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\title{A Note on Minimization Interdiction}
\author{Yu Cong \and Kangyi Tian}
\date{\today}

\DeclareMathOperator*{\opt}{OPT}

\begin{document}
\maketitle

\begin{abstract}
Motivated by the FPTAS for connectivity interdiction of Huang \etal{}
\cite{huang_fptas_2024}, we isolate the part of the argument that does not use
cuts.  The setting is a minimization problem over a feasible-set family
$\mathcal F$ with a linear objective $w(S)=\sum_{e\in S}w(e)$.  After dualizing
the interdiction budget, deletion can be absorbed into truncated weights
$w_\lambda(e)=\min\{w(e),\lambda c(e)\}$.  At an optimal Lagrange multiplier
$\lambda^*$, the unknown optimal interdiction witness is a strict
$2$-approximate minimizer of the reweighted problem.  Thus an exact algorithm
can be obtained whenever one can optimize $w_{\lambda^*}$ over $\mathcal F$,
enumerate all its $2$-approximate minimizers, and solve the remaining knapsack problem.
\end{abstract}

\section{Model}

Let $E$ be a finite ground set and let $\mathcal F\subseteq 2^E$ be the family
of feasible sets of an underlying linear minimization problem
\[
    \min_{S\in\mathcal F} w(S),
\]
where $w(S)=\sum_{e\in S}w(e)$ for weights $w:E\to \Z_+$.  We are also given
an interdiction cost $c:E\to \Z_+$ and a budget $b\in \Z_+$.

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),
\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,
even though we do not write variables for the feasible-set constraints:
\[
\begin{aligned}
\opt=\min&\quad w(S\setminus R)\\
\text{s.t.}&\quad S\in\mathcal F,\quad R\subseteq S,\\
&\quad c(R)\leq b.
\end{aligned}
\tag{IP}
\]
The set $S$ is the object chosen by the original minimization problem, and
$R$ is the part of $S$ removed by the interdiction budget.  For a fixed $S$,
choosing $R$ is a knapsack-like deletion problem; the global difficulty is that
we do not know which feasible set $S$ will be optimal after deletion.

The Lagrangian relaxation prices the single budget constraint.  For a multiplier
$\lambda\geq 0$, move the constraint $c(R)\leq b$ into the objective:
\[
    \Phi(\lambda)
    =
    \min_{\substack{S\in\mathcal F\\R\subseteq S}}
        \bigl(w(S\setminus R)+\lambda(c(R)-b)\bigr).
\]
For every fixed $\lambda$, this is a lower bound on the interdiction optimum:
budget-feasible pairs get a non-positive correction term
$\lambda(c(R)-b)$.  The Lagrangian dual is $\Lambda=\max_{\lambda\geq 0}\Phi(\lambda)$ and we denote a maximizer by $\lambda^*$. We assume $\Lambda>0$ since otherwise every feasible set $S$ would satisfy $c(S)\leq b$, so the interdiction problem would reduce to the original optimization problem.

Now remove the constant term $-\lambda b$ from $\Phi(\lambda)$ and focus on the
inner minimization
\[
    L(\lambda)
    =
    \min_{\substack{S\in\mathcal F\\R\subseteq S}}
        \bigl(w(S\setminus R)+\lambda c(R)\bigr),
\]
so that $\Phi(\lambda)=L(\lambda)-\lambda b$.  For a fixed $S$ and $\lambda$,
each element $e\in S$ has only two relevant choices: If $e$ is included in $R$, its cost will be $\lambda c(e)$; Otherwise the cost is $w(e)$.
Thus the best truncated weight of whether including $e$ in $R$ is $w_\lambda(e)=\min\{w(e),\lambda c(e)\}$.
Because both $w$ and $c$ are linear,
\[
\begin{aligned}
    \min_{R\subseteq S}\bigl(w(S\setminus R)+\lambda c(R)\bigr)
    &=\sum_{e\in S}\min\{w(e),\lambda c(e)\}\\
    &=w_\lambda(S).
\end{aligned}
\]
Therefore we have $L(\lambda)=\min_{S\in\mathcal F} w_\lambda(S)$, which is the original linear optimization problem under the truncated weight.

The function $\Phi$ is the lower envelope of finitely many lines in $\lambda$,
so it is piecewise-linear and concave.

\begin{lemma}\label{lem:lagrangian-lower-bound}
For every $\lambda\geq 0$, $\Phi(\lambda)\leq \opt$.  Hence
$\Lambda\leq \opt$.
\end{lemma}

\begin{proof}
Let $(S,R)$ be feasible for \eqref{eq:interdiction}.  Since $c(R)\leq b$, $w(S\setminus R)+\lambda(c(R)-b)\leq w(S\setminus R)$.
Minimizing the left-hand side over all pairs $(S,R)$, and then minimizing the
right-hand side only over budget-feasible pairs, gives
$\Phi(\lambda)\leq \opt$.
\end{proof}

\begin{lemma}\label{lem:feasible-active}
Assume $\lambda^*$ is a finite maximizer of $\Phi$.  Then there is a pair
$(S^{LD},R^{LD})$ attaining $L(\lambda^*)$ such that $c(R^{LD})\leq b$.
Consequently,
\[
    L(\lambda^*)\geq \opt.
\]
\end{lemma}

\begin{proof}
If every pair attaining $L(\lambda^*)$ had $c(R)>b$, then every active line in
the lower envelope defining $\Phi$ would have positive slope at $\lambda^*$.
For sufficiently small $\delta>0$, the value of the lower envelope would then
increase from $\lambda^*$ to $\lambda^*+\delta$, contradicting the optimality of
$\lambda^*$.

Thus some active pair $(S^{LD},R^{LD})$ has $c(R^{LD})\leq b$.  This pair is
feasible for \eqref{eq:interdiction}, so
$w(S^{LD}\setminus R^{LD})\geq \opt$.  Therefore
\[
    L(\lambda^*)
    =
    w(S^{LD}\setminus R^{LD})+\lambda^*c(R^{LD})
    \geq \opt.
\]
\end{proof}

\section{The main observation}

\begin{theorem}\label{thm:two-approx-witness}
Let $(S^*,R^*)$ be an optimal solution to \eqref{eq:interdiction}.  If
$\Lambda>0$, then
\[
    L(\lambda^*)
    \leq
    w_{\lambda^*}(S^*)
    \leq
    L(\lambda^*)+b\lambda^*
    <
    2L(\lambda^*).
\]
In particular, $S^*$ is a strict $2$-approximate minimizer of
$\min_{S\in\mathcal F} w_{\lambda^*}(S)$.
\end{theorem}

\begin{proof}
The lower bound follows immediately from the definition of $L(\lambda^*)$:
\[
    L(\lambda^*)=\min_{S\in\mathcal F}w_{\lambda^*}(S)
    \leq w_{\lambda^*}(S^*).
\]
For the upper bound, use the particular deletion set $R^*$ inside the definition
of $w_{\lambda^*}(S^*)$:
\[
\begin{aligned}
    w_{\lambda^*}(S^*)
    &\leq w(S^*\setminus R^*)+\lambda^*c(R^*) \\
    &= \opt+\lambda^*c(R^*) \\
    &\leq \opt+\lambda^*b \\
    &\leq L(\lambda^*)+\lambda^*b,
\end{aligned}
\]
where the last inequality is \autoref{lem:feasible-active}.  Finally,
$\Lambda=L(\lambda^*)-\lambda^*b>0$, so
$L(\lambda^*)+\lambda^*b<2L(\lambda^*)$.
\end{proof}

\begin{remark}
    For connectivity interdiction, $S^*$ is the optimal interdiction cut, so it is
    among the strict $2$-approximate min-cuts in the graph with capacities
    $w_{\lambda^*}$.
\end{remark}

\section{Algorithmic template}

The theorem gives the following general template.

\begin{algo}
\underbar{\textsc{Linear-Minimization-Interdiction}}$(E,\mathcal F,w,c,b)$:\\
compute a maximizer $\lambda^*$ of $\Phi(\lambda)=L(\lambda)-\lambda b$\\
compute the truncated weight $w_{\lambda^*}$\\
enumerate every $S\in\mathcal F$ with $w_{\lambda^*}(S)<2L^*$\\
for each enumerated $S$:\\
\;\; compute $g_b(S)=\min\{w(S\setminus R):R\subseteq S,\ c(R)\leq b\}$\\
return the pair $(S,R)$ with minimum value
\end{algo}

$\lambda^*$ can be found using parametric search techniques.

\begin{lemma}[\cite{salowe_parametric}]\label{lem:para}
Let $T(n)$ be the complexity of computing $L(\lambda)=\min_{S\in\mathcal F} w_\lambda(S)$ for fixed $\lambda$ (where $n$ is the size of the input), then one can compute $\lambda^*$ using parametric search in $O(T(n)^2)$ time.
\end{lemma}

Computing $g_b(S)$ is essentially solving a knapsack problem on groundset $S$ and takes $\tilde O(m+\frac{1}{\e^2})$ time for an $(1+\e)$-approximation \cite{10.1145/3618260.3649730}.

\paragraph{Application on Connecticity Interdiction} $L(\lambda)$ can be obtained via min-cut in deterministic $\tilde O(m)$ time \cite{Li_2021}. Combining with \autoref{lem:para} gives a $\tilde O(m^2)$-time algorithm for $\lambda^*$. Karger \cite{Karger2000} showed the number of $\alpha$-approximate min-cut is $O(n^{\floor{2\alpha}})$. Thus the number of enumerated $S$ is $O(n^3)$ and we can enumerate them in ...

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