wanyu/IteraTeR_v2 huggingface dataset:
Original: We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G . This problem is #W 1 -hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)\cdot n^{O(1) 1%DIFDELCMD < ]%%% -hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side.
Revised: We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G . This problem is #W 1 -hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)\cdot n^{O(1) %DIFDELCMD < ]%%% -hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side.
Edit intent: fluency
Original: We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G . This problem is #W 1 -hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)\cdot n^{O(1) 1%DIFDELCMD < ]%%% -hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side.
Revised: We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G . This problem is #W 1 -hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k)\cdot n^{O(1) 1%DIFDELCMD < ]%%% on graphs G that are line graphs or bipartite graphs with degree 2 on one side.
Edit intent: clarity
Original:  To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs.
Revised:  In our proofs, we use that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs.
Edit intent: clarity
Original:  To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two into (arbitrary) host graphs.
Revised:  To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k length-2 paths into (arbitrary) host graphs.
Edit intent: clarity
Original:  For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms , restricted to patterns from H , is #W 1 -hard if no such bound exists .
Revised:  For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms with patterns from H , is #W 1 -hard if no such bound exists .
Edit intent: clarity
Original:  For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms , restricted to patterns from H , is #W 1 -hard if no such bound exists .
Revised:  For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms , restricted to patterns from H 1 -hard if no such bound exists .
Edit intent: clarity
Original:  For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms , restricted to patterns from H , is #W 1 -hard if no such bound exists .
Revised:  For hereditary classes H of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms , restricted to patterns from H , is #W 1 -hard .
Edit intent: clarity
Original:  Despite a number of efforts, the bound remained unchanged for about 34 years.
Revised:  Despite a number of efforts, the bound remained unchanged for about 35 years.
Edit intent: fluency
Original: We improve the best  upper bound on the length of the shortest reset words of synchronizing automata.
Revised: We improve the best known upper bound on the length of the shortest reset words of synchronizing automata.
Edit intent: clarity
Original:  Despite a number of efforts, the bound remained unchanged for about 35 years.
Revised:  Despite a number of efforts, it remained unchanged for about 35 years.
Edit intent: clarity
Original:  The new upper bound improves the coefficient 1/6 at n^3 by 4/46875.
Revised:  To obtain the new upper bound improves the coefficient 1/6 at n^3 by 4/46875.
Edit intent: clarity
Original:  A word is avoiding for a state q if after reading the word , the automaton cannot be in q. We prove an upper bound on the length of a word that is either avoiding or compressing a subset.
Revised:  A word is avoiding for a state q if after reading the word the automaton cannot be in q. We prove an upper bound on the length of a word that is either avoiding or compressing a subset.
Edit intent: fluency
Original:  A word is avoiding for a state q if after reading the word , the automaton cannot be in q. We prove an upper bound on the length of a word that is either avoiding or compressing a subset.
Revised:  A word is avoiding for a state q if after reading the word , the automaton cannot be in q. We obtain upper bounds on the length of a word that is either avoiding or compressing a subset.
Edit intent: clarity
Original:  Then we show a quadratic bound on the length of the shortest avoiding words, and , using the approach of Trahtman from 2011  , we improve the general upper bound on the length of the shortest reset words.
Revised:  Then we show a quadratic bound on the length of the shortest avoiding words, and using the approach of Trahtman from 2011 , we improve the general upper bound on the length of the shortest reset words.
Edit intent: fluency
Original:  With the help of this technique, we study several concrete problems , also in more general settings.
Revised:  With the help of problems , also in more general settings.
Edit intent: clarity
Original: The stochastic dynamics of networks of biochemical reactions in living cells are typically modelled using chemical master equations (CMEs ).
Revised: The stochastic dynamics of biochemical networks is usually modelled with the chemical master equation (CME ).
Edit intent: clarity
Original:  First, we use semidefinite programming to obtain increasingly tighter upper and lower bounds on the moments of the stationary distribution for networks with rational propensities.
Revised:  First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distribution for networks with rational propensities.
Edit intent: clarity
Original:  First, we use semidefinite programming to obtain increasingly tighter upper and lower bounds on the moments of the stationary distribution for networks with rational propensities.
Revised:  First, we use semidefinite programming to obtain increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities.
Edit intent: fluency
Original:  The bounds obtained provide a computational test for the uniqueness of the stationary distribution .
Revised:  The bounds obtained provide a computational test for the uniqueness of these distributions .
Edit intent: clarity
Original:  In the unique case, the bounds collectively form an approximation of the stationary distribution accompanied with a computable \ell^1-error bound .
Revised:  In the unique case, the bounds form an approximation of the stationary distribution accompanied with a computable \ell^1-error bound .
Edit intent: clarity
Original:  In the unique case, the bounds collectively form an approximation of the stationary distribution accompanied with a computable \ell^1-error bound .
Revised:  In the unique case, the bounds collectively form an approximation of the stationary distribution with a computable \ell^1-error bound .
Edit intent: clarity
Original:  In the unique case, the bounds collectively form an approximation of the stationary distribution accompanied with a computable \ell^1-error bound .
Revised:  In the unique case, the bounds collectively form an approximation of the stationary distribution accompanied with a computable bound on its error .
Edit intent: clarity
Original:  In the non-unique case, we explain how to adapt our approach so that it yields approximations of the ergodic distributions , also accompanied with computable error bounds .
Revised:  In the non-unique case, we explain how to adapt our approach so that it yields approximations of the ergodic distributions .
Edit intent: clarity
Original: The stochastic dynamics of biochemical networks is usually modelled with the chemical master equation (CME).
Revised: The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME).
Edit intent: fluency
Original:  To fill this gap , we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy.
Revised:  Here , we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy.
Edit intent: coherence
Original:  The bounds obtained  provide a computational test for the uniqueness of these distributions .
Revised:  The bounds obtained also provide a computational test for the uniqueness of these distributions .
Edit intent: clarity
Original:  The bounds obtained  provide a computational test for the uniqueness of these distributions .
Revised:  The bounds obtained provide a computational test for the uniqueness of the distribution .
Edit intent: clarity
Original: The complexity of approximately counting independent sets in bipartite graphs (#BIS) is a central open problem in approximate counting , and it is widely believed to be neither easy nor NP-hard.
Revised: The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is a central open problem in approximate counting , and it is widely believed to be neither easy nor NP-hard.
Edit intent: clarity
Original:  We study several natural parameterised variants of #BIS , both from the polynomial-time and from the fixed-parameter viewpoint: counting independent sets of a given size ;
Revised:  We study several natural parameterised variants of #BIS , both from the polynomial-time and from the fixed-parameter viewpoint: counting independent sets of a given size ,
Edit intent: fluency
Original:  counting independent sets with a given number of vertices in one vertex class ;
Revised:  counting independent sets with a given number of vertices in one vertex class 
Edit intent: fluency
Original:  and counting maximum independent sets among those with a given number of vertices in one vertex class.
Revised:  and counting maximum independent sets amongst those with a given number of vertices in one vertex class.
Edit intent: fluency
Original:  Among other things, we show that all  these problems are NP-hard to approximate within any polynomial ratio.
Revised:  Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio.
Edit intent: fluency
Original:   We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, and the other two are W[1]-hard to approximate  within any polynomial ratio.
Revised:  We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate within any polynomial ratio.
Edit intent: coherence
Original:   We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, and the other two are W[1]-hard to approximate  within any polynomial ratio.
Revised:  We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, and the other two are W[1]-hard to approximate even within any polynomial ratio.
Edit intent: clarity
Original:  Finally, we show that  when restricted to graphs of bounded degree, all three problems admit exact fixed-parameter algorithms with reasonable time complexity .
Revised:  Finally, we show that , when restricted to graphs of bounded degree, all three problems admit exact fixed-parameter algorithms with reasonable time complexity .
Edit intent: coherence
Original:  Finally, we show that  when restricted to graphs of bounded degree, all three problems admit exact fixed-parameter algorithms with reasonable time complexity .
Revised:  Finally, we show that when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms with reasonable time complexity .
Edit intent: clarity
Original:  Finally, we show that  when restricted to graphs of bounded degree, all three problems admit exact fixed-parameter algorithms with reasonable time complexity .
Revised:  Finally, we show that when restricted to graphs of bounded degree, all three problems admit exact fixed-parameter algorithms .
Edit intent: clarity
Original: Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors.
Revised: Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurringin the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors.
Edit intent: fluency
Original:  Though an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that  is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media.
Revised:  Although an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media.
Edit intent: fluency
Original:  Though an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that  is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media.
Revised:  Though an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that it is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media.
Edit intent: fluency
Original:  The paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, full explanations of the object considered and detailed lists of references are provided.
Revised:  This paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, full explanations of the object considered and detailed lists of references are provided.
Edit intent: fluency
Original:  The paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, full explanations of the object considered and detailed lists of references are provided.
Revised:  The paper is written in such a way as to be of interest to both biologists and mathematician alike. In order to accomplish this aim, full explanations of the object considered and detailed lists of references are provided.
Edit intent: clarity
Original:  The paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, full explanations of the object considered and detailed lists of references are provided.
Revised:  The paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, both complete explanations of the object considered and detailed lists of references are provided.
Edit intent: clarity
Original:  The paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, full explanations of the object considered and detailed lists of references are provided.
Revised:  The paper is written in such a way to be usable by both the communities of biologists and mathematicians: to this aim, full explanations of the objects considered and detailed lists of references are provided.
Edit intent: fluency
Original: This paper focuses on the development of randomized approaches for building deep neural networks .
Revised: This paper develops a randomized approach for incrementally building deep neural networks .
Edit intent: clarity
Original:  A supervisory mechanism is proposed to constrain the random assignment of the hidden parameters (i.e., all biasesand weights within the hidden layers ).
Revised:  A supervisory mechanism is proposed to constrain the random assignment of the weights and biases, and all the hidden layers ).
Edit intent: clarity
Original:  The read-out weights attached with all direct links from each hidden layer to the output layer are incrementally evaluated by the least squares method.
Revised:  The read-out weights attached with all direct links from each hidden layer to the output layer are evaluated by the least squares method.
Edit intent: clarity
Original:  Simulation results with comparisons on function approximation align with the theoretical findings .
Revised:  An empirical study on a function approximation is carried out to demonstrate some properties of the proposed deep learner model .
Edit intent: clarity
Original:  A major evolution would be to leverage the characteristic of OSNs to evaluate the effectiveness of the many routing schemes developed by the research community in real-world scenarios.
Revised:  A major evolution would be to leverage the characteristics of OSNs to evaluate the effectiveness of the many routing schemes developed by the research community in real-world scenarios.
Edit intent: fluency
Original: Assume that the ]  edges of the complete bipartite graph K_{n,n} are labeled with elements of F _2^d , such that the sum over any simple cycle  is nonzero.
Revised: Assume that the ] edges of the complete bipartite graph K_{n,n} with labels coming from F _2^d , such that the sum over any simple cycle is nonzero.
Edit intent: clarity
Original:  SODA 2017%DIFDELCMD < ] %%% as it characterizes the alphabet size needed for maximally recoverable codes in  grid topologies.
Revised:  %DIFDELCMD < ] %%% as it characterizes the alphabet size needed for maximally recoverable codes in grid topologies.
Edit intent: coherence
Original:  The upper bound is an explicit construction which improves upon the random construction.
Revised:  The upper bound is a recursive construction which beats the random construction.
Edit intent: clarity
Original:  We propose novel CPU-GPU methods that \emph{scale to networks with billions of edges , yet, possess rigorous theoretical guarantee on the solution quality.
Revised:  We propose novel CPU-GPU methods that \emph{scale to networks with billions of edges , yet, possess rigorous theoretical guarantee on the solution quality.
Edit intent: fluency
Original:  At the core of our methods is an O(1)-space out-of-core algorithm to generate a new type of random walks, called Hitting Self-avoiding Walks (%DIFDELCMD < %%% s ).
Revised:  At the core of our methods is an O(1)-space out-of-core algorithm to generate a new type of random walks, called Hitting Self-avoiding Walks (%DIFDELCMD < %%% s ).
Edit intent: fluency
Original:  At the core of our methods is an O(1)-space out-of-core algorithm to generate a new type of random walks, called Hitting Self-avoiding Walks (%DIFDELCMD < %%% s ).
Revised:  At the core of our methods is an O(1)-space out-of-core algorithm to generate a new type of random walks, called (%DIFDELCMD < %%% s ).
Edit intent: coherence
Original:  At the core of our methods is an O(1)-space out-of-core algorithm to generate a new type of random walks, called Hitting Self-avoiding Walks (%DIFDELCMD < %%% s ).
Revised:  At the core of our methods is an O(1)-space out-of-core algorithm to generate a new type of random walks, called Hitting Self-avoiding Walks %DIFDELCMD < %%% s ).
Edit intent: fluency
Original: The apparent paradox, devised by Juan M. R. Parrondo, that two losing games A and B can produce an winning outcome has been adapted in many physical and biological systems to explain their working.
Revised: The apparent paradox, devised by J. M. R. Parrondo, that two losing games A and B can produce an winning outcome has been adapted in many physical and biological systems to explain their working.
Edit intent: fluency
Original:   The implications of our results for observing quantum ratchet like behavior using quantum walks is also discussed.
Revised:  The implications of our work for observing quantum ratchet like behavior using quantum walks is also discussed.
Edit intent: clarity
Original: The apparent paradox, devised by J. M. R. Parrondo, that two losing games A and B can produce an winning outcome has been adapted in many physical and biological systems to explain their working.
Revised: The apparent paradox, devised by J. ~M.~R.~ Parrondo, that two losing games A and B can produce an winning outcome has been adapted in many physical and biological systems to explain their working.
Edit intent: coherence
Original:  In this paper  we explore the behaviour of the learning rate for a number of simple model systems, establishing when and how its behaviour is distinct from the instantaneous mutual information between subsystems.
Revised:  In this paper , we explore the behaviour of the learning rate for a number of simple model systems, establishing when and how its behaviour is distinct from the instantaneous mutual information between subsystems.
Edit intent: fluency
Original:  We consider applications to [{18}F]-fluorodeoxyglucose Positron Emission Tomography ( FDG--PET ) data and analyze the two--compartment catenary model describing the standard FDG metabolization by an homogeneous tissue and the three--compartment non--catenary model representing the renal physiology.
Revised:  We consider applications to [{18}F]-fluorodeoxyglucose Positron Emission Tomography ( FDG-PET ) data and analyze the two--compartment catenary model describing the standard FDG metabolization by an homogeneous tissue and the three--compartment non--catenary model representing the renal physiology.
Edit intent: fluency
Original:  We consider applications to [{18}F]-fluorodeoxyglucose Positron Emission Tomography ( FDG--PET ) data and analyze the two--compartment catenary model describing the standard FDG metabolization by an homogeneous tissue and the three--compartment non--catenary model representing the renal physiology.
Revised:  We consider applications to [{18}F]-fluorodeoxyglucose Positron Emission Tomography ( FDG--PET ) data and analyze the two-compartment catenary model describing the standard FDG metabolization by an homogeneous tissue and the three--compartment non--catenary model representing the renal physiology.
Edit intent: fluency