diff --git a/main.tex b/main.tex index 0f980e2..8d612a7 100644 --- a/main.tex +++ b/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} diff --git a/ref.bib b/ref.bib index cdab67c..67ba471 100644 --- a/ref.bib +++ b/ref.bib @@ -1,113 +1,104 @@ - -@inproceedings{vygen_fptas_2024, - address = {Cham}, - title = {An {FPTAS} for {Connectivity} {Interdiction}}, - volume = {14679}, - isbn = {978-3-031-59834-0 978-3-031-59835-7}, - url = {https://link.springer.com/10.1007/978-3-031-59835-7_16}, - language = {en}, - urldate = {2024-11-04}, - booktitle = {Integer {Programming} and {Combinatorial} {Optimization}}, - publisher = {Springer Nature Switzerland}, - author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai}, - editor = {Vygen, Jens and Byrka, Jarosław}, - year = {2024}, - doi = {10.1007/978-3-031-59835-7_16}, - pages = {210--223}, +@inproceedings{huang_fptas_2024, + address = {Cham}, + title = {An {FPTAS} for {Connectivity} {Interdiction}}, + volume = {14679}, + isbn = {978-3-031-59834-0 978-3-031-59835-7}, + url = {https://link.springer.com/10.1007/978-3-031-59835-7_16}, + language = {en}, + urldate = {2024-11-04}, + booktitle = {Integer {Programming} and {Combinatorial} {Optimization}}, + publisher = {Springer Nature Switzerland}, + author = {Huang, Chien-Chung and Obscura Acosta, Nidia and Yingchareonthawornchai, Sorrachai}, + editor = {Vygen, Jens and Byrka, Jaros\l{}aw}, + year = {2024}, + doi = {10.1007/978-3-031-59835-7_16}, + pages = {210--223} } - @article{chalermsook_approximating_2022, - title = {Approximating k-{Edge}-{Connected} {Spanning} {Subgraphs} via a {Near}-{Linear} {Time} {LP} {Solver}}, - volume = {229}, - copyright = {Creative Commons Attribution 4.0 International license, info:eu-repo/semantics/openAccess}, - issn = {1868-8969}, - url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.37}, - doi = {10.4230/LIPICS.ICALP.2022.37}, - language = {en}, - urldate = {2025-03-09}, - journal = {LIPIcs, Volume 229, ICALP 2022}, - author = {Chalermsook, Parinya and Huang, Chien-Chung and Nanongkai, Danupon and Saranurak, Thatchaphol and Sukprasert, Pattara and Yingchareonthawornchai, Sorrachai}, - editor = {Bojańczyk, Mikołaj and Merelli, Emanuela and Woodruff, David P.}, - year = {2022}, - keywords = {Approximation Algorithms, Data Structures, Theory of computation → Routing and network design problems}, - pages = {37:1--37:20}, + title = {Approximating k-{Edge}-{Connected} {Spanning} {Subgraphs} via a {Near}-{Linear} {Time} {LP} {Solver}}, + volume = {229}, + copyright = {Creative Commons Attribution 4.0 International license, info:eu-repo/semantics/openAccess}, + issn = {1868-8969}, + url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.37}, + doi = {10.4230/LIPICS.ICALP.2022.37}, + language = {en}, + urldate = {2025-03-09}, + journal = {LIPIcs, Volume 229, ICALP 2022}, + author = {Chalermsook, Parinya and Huang, Chien-Chung and Nanongkai, Danupon and Saranurak, Thatchaphol and Sukprasert, Pattara and Yingchareonthawornchai, Sorrachai}, + editor = {Boja\'{n}czyk, Miko\l{}aj and Merelli, Emanuela and Woodruff, David P.}, + year = {2022}, + keywords = {Approximation Algorithms, Data Structures, Theory of computation \rightarrow{} Routing and network design problems}, + pages = {37:1--37:20} } - @article{garg_faster_nodate, -author = {Garg, Naveen and K\"{o}nemann, Jochen}, -title = {Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems}, -journal = {SIAM Journal on Computing}, -volume = {37}, -number = {2}, -pages = {630-652}, -year = {2007}, -doi = {10.1137/S0097539704446232}, -URL = { https://doi.org/10.1137/S0097539704446232}, -eprint = {https://doi.org/10.1137/S0097539704446232}, -abstract = { This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new, faster, and much simpler algorithms for these problems. } + author = {Garg, Naveen and K\"{o}nemann, Jochen}, + title = {Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems}, + journal = {SIAM Journal on Computing}, + volume = {37}, + number = {2}, + pages = {630--652}, + year = {2007}, + doi = {10.1137/S0097539704446232}, + url = {https://doi.org/10.1137/S0097539704446232}, + eprint = {https://doi.org/10.1137/S0097539704446232}, + abstract = {This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new, faster, and much simpler algorithms for these problems.} } - @article{cunningham_optimal_1985, - title = {Optimal attack and reinforcement of a network}, - volume = {32}, - issn = {0004-5411, 1557-735X}, - url = {https://dl.acm.org/doi/10.1145/3828.3829}, - doi = {10.1145/3828.3829}, - language = {en}, - number = {3}, - urldate = {2025-04-11}, - journal = {Journal of the ACM}, - author = {Cunningham, William H.}, - month = jul, - year = {1985}, - pages = {549--561}, + title = {Optimal attack and reinforcement of a network}, + volume = {32}, + issn = {0004-5411, 1557-735X}, + url = {https://dl.acm.org/doi/10.1145/3828.3829}, + doi = {10.1145/3828.3829}, + language = {en}, + number = {3}, + urldate = {2025-04-11}, + journal = {Journal of the ACM}, + author = {Cunningham, William H.}, + month = jul, + year = {1985}, + pages = {549--561} } - - @article{chekuri_lp_2020, - title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}}, - volume = {34}, - issn = {0895-4801, 1095-7146}, - url = {https://epubs.siam.org/doi/10.1137/19M1299359}, - doi = {10.1137/19M1299359}, - language = {en}, - number = {2}, - urldate = {2022-04-10}, - journal = {SIAM Journal on Discrete Mathematics}, - author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao}, - month = jan, - year = {2020}, - keywords = {Approximation, K-cut, Minimum cut, Tree packing}, - pages = {1334--1353}, + title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} $k$-{Cut}}, + volume = {34}, + issn = {0895-4801, 1095-7146}, + url = {https://epubs.siam.org/doi/10.1137/19M1299359}, + doi = {10.1137/19M1299359}, + language = {en}, + number = {2}, + urldate = {2022-04-10}, + journal = {SIAM Journal on Discrete Mathematics}, + author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao}, + month = jan, + year = {2020}, + keywords = {Approximation, K-cut, Minimum cut, Tree packing}, + pages = {1334--1353} } - @inproceedings{10.1145/3618260.3649730, -author = {Chen, Lin and Lian, Jiayi and Mao, Yuchen and Zhang, Guochuan}, -title = {A Nearly Quadratic-Time FPTAS for Knapsack}, -year = {2024}, -isbn = {9798400703836}, -publisher = {Association for Computing Machinery}, -address = {New York, NY, USA}, -url = {https://doi.org/10.1145/3618260.3649730}, -doi = {10.1145/3618260.3649730}, -abstract = {We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in O(n + (1/)2) time. Prior to our work, the best running time is O(n + (1/)11/5) [Deng, Jin, and Mao’23]. Our algorithm is the best possible (up to a polylogarithmic factor), as Knapsack has no O((n + 1/)2−δ)-time FPTAS for any constant δ > 0, conditioned on the conjecture that (min, +)-convolution has no truly subquadratic-time algorithm.}, -booktitle = {Proceedings of the 56th Annual ACM Symposium on Theory of Computing}, -pages = {283–294}, -numpages = {12}, -keywords = {Approximation scheme, Knapsack}, -location = {Vancouver, BC, Canada}, -series = {STOC 2024} + author = {Chen, Lin and Lian, Jiayi and Mao, Yuchen and Zhang, Guochuan}, + title = {A Nearly Quadratic-Time FPTAS for Knapsack}, + year = {2024}, + isbn = {9798400703836}, + publisher = {Association for Computing Machinery}, + address = {New York, NY, USA}, + url = {https://doi.org/10.1145/3618260.3649730}, + doi = {10.1145/3618260.3649730}, + abstract = {We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in O(n + (1/)2) time. Prior to our work, the best running time is O(n + (1/)11/5) [Deng, Jin, and Mao'23]. Our algorithm is the best possible (up to a polylogarithmic factor), as Knapsack has no O((n + 1/)2-\ensuremath{\delta})-time FPTAS for any constant \ensuremath{\delta} > 0, conditioned on the conjecture that (min, +)-convolution has no truly subquadratic-time algorithm.}, + booktitle = {Proceedings of the 56th Annual ACM Symposium on Theory of Computing}, + pages = {283–294}, + numpages = {12}, + keywords = {Approximation scheme, Knapsack}, + location = {Vancouver, BC, Canada}, + series = {STOC 2024} } - - @incollection{salowe_parametric, -author = {Salowe, Jeffrey S.}, -title = {Parametric search}, -year = {1997}, -isbn = {0849385245}, -publisher = {CRC Press, Inc.}, -address = {USA}, -booktitle = {Handbook of Discrete and Computational Geometry}, -pages = {683–695}, -numpages = {13} -} \ No newline at end of file + author = {Salowe, Jeffrey S.}, + title = {Parametric search}, + year = {1997}, + isbn = {0849385245}, + publisher = {CRC Press, Inc.}, + address = {USA}, + booktitle = {Handbook of Discrete and Computational Geometry}, + pages = {683–695}, + numpages = {13} +}