z

main.tex

@@ -35,7 +35,7 @@
 
 \section{``Cut-free'' Proof}
 \begin{problem}[b-free knapsack]\label{bfreeknap}
-    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 $c(F)\leq b$.
+    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}
 Note that $\mathcal F$ is usually not explicitly given.
 
@@ -129,8 +129,10 @@ 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}.
+
 \section{Connections}
-For unit weight and capacity, 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 
 problem as finding the minimum weighted edge set whose removal breaks $k$-edge 
 connectivity.
 
@@ -192,7 +194,7 @@ We have shown that the budget $B$ in normalized min-cut does not really matter a
 
 \subsection{graph strength}
 % Assume that the graph $G$ is connected. 
-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.
+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 $\sigma(G)=\min_{F\subset E} \frac{c(E-F)}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing.
 
 The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. (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}
@@ -216,6 +218,16 @@ Now we focus on $L(\lambda)=\min \{w(C\setminus F)-\lambda(b-c(F)) | \forall \te
 % 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 that $B-c(F)$ is always positive since otherwise the curve is not concave.
+
+Now we depict the curve. The first segment is determined by $C=F\in \argmin_{X\subset V} c(\delta(X))$. The first breakpoint is $\e=\min \frac{w(C\setminus F)}{B-c(F)}$ and for the second segment we have $(C,F)\in \argmin\frac{w(C\setminus F)}{B-c(F)}$ with minimum $c(F)$.
+Now there are two cases:
+\begin{enumerate}
+\item if $c(F)\leq b$, then the first breakpoint is the optimal $\lambda$.
+\item if $c(F)\in (b,B)$, we have to look at more segments.
+\end{enumerate}
+Unfortunately, the seconds case is possible (consider a path with parallel edges) and the number of segments can be exponential.
+
 
 % \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. I think normalized min-cut should not require $c(F)\leq b$. Further checks are needed. What we can guarantee is that $c(F)\leq B$.}
 
@@ -224,7 +236,7 @@ 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.
+I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem. \textcolor{red}{Which is wrong}.
 
 First consider the dual of linear relaxation of \autoref{ip:interdiction}.
 
@@ -236,14 +248,14 @@ 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 theorem for \autoref{lp:dualcutint}.
+% We want to prove something like tree packing theorem for \autoref{lp:dualcutint}.
-\begin{conjecture}
+% \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.
+%     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}
+% \end{conjecture}
 
-I believe the previous conjecture is not likely to be true.
+% I believe the previous conjecture is not likely to be true.
 
-\paragraph{Weight truncation} Assuming we know the optimal $\lambda$ to the LP dual, \autoref{lp:dualcutint} in fact gives the idea of weight truncation. The capacity of each edge $e$ in the ``tree packing'' is $\min\{c(e)\lambda,w(e)\}$. Therefore, the optimum of \autoref{lp:dualcutint} is $\Lambda_{w_\tau}^{fr}-b\lambda$, where $\Lambda_{w_\tau}^{fr}$ is the fractional mincut on $G$ with weights $w_\tau$.
+\paragraph{Weight truncation} Assuming that the optimal $\lambda$ to the LP dual is known, \autoref{lp:dualcutint} in fact gives the idea of weight truncation. The capacity of each edge $e$ in the ``tree packing'' is $\min\{c(e)\lambda,w(e)\}$. Therefore, the optimum of \autoref{lp:dualcutint} is $\Lambda_{w_\tau}^{fr}-b\lambda$, where $\Lambda_{w_\tau}^{fr}$ is the fractional mincut on $G$ with weights $w_\tau$.
 
 \paragraph{The optimal $\lambda$} Denote by $\lambda^*$ the optimal $\lambda$ that maximizes $L(\lambda)$. From the previous argument on the first segment of $L(\lambda)$ we know that $\lambda^* \geq \min \frac{w(C\setminus F)}{B-c(F)}$. Now assume $\lambda^* > \min_{c(F)\leq b} \frac{w(C\setminus F)}{b-c(F)}$. We have $\min w(C\setminus F)-\lambda^*(b-c(F))<w(C\setminus F)-\min_{c(F)\leq b} \frac{w(C\setminus F)}{b-c(F)}(b-c(F))=0$ since the optimum must be achieved by $F$ such that $0\leq b-c(F)$(the slope). The negative optimum of $L(\lambda)$ contradicts the fact that $L(0)=0$ and $L$ is concave. Hence, the optimal solution $\lambda^*$ is in the range $[\min\frac{w(C\setminus F)}{B-c(F)},\min_{c(F)\leq b}\frac{w(C\setminus F)}{b-c(F)}]$.