zzz

notes.tex

@@ -5,7 +5,6 @@
 \geometry{margin=2cm}
 
 \title{A Note on Interdiction of Linear Minimization Problems}
-\author{Yu Cong \and Kangyi Tian}
 \date{\today}
 
 \DeclareMathOperator*{\opt}{OPT}
@@ -40,26 +39,13 @@ The linear minimization interdiction problem considered here is
 \begin{equation}\label{eq:interdiction}
     \opt
     =
-    \min_{\substack{S\in\mathcal F,\; R\subseteq S\\ c(R)\leq b}}
-        w(S\setminus R),
+    \min \set{w(S\setminus R)\;|\;S\in\mathcal F,\; R\subseteq S,c(R)\leq b},
 \end{equation}
 where $c(R)=\sum_{e\in R}c(e)$.  This is equivalent to first deleting an
 arbitrary set $R$ with $c(R)\leq b$ and then minimizing $w(S\setminus R)$ over
 $S\in\mathcal F$, because only $R\cap S$ affects the value of a chosen feasible
 set $S$.
 
-Connectivity interdiction is the special case where $\mathcal F$ is the family
-of cuts of a graph.  In that case \eqref{eq:interdiction} is exactly the
-$b$-free min-cut formulation
-\[
-    \min_{\substack{\text{cut } C,\;R\subseteq C\\c(R)\leq b}}
-        w(C\setminus R).
-\]
-
-The argument below uses both minimization and linearity.  Minimization gives the
-near-minimizer certificate in \autoref{thm:two-approx-witness}; linearity gives
-the truncation formula $w_\lambda(e)=\min\{w(e),\lambda c(e)\}$.
-
 \section{Lagrangian relaxation}
 
 It is helpful to first look at \eqref{eq:interdiction} as an integer program,