++++++++++++------------ 2 files changed, 24 insertions(+), 12 deletions(-) diff --git a/main.pdf b/main.pdf index 7f1b3881bcd641ddca2090e2515e1b6f57116633..1b4b03193ad4154da1edcc7af1d1988cd792d168 100644 GIT binary patch delta 72915 zcmZU)Q*b3*)U~@~+qP|^W1Ah@Hg{}vjE-&FHaoU$J2~%nPJMsXssC=)yj&M^j8V^6 z(QSxf!-#*#z}UE1)9MAlXn;$)TS*%&D8Ltk{4QlQxM8k7xx|e1=A@GB$4R#7#k*i% z$sH@-M11MA=#Lc-EC5XaIJHcD_VKV}0sym9>+=~5ZeeXYW~RqW3=Wvv<@p$X5_68j z7Q#kh!L&UEpNm5qsp0;-@BQ3h{GiFybKyOu&xl>z^q%1S{`g(z32d~%^)!EgsMUUY z*>dMQDe+EvgdxW+PuZ6Wn!(IMvyLeRbnBbccJdUeH$IDoq5zAcX0Qe{Y6)n5?yt;y z46A<6WK?forD;0kV_YPe@{>qg+w7rb*W+MCY0+A&I~UOZ?baG3&zZq0S)a)$ian1D*!ZAZ(8Q4Zu4vkD zcDafi^d+w)JnjnPY3I7Fsx)_V;SVc*?&rEJS3&(v3Ke-qgAszmJ1ZSWx7s 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remember that $\mathcal F$ is usually not explicitly given. \begin{problem}[Normalized knapsack]\label{nknap} Given the same input as \autoref{bfreeknap}, find $\min \limits_{X - \in \mathcal F, F\subset E} \frac{w(X\setminus F)}{b+1-c(F)}$ such that + \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 $w_\tau:\E\to \R$, @@ -39,12 +40,14 @@ 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} -proof is exactly the same as lemma 1 in \cite{vygen_fptas_2024}. +The proof is exactly 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 $w_\tau$. \begin{lemma}\label{lem:lb} - For any $X\in \mathcal F$, $w_\tau(X)\ge \tau(1+b)$. + For any $X\in \mathcal F$, $w_\tau(X)\ge \tau B$. \end{lemma} -proof is the same. + \begin{lemma} % $w_\tau (X^N)\le \tau(b+1)$. @@ -53,23 +56,32 @@ proof is the same. \begin{proof} \begin{align*} w_\tau (X^N) & \le w(X^N\setminus F^N) + w_\tau(F^N)\\ - & = \tau \cdot(b+1-c(F^N)) + \tau\cdot c(F^N)\\ - & = \tau(b+1) + & = \tau \cdot(B-c(F^N)) + \tau\cdot c(F^N)\\ + & = \tau B \end{align*} Thus by \autoref{lem:lb}, $X^N$ gets the minimum. \end{proof} -Let $(X^*,F^*)$ be the optimal solution to \autoref{bfreeknap}. +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} +\begin{lemma}[Lemma 4 in \cite{vygen_fptas_2024}]\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 $w(X^*\setminus F^*)\geq - \tau(\alpha+(\alpha-1)b)$. + \tau(\alpha B-b)$. \end{lemma} -The proof is the same. -In fact, corollary 1 and theorem 5 are also the same as those in -\cite{vygen_fptas_2024}. Finally we get a knapsack version of Theorem 4: +% 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 $\min_{X\in\mathcal F} + w_\tau(X)$ for some $\alpha>1$. We have +\[ +\frac{w(C^N\setminus F^N)}{w(C^*\setminus F^*)}\leq \frac{\tau(B-c(F^N))}{\tau(\alpha B-b)}\leq \frac{B}{\alpha B-b}, +\] +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}] 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