unit cost gap

main.tex

@@ -14,7 +14,7 @@
 \begin{document}
 \maketitle
 
-\section{``Cut-free'' Proof}
+\section{Generalize proof in [Huang \etal{}, IPCO'24]}
 \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 $c(F)\leq b$.
 \end{problem}
@@ -23,7 +23,7 @@ Note that $\mathcal F$ is usually not explicitly given.
 \begin{problem}[Normalized knapsack]\label{nknap}
     Given the same input as \autoref{bfreeknap}, find $\min \limits_{X\in \mathcal F, F\subset E} \frac{w(X\setminus F)}{B-c(F)}\; s.t.\; c(F)\leq b$.
 \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.
+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 $w_\tau:\E\to \R$,
 
@@ -38,7 +38,7 @@ 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 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 $w_\tau$.
 
@@ -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.
 \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}
     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}
 
 % In fact, corollary 1 and theorem 5 are also the same as those in 
-% \cite{vygen_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} 
+% \cite{huang_fptas_2024}. 
+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
 \[
 \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}.
 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}
     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 
@@ -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 $\min_{X\in\mathcal F} w_\tau(X)$.
 
-\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 
 an enumeration algorithm. But I will use this term anyway.) In their settings, 
 $\mathcal F$ 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.
 \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}
 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.
 
 \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}.
 
 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 $c(F^*)\leq b$.
 
 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}
 
 \subsection{general objective function}