wanyu/IteraTeR_v2 huggingface dataset:
Original: For a fixed connected graph H, the \{H\{\cal  {\cal  -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal   this problem can be solved in time f  (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal  }.
Revised: For a fixed connected graph H, the \{H\{\cal F {\cal -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal this problem can be solved in time f (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal }.
Edit intent: clarity
Original: For a fixed connected graph H, the \{H\{\cal  {\cal  -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal   this problem can be solved in time f  (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal  }.
Revised: For a fixed connected graph H, the \{H\{\cal , the{\cal -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal this problem can be solved in time f (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal }.
Edit intent: fluency
Original: For a fixed connected graph H, the \{H\{\cal  {\cal  -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal   this problem can be solved in time f  (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal  }.
Revised: For a fixed connected graph H, the \{H\{\cal {\cal F -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal this problem can be solved in time f (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal }.
Edit intent: clarity
Original: For a fixed connected graph H, the \{H\{\cal  {\cal  -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal   this problem can be solved in time f  (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal  }.
Revised: For a fixed connected graph H, the \{H\{\cal {\cal -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal F this problem can be solved in time f (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal }.
Edit intent: clarity
Original: For a fixed connected graph H, the \{H\{\cal  {\cal  -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal   this problem can be solved in time f  (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal  }.
Revised: For a fixed connected graph H, the \{H\{\cal {\cal -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal this problem can be solved in time f (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal F }.
Edit intent: clarity
Original: For a fixed connected graph H, the \{H\{\cal  {\cal  -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal   this problem can be solved in time f  (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal  }.
Revised: For a fixed connected graph H, the \{H\{\cal {\cal -M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G . It is known that{\cal this problem can be solved in time f (tw)\cdot n^{O(1)}, where tw is the treewidth of G . We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^{\Theta(tw)} depending on }{\cal In a recent series of articles, we have initiated the programme of optimizing asymptotically the function f_{{\cal F}.
Edit intent: coherence
Original:  For the upper bounds, we present an algorithm in time 2^{\Theta( tw\cdot \log tw) \cdot n^{O(1)} for the more general problem where all minor models of connected graphs in a finite family }  ]  { \mathcal \cal F} need to be hit.
Revised:  For the upper bounds, we present an algorithm in time 2^{\Theta( tw\cdot \log tw) \cdot n^{O(1)} for the more general problem where all minor models of connected graphs in a finite family } ] { \cal F} need to be hit.
Edit intent: coherence
Original: \} and of Kociumaka and Pilipczuk }[ [  We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement.
Revised: \} and of Kociumaka and Pilipczuk }[ [ . We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement.
Edit intent: fluency
Original: {\cal  [ SODA 2014 ] for the particular case ] { \mathcal \cal F} = \{ K_5,K_{3,3\} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 , which uses the framework introduced by Lokshtanov et al.
Revised: {\cal F [ SODA 2014 ] for the particular case ] { \mathcal \cal F} = \{ K_5,K_{3,3\} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 , which uses the framework introduced by Lokshtanov et al.
Edit intent: fluency
Original: {\cal  [ SODA 2014 ] for the particular case ] { \mathcal \cal F} = \{ K_5,K_{3,3\} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 , which uses the framework introduced by Lokshtanov et al.
Revised: {\cal [ SODA 2014 ] for the particular case ] { \cal F} = \{ K_5,K_{3,3\} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 , which uses the framework introduced by Lokshtanov et al.
Edit intent: coherence
Original: {\cal  [ SODA 2014 ] for the particular case ] { \mathcal \cal F} = \{ K_5,K_{3,3\} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 , which uses the framework introduced by Lokshtanov et al.
Revised: {\cal [ SODA 2014 ] for the particular case ] { \mathcal \cal F} = \{ \} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 , which uses the framework introduced by Lokshtanov et al.
Edit intent: coherence
Original: {\cal  [ SODA 2014 ] for the particular case ] { \mathcal \cal F} = \{ K_5,K_{3,3\} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 , which uses the framework introduced by Lokshtanov et al.
Revised: {\cal [ SODA 2014 ] for the particular case ] { \mathcal \cal F} = \{ K_5,K_{3,3\} . For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in }%DIFDELCMD < [%%% IPEC 2018 
Edit intent: coherence
Original:  More precisely, we prove the following result: for any \delta_0 > 0, there exists a k_0 such that if k \geq k_0, \delta \in (\delta_0,1 - 2/k], and {\delta}k + k - 1 is an odd integer then the presidential-type predicate P(x) = sign({\delta}k{x_1} + i=2^{k}{x_i}) is approximable  .
Revised:  More precisely, we prove the following result: for any \delta_0 > 0, there exists a k_0 such that if k \geq k_0, \delta \in (\delta_0,1 - 2/k], and {\delta}k + k - 1 is an odd integer then the presidential type predicate P(x) = sign({\delta}k{x_1} + i=2^{k}{x_i}) is approximable .
Edit intent: fluency
Original: We will find a lower bound on the recognition complexity of the theories that are nontrivial relative to equality (or equational-nontrivial ), namely, each of these theories is consistent with the formula, whose sense is that there exist two various elementsat least .
Revised: We will find a lower bound on the recognition complexity of the theories that are nontrivial relative to some equivalence relation (this relation may be equality ), namely, each of these theories is consistent with the formula, whose sense is that there exist two various elementsat least .
Edit intent: clarity
Original: We will find a lower bound on the recognition complexity of the theories that are nontrivial relative to equality (or equational-nontrivial ), namely, each of these theories is consistent with the formula, whose sense is that there exist two various elementsat least .
Revised: We will find a lower bound on the recognition complexity of the theories that are nontrivial relative to equality (or equational-nontrivial ), namely, each of these theories is consistent with the formula, whose sense is that there exist two non-equivalent elements .
Edit intent: clarity
Original:  Then, we will construct a polynomial reduction of the theory of this Boolean algebra to the  first-order theory of pure equality . Keywords: Computational complexity, the theory of equality, the coding of computations, simulation by means formulae, polynomial time, polynomial space, lower complexity bound
Revised:  Then, we will transform the modeling formulae of the theory of this Boolean algebra to the first-order theory of pure equality . Keywords: Computational complexity, the theory of equality, the coding of computations, simulation by means formulae, polynomial time, polynomial space, lower complexity bound
Edit intent: clarity
Original:  Then, we will construct a polynomial reduction of the theory of this Boolean algebra to the  first-order theory of pure equality . Keywords: Computational complexity, the theory of equality, the coding of computations, simulation by means formulae, polynomial time, polynomial space, lower complexity bound
Revised:  Then, we will construct a polynomial reduction of the theory of this Boolean algebra to the simulation ones of the first-order theory of pure equality . Keywords: Computational complexity, the theory of equality, the coding of computations, simulation by means formulae, polynomial time, polynomial space, lower complexity bound
Edit intent: clarity
Original:  In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n^{1-\varepsilon} for any \varepsilon > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction.
Revised:  In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n^{1-\varepsilon} for any \varepsilon > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction.
Edit intent: fluency
Original:  Such clustering problems (on graph metrics) are essential to data analysis and used for example in road networks and data visualization.
Revised:  Such clustering problems are essential to data analysis and used for example in road networks and data visualization.
Edit intent: clarity
Original:  Specifically, we consider (k, z)-Clustering, where given a metric space (V, d), the goal is to minimize, over all k-point center sets C, the objective x \in V{d^z(x, C) A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set . Coresets offer significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings.
Revised:  A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set . Coresets offer significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings.
Edit intent: coherence
Original:  Specifically, we consider (k, z)-Clustering, where given a metric space (V, d), the goal is to minimize, over all k-point center sets C, the objective x \in V{d^z(x, C) A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set . Coresets offer significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings.
Revised:  Specifically, we consider (k, z)-Clustering, where given a metric space (V, d), the goal is to minimize, over all k-point center sets C, the objective x \in V{d^z(x, C) A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set , and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings.
Edit intent: coherence
Original:  Our main result is a near-linear time construction of a coreset of size O _{\epsilon, k , z }(tw(G)) for (k, z)-Clustering in a graph G whose treewidth is tw(G)  . The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of O(tw(G)) on the shattering dimension under any point weights.
Revised:  Our main result is a near-linear time construction of a coreset of size O _{\epsilon, k }(tw(G)) for (k, z)-Clustering in a graph G whose treewidth is tw(G) . The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of O(tw(G)) on the shattering dimension under any point weights.
Edit intent: fluency
Original:  Our main result is a near-linear time construction of a coreset of size O _{\epsilon, k , z }(tw(G)) for (k, z)-Clustering in a graph G whose treewidth is tw(G)  . The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of O(tw(G)) on the shattering dimension under any point weights.
Revised:  Our main result is a near-linear time construction of a coreset of size O _{\epsilon, k , z }(tw(G)) where tw(G) . The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of O(tw(G)) on the shattering dimension under any point weights.
Edit intent: coherence
Original:  Previously, the only construction applicable to graph metrics, even for z=1, was a generic one with size O_{\epsilon, k(\log n) where n=|V| }%DIFDELCMD < [%%% Feldman and Langberg, STOC 2011 . We complement our construction with an \epsilon, k(tw(G)) size lower bound, which matches our construction's linear dependence on tw(G).
Revised:  (\log n) where n=|V| }%DIFDELCMD < [%%% Feldman and Langberg, STOC 2011 . We complement our construction with an \epsilon, k(tw(G)) size lower bound, which matches our construction's linear dependence on tw(G).
Edit intent: coherence
Original:  Previously, the only construction applicable to graph metrics, even for z=1, was a generic one with size O_{\epsilon, k(\log n) where n=|V| }%DIFDELCMD < [%%% Feldman and Langberg, STOC 2011 . We complement our construction with an \epsilon, k(tw(G)) size lower bound, which matches our construction's linear dependence on tw(G).
Revised:  Previously, the only construction applicable to graph metrics, even for z=1, was a generic one with size O_{\epsilon, k(\log n) where n=|V| }%DIFDELCMD < [%%% . We complement our construction with an \epsilon, k(tw(G)) size lower bound, which matches our construction's linear dependence on tw(G).
Edit intent: coherence
Original:  In view of much recent attention to the RIS concept as a promising technology for 6G wireless communications, we present a survey of RISs in this paper.
Revised:  In view of much recent attention to the IRS concept as a promising technology for 6G wireless communications, we present a survey of RISs in this paper.
Edit intent: fluency
Original:  In view of much recent attention to the RIS concept as a promising technology for 6G wireless communications, we present a survey of RISs in this paper.
Revised:  In view of much recent attention to the RIS concept as a promising technology for 6G wireless communications, we present a survey of IRSs in this paper.
Edit intent: fluency
Original:  Specifically, we categorize recent research studies of RISs as follows.
Revised:  Specifically, we categorize recent research studies of IRSs as follows.
Edit intent: fluency
Original:  For RIS-aided communications, the summary includes capacity/data rate analyses, power/spectral optimizations, channel estimation, deep learning-based design, and reliability analysis.
Revised:  For IRS-aided communications, the summary includes capacity/data rate analyses, power/spectral optimizations, channel estimation, deep learning-based design, and reliability analysis.
Edit intent: fluency
Original:  Then we review RISs implementations as well as the use of RISs in secure communications, terminal-positioning, and other novel applications.
Revised:  Then we review IRSs implementations as well as the use of RISs in secure communications, terminal-positioning, and other novel applications.
Edit intent: fluency
Original:  Then we review RISs implementations as well as the use of RISs in secure communications, terminal-positioning, and other novel applications.
Revised:  Then we review RISs implementations as well as the use of IRSs in secure communications, terminal-positioning, and other novel applications.
Edit intent: fluency
Original:  We further identify future research directions for RISs , with an envision of the RIS technology playing a critical role in 6G communication networks similar to that of massive MIMO in 5G networks.
Revised:  We further identify future research directions for IRSs , with an envision of the RIS technology playing a critical role in 6G communication networks similar to that of massive MIMO in 5G networks.
Edit intent: fluency
Original:  We further identify future research directions for RISs , with an envision of the RIS technology playing a critical role in 6G communication networks similar to that of massive MIMO in 5G networks.
Revised:  We further identify future research directions for RISs , with an envision of the IRS technology playing a critical role in 6G communication networks similar to that of massive MIMO in 5G networks.
Edit intent: clarity
Original:  As a timely summary of RISs , our work will be of interest to both researchers and practitioners working on RISs for 6G networks.
Revised:  As a timely summary of IRSs , our work will be of interest to both researchers and practitioners working on RISs for 6G networks.
Edit intent: fluency
Original:  As a timely summary of RISs , our work will be of interest to both researchers and practitioners working on RISs for 6G networks.
Revised:  As a timely summary of RISs , our work will be of interest to both researchers and practitioners working on IRSs for 6G networks.
Edit intent: fluency
Original:  In this article, we consider different colour corrections of compressed videos which increase the values of full-reference metric VMAF and almost don't decrease other widely-used metric SSIM. The proposed video contrast enhancement approach shows the metric inapplicability  for video codecs comparisons, as it may be used for cheating in the comparisons via tuning to improve this metric values.
Revised:  In this article, we consider different color corrections of compressed videos which increase the values of full-reference metric VMAF and almost don't decrease other widely-used metric SSIM. The proposed video contrast enhancement approach shows the metric inapplicability for video codecs comparisons, as it may be used for cheating in the comparisons via tuning to improve this metric values.
Edit intent: fluency
Original:  Instead, mobile phone data are becoming a go to proxy to evaluate population density, activity and social characteristics.
Revised:  Instead, mobile phone data are becoming a trending proxy to evaluate population density, activity and social characteristics.
Edit intent: clarity
Original:  They offer additional advantages for infrastructure planning such as being updated in real-time, including mobility information and recording temporary visitors' activity.
Revised:  They offer additional advantages for infrastructure planning such as being updated in real-time, including mobility information and recording visitors' activity.
Edit intent: clarity
Original:  As an applied case, we test their ability at predicting accurately domestic electricity consumption.
Revised:  As an applied case, we test their ability at predicting domestic electricity consumption.
Edit intent: clarity
Original:  We show that, contrary to common belief, average mobile phone activity is not well correlated with population density.
Revised:  We show that, contrary to common belief, average mobile phone activity does not correlate well with population density.
Edit intent: fluency
Original:  More importantly, we successfully use curve and network clustering techniques to enhance the accuracy of the predictions, to recover good population mapping potential and to reduce the collection of informative data for planning to substantially smaller samples.
Revised:  More importantly, we use curve and network clustering techniques to enhance the accuracy of the predictions, to recover good population mapping potential and to reduce the collection of informative data for planning to substantially smaller samples.
Edit intent: clarity