z

notes.tex

@@ -2,10 +2,8 @@
 \usepackage{chao}
 \usepackage{algo}
 
-\geometry{margin=2cm}
+\title{A Note on Interdiction of Linear Minimization Problems}
-
+\author{Yu Cong \footnote{\email{yucong143@gmail.com}, University of Electronic Science and Technology of China.} \and Kangyi Tian\footnote{\email{kangyitian947@gmail.com}, University of Electronic Science and Technology of China.}}
-\title{A Note on Minimization Interdiction}
-\author{Yu Cong \and Kangyi Tian}
 \date{\today}
 
 \DeclareMathOperator*{\opt}{OPT}
@@ -40,26 +38,13 @@ 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}}
+    \min \set{w(S\setminus R)\;|\;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,
@@ -193,6 +178,7 @@ $L(\lambda^*)+\lambda^*b<2L(\lambda^*)$.
 \end{proof}
 
 \begin{remark}
+    The strict $2$-approximation analysis generalizes to non-negative set function objectives.
     For connectivity interdiction, $S^*$ is the optimal interdiction cut, so it is
     among the strict $2$-approximate min-cuts in the graph with capacities
     $w_{\lambda^*}$.
@@ -200,27 +186,72 @@ $L(\lambda^*)+\lambda^*b<2L(\lambda^*)$.
 
 \section{Algorithmic template}
 
-The theorem gives the following general template.
+\begin{figure}  
-
 \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 a maximizer $\lambda^*$ of $\Phi(\lambda)=L(\lambda)-\lambda b$\\
-compute the truncated weight $w_{\lambda^*}$\\
+\;compute the truncated weight $w_{\lambda^*}$\\
-enumerate every $S\in\mathcal F$ with $w_{\lambda^*}(S)<2L^*$\\
+\;enumerate every $S\in\mathcal F$ with $w_{\lambda^*}(S)<2L(\lambda^*)$\\
-for each enumerated $S$:\\
+\;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
+\;return the pair $(S,R)$ with minimum value
 \end{algo}
+\caption{Template for solving interdiction version of linear minimization problem.}
+\label{fig:alg}
+\end{figure}
 
+The theorem gives the a general template shown in \autoref{fig:alg}.
 $\lambda^*$ 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.
+Let $S(n)$ be the complexity of computing $L(\lambda)=\min_{H\in\mathcal F} w_\lambda(H)$ for fixed $\lambda$ (where $n$ is the size of the input), then one can compute $\lambda^*$ using parametric search in $O(S(n)^2)$ time.
+
+If there is a parallel algorithm that solves $L(\lambda)$ for fixed $\lambda$ using $P(n)$ processors in time $T(n)$, then one can compute $\lambda^*$ using parametric search in $O( S(n)T(n)\log P(n)+T(n)P(n) )$ 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}.
+This algorithmic template shows the interdiction version of a minimization problem can be reduced to polynomially many knapsack problem if all strict $2$-approximations of a reweighted original problem can be enumerated in polynomial time.
 
-\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 ...
+\subsection{Application on Connectivity Interdiction}
+
+Let $G=(V,E)$ be an undirected multigraph, and let $\mathcal F$ be the family of
+all nontrivial cuts $\delta(U)$, where $\emptyset\neq U\subsetneq V$.  The
+interdiction problem is
+\[
+    \min_{\substack{C\in\mathcal F,\;R\subseteq C\\c(R)\leq b}}
+        w(C\setminus R).
+\]
+If there is a cut $C$ with $c(C)\leq b$, then the optimum is $0$: remove all
+edges in $C$.  This case is detected by one min-cut computation under capacities
+$c$, so below assume every cut has $c$-cost larger than $b$.
+
+For a fixed $\lambda$, put capacity $w_\lambda(e)=\min\{w(e),\lambda c(e)\}$
+on every edge.  Then
+$
+    L(\lambda)
+    =
+    \min_{C\in\mathcal F} w_\lambda(C),
+$
+so evaluating $L(\lambda)$ is just a global min-cut computation in
+$(G,w_\lambda)$.  Using the deterministic almost-linear time min-cut algorithm
+of Li \cite{Li_2021}, this takes $\tilde O(m)$ time.  Therefore,
+\autoref{lem:para} gives a $\tilde O(m^2)$-time algorithm for computing $\lambda^*$.
+If randomization is allowed, we can find $\lambda^*$ in near linear time.
+Min-cut can be computed with high probability in $O(m\log^2 n)$ work and $O(\log^3 n)$ depth \cite{Anderson_Blelloch_2021}. Brent's law \cite{Gustafson_2011} shows that the running time is $T(n,P)=O(\frac{m\log^2 n}{P}+\log^3 n)$ with $P$ processors. Setting $P=O(m)$ and plugging the parallel algorithm into \autoref{lem:para} give a randomized $\tilde O(m)$-time algorithm for $\lambda^*$.
+
+Let $L^*=L(\lambda^*)$.  By \autoref{thm:two-approx-witness}, the optimal
+interdiction cut $C^*$ satisfies
+$
+    w_{\lambda^*}(C^*)<2L^*
+$.
+Thus $C^*$ is one of the strict $2$-approximate min-cuts in
+$(G,w_{\lambda^*})$.  Karger \cite{Karger2000} showed that the number of
+$\alpha$-approximate min-cuts is $O(n^{\floor{2\alpha}})$.  Hence the number of
+cuts with value strictly smaller than $2L^*$ is $O(n^3)$, and they can be
+enumerated in randomized $O(n^3)$ time\footnote{We note that only succinct representation of cuts (with respect to Karger's tree packing) are needed and the cut edges can be recovered while solving the knapsack. However, it is not known if one can deterministically enumerate all $\alpha$-approximate min-cuts in time $O(n^{\floor{2\alpha}})$.}\cite{Karger2000}.
+To sum up, our framework gives an FPRAS for connectivity interdiction with running time $\tilde O(m+n^3(m+\frac{1}{\e^2}))$.
+
+Compared with \cite{huang_fptas_2024}, our analyais is based on LP methods and is more intuitive. An algorithmic improvement is that we use parametric search thus avoiding the enumeration of $\lambda^*$ (the optimum of a normalized min-cut problem in \cite{huang_fptas_2024}).
 
 \bibliographystyle{plain}
 \bibliography{ref}

ref.bib

@@ -116,4 +116,7 @@
 	year = {2000},
 	pages = {46--76},
 }
-@inproceedings{Li_2021, address={Virtual Italy}, title={Deterministic mincut in almost-linear time}, ISBN={9781450380539}, url={https://dl.acm.org/doi/10.1145/3406325.3451114}, DOI={10.1145/3406325.3451114}, booktitle={Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing}, publisher={ACM}, author={Li, Jason}, year={2021}, month=june, pages={384–395}, language={en} }
+@inproceedings{Li_2021, address={Virtual Italy}, title={Deterministic mincut in almost-linear time}, ISBN={9781450380539}, url={https://dl.acm.org/doi/10.1145/3406325.3451114}, DOI={10.1145/3406325.3451114}, booktitle={Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing}, publisher={ACM}, author={Li, Jason}, year={2021}, month={june}, pages={384–395}, language={en} }
+@inproceedings{Anderson_Blelloch_2021, address={Virtual Event USA}, title={Parallel Minimum Cuts in {$O ( m \log^2 n )$} Work and Low Depth}, ISBN={9781450380706}, url={https://dl.acm.org/doi/10.1145/3409964.3461797}, DOI={10.1145/3409964.3461797}, booktitle={Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures}, publisher={ACM}, author={Anderson, Daniel and Blelloch, Guy E.}, year={2021}, month={july}, pages={71–82}, language={en} }
+
+@incollection{Gustafson_2011, title={Brent’s Theorem}, ISBN={9780387097664}, url={https://link.springer.com/rwe/10.1007/978-0-387-09766-4_80}, DOI={10.1007/978-0-387-09766-4_80}, abstractNote={“Brent’s Theorem” published in “Encyclopedia of Parallel Computing”}, booktitle={Encyclopedia of Parallel Computing}, publisher={Springer, Boston, MA}, author={Gustafson, John L.}, year={2011}, pages={182–185}, language={en} }