From 66e38e77adb9a7e5228a9763e513f97c70b6e14c Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sat, 19 Apr 2025 23:32:22 +0800 Subject: [PATCH] =?UTF-8?q?c(F)=E2=89=A4b=3F?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- main.pdf | Bin 258904 -> 259739 bytes main.tex | 29 +++++++++++++++++++++++++++-- 2 files changed, 27 insertions(+), 2 deletions(-) diff --git a/main.pdf b/main.pdf index 1e59ebb1cda17ecc748eb72947e74314e64d5676..e752fd1c3febbc4bbe2b0c350e4750dd98c8b47e 100644 GIT binary patch delta 16878 zcmafZQ*b6gw{4Oywrxyo+xEn^ZRd+^JDJ$For!HvY#Vdu-2a~Qa&OgrTfJ*Nba!=C z@3q!mJ4VXBLPDWX5|f~3X5fUSm|GfIgJoxGK%qkd<6=ofL2LSnT-9B?H8XAO)VCnwrl({_otrgn^Q zb9Z|1@R>LfE=70wxFR?1}6veG#(`?M_D9+D!=Fb)Ho|{uxa@p3rWv$-8 zEnW7SzkY%io$r@*bsOf~Wu@;&`~C{zEj!m9+p;O5llg+*crg;}aX*?z2L$Nbla`V8 zQB20kq$~fDCb?dzekKf({gB)=BHQ4X*8Dgu*Mh1{^ZDx$Oa~HKy>SJ{2F6M77DFCi zuFOPu7Rk1*qK6?>Vcor3H*70xMxT-cq|YdaXz*txz5(b!g*0g8& 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We can still assume that $G$ is connected and see that $L(\lambda)$ is pwl concave (1 and 2 still hold). Let $\lambda^*$ be a breakpoint on $L$. Suppose that there are two optimal solutions $(C_1,F_1)$ and $(C_2,F_2)$ at $\lambda^*$. For fixed $C$ ($C_1=C_2$), the same argument for principal partition still works. However, the difficult part is that $C$ might not be the same. So it's unlikely that 3 and 4 hold. For cut interdiction problem, 5 shows connections between normalized mincut and the original interdiction problem. Recall that we observe the denominator in normalized min-cut can be relaxed (that is, we can use $\frac{w(C\setminus F)}{B-c(F)}$ for any $B>b$, instead of restricting to $B=b+1$) and the analysis still works. Now following the previous argument for 5, we assume $\lambda\in [0,\e]$ for small enough positive $\e$. For any $C$, we have $F=C$ since $w(C\setminus F)$ is dominating. For the remaining term $-\lambda(b-c(F))$ we are selecting a cut $F$ with smallest cose with respect to $c$. We can assume that any cut in $G$ has larger cost than $b$ since otherwise the optimum is simply 0. Now we can see that $B$ in the denominator $B-c(F)$ should be the cost of mincut in $G$. +Now we focus on $L(\lambda)=\min \{w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut } C\;\forall F\subset C\}$. We can still assume that $G$ is connected and see that $L(\lambda)$ is pwl concave (1 and 2 still hold). Let $\lambda^*$ be a breakpoint on $L$. Suppose that there are two optimal solutions $(C_1,F_1)$ and $(C_2,F_2)$ at $\lambda^*$. For fixed $C$ ($C_1=C_2$), the same argument for principal partition still works. However, the difficult part is that $C$ might not be the same. So it's unlikely that 3 and 4 hold. For cut interdiction problem, 5 shows connections between normalized mincut and the original interdiction problem. Recall that we observe the denominator in normalized min-cut can be relaxed (that is, we can use $\frac{w(C\setminus F)}{B-c(F)}$ for any $B>b$, instead of restricting to $B=b+1$) and the analysis still works. Now following the previous argument for 5, we assume $\lambda\in [0,\e]$ for small enough positive $\e$. For any $C$, we have $F=C$ since $w(C\setminus F)$ is dominating. For the remaining term $-\lambda(b-c(F))$ we are selecting a cut $F$ with smallest cose with respect to $c$. Note that we can assume that any cut in $G$ has larger cost than $b$ since otherwise the optimum is simply 0. +% Now we can see that $B$ in the denominator $B-c(F)$ should be the cost of mincut in $G$. +Let $B$ be the minimum cost of cuts in $G$. +We have $-\lambda(b-B)\leq w(C\setminus F)-\lambda(b-c(F))$ for any cut $C$ and $F\subsetneq C$. Thus the upperbound is $\e=\min \frac{w(C\setminus F)}{B-c(F)}$. + +\note{It remains to show that the optimal solution at $\e$ guarantees $c(F)\leq b$? or maybe we don't need this for normalized mincut.} \subsection{integrality gap} I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem. @@ -215,7 +240,7 @@ s.t.& & \sum_{T\ni e} z_T &\leq w(e) & &\forall e\in E\\ & & z_T,\lambda &\geq 0 & & \end{aligned} \end{equation} -We want to prove something like tree packing for \autoref{lp:dualcutint}. +We want to prove something like tree packing theorem for \autoref{lp:dualcutint}. \begin{conjecture} The optimum of \autoref{lp:dualcutint} is $\min \set{\frac{w(C\setminus F)}{B-c(F)}| \forall \text{cut $C$}, c(F)\leq b}$, where $B$ is the cost of mincut in $G$ and $b$ is the budget. \end{conjecture}