wanyu/IteraTeR_v2 huggingface dataset:
Original:  1. For an appropriate extension of depth-2 monotone circuits with local oracles, we show that the size of the smallest circuits separating U_{n,3} (triangles) and V_{n,3} (complete bipartite graphs) undergoes two phase transitions according to \mu. 2. For 5 \leq k(n) \leq n^{1/4}, arbitrary depth, and \mu \leq 1/50, we prove that the monotone circuit size complexity of separating the sets U_{n,k} and V_{n,k} is n^{\Theta(k)}, under a certain technical assumption on the local oracle gates.
Revised:  1. For an appropriate extension of depth-2 monotone circuits with local oracles, we show that the size of the smallest circuits separating U_{n,3} (triangles) and V_{n,3} (complete bipartite graphs) undergoes two phase transitions according to \mu. 2. For 5 \leq k(n) \leq n^{1/4}, arbitrary depth, and \mu \leq 1/50, we prove that the monotone circuit size complexity of separating the sets U_{n,k} and V_{n,k} is n^{\Theta(k)}, under a certain restrictive assumption on the local oracle gates.
Edit intent: clarity
Original:  We consider this in the context of resource-allocation problems ;
Revised:  We consider this in the context of resource-allocation problems ,
Edit intent: fluency
Original: In this work, we study the dynamic portfolio optimization problem related to the pairs trading, which is an investment strategy that matches a long position in one security with a short position in an another security with similar characteristics.
Revised: In this work, we study a dynamic portfolio optimization problem related to the pairs trading, which is an investment strategy that matches a long position in one security with a short position in an another security with similar characteristics.
Edit intent: fluency
Original: In this work, we study the dynamic portfolio optimization problem related to the pairs trading, which is an investment strategy that matches a long position in one security with a short position in an another security with similar characteristics.
Revised: In this work, we study the dynamic portfolio optimization problem related to pairs trading, which is an investment strategy that matches a long position in one security with a short position in an another security with similar characteristics.
Edit intent: fluency
Original: In this work, we study the dynamic portfolio optimization problem related to the pairs trading, which is an investment strategy that matches a long position in one security with a short position in an another security with similar characteristics.
Revised: In this work, we study the dynamic portfolio optimization problem related to the pairs trading, which is an investment strategy that matches a long position in one security with a short position in another security with similar characteristics.
Edit intent: fluency
Original:  The relation between pairs, called  spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time finite state Markov chain.
Revised:  The relationship between pairs, called spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time finite state Markov chain.
Edit intent: fluency
Original:  The relation between pairs, called  spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time finite state Markov chain.
Revised:  The relation between pairs, called a spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time finite state Markov chain.
Edit intent: fluency
Original:  We characterize optimal dollar-neutral strategies as well as optimal value functions under both full and partial information and show that the certainty  principle holds for the optimal portfolio strategy.
Revised:  We characterize optimal dollar-neutral strategies as well as optimal value functions under full and partial information and show that the certainty principle holds for the optimal portfolio strategy.
Edit intent: fluency
Original: Genome-wide association studies (GWAS ) in humans are revealing the genetic architecture of biomedical , life history and anthropomorphic traits, i.e., the frequencies and effect sizes of variants contributing to heritable variation in a trait.
Revised: GWAS in humans are revealing the genetic architecture of biomedical , life history and anthropomorphic traits, i.e., the frequencies and effect sizes of variants contributing to heritable variation in a trait.
Edit intent: clarity
Original: Genome-wide association studies (GWAS ) in humans are revealing the genetic architecture of biomedical , life history and anthropomorphic traits, i.e., the frequencies and effect sizes of variants contributing to heritable variation in a trait.
Revised: Genome-wide association studies (GWAS ) in humans are revealing the genetic architecture of biomedical and anthropomorphic traits, i.e., the frequencies and effect sizes of variants contributing to heritable variation in a trait.
Edit intent: clarity
Original: Genome-wide association studies (GWAS ) in humans are revealing the genetic architecture of biomedical , life history and anthropomorphic traits, i.e., the frequencies and effect sizes of variants contributing to heritable variation in a trait.
Revised: Genome-wide association studies (GWAS ) in humans are revealing the genetic architecture of biomedical , life history and anthropomorphic traits, i.e., the frequencies and effect sizes of variants that contribute to heritable variation in a trait.
Edit intent: fluency
Original:  To interpret these findings, we need to understand how genetic architecture is shaped by basic population genetics processes - notably , by mutation, natural selection and genetic drift.
Revised:  To interpret these findings, we need to understand how genetic architecture is shaped by basic population genetics processes-notably , by mutation, natural selection and genetic drift.
Edit intent: fluency
Original:  We solve the model for the phenotypic distribution and allelic dynamics at steady state and derive robust, closed form solutions for summaries of genetic architecture.
Revised:  We solve the model for the phenotypic distribution and allelic dynamics at steady state and derive robust, closed form solutions for summary statistics of the genetic architecture.
Edit intent: clarity
Original:  Our results suggest that the distribution of genetic variance among the loci discovered in GWAS take a simple form that depends on one evolutionary parameter, and provide a simple interpretation for missing heritability and why it varies among traits .
Revised:  Our results suggest that the distribution of variances contributed by loci identified in GWAS is well approximated by a simple functional form that depends on one evolutionary parameter, and provide a simple interpretation for missing heritability and why it varies among traits .
Edit intent: clarity
Original:  Our results suggest that the distribution of genetic variance among the loci discovered in GWAS take a simple form that depends on one evolutionary parameter, and provide a simple interpretation for missing heritability and why it varies among traits .
Revised:  Our results suggest that the distribution of genetic variance among the loci discovered in GWAS take a simple form that depends on a single parameter: the expected contribution to genetic variance of a strongly selected site affecting the trait .
Edit intent: clarity
Original:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size  .
Revised:  We test this prediction against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size .
Edit intent: clarity
Original:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size  .
Revised:  We test our predictions against the results of GWAS for height and BMI and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size .
Edit intent: clarity
Original:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size  .
Revised:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that it fits the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size .
Edit intent: clarity
Original:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size  .
Revised:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and mutational target size .
Edit intent: fluency
Original:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size  .
Revised:  We test our predictions against the results of GWAS for height and body mass index (BMI ) and find that they fit the data well, allowing us to make inferences about the degree of pleiotropy and the mutational target size for these traits .
Edit intent: clarity
Original:  Our findings help to understand why GWAS for height explain more of the heritable variance than similarly-sized GWAS for BMI, and to predict how future increases in sample size will translate into explained heritability .
Revised:  Our findings help to explain why the GWAS for height explain more of the heritable variance than similarly-sized GWAS for BMI, and to predict how future increases in sample size will translate into explained heritability .
Edit intent: style
Original:  Our findings help to understand why GWAS for height explain more of the heritable variance than similarly-sized GWAS for BMI, and to predict how future increases in sample size will translate into explained heritability .
Revised:  Our findings help to understand why GWAS for height explains more of the heritable variance than similarly-sized GWAS for BMI, and to predict how future increases in sample size will translate into explained heritability .
Edit intent: fluency
Original: It has long been known that Feedback Vertex Set can be solved in time 2^{O(w\log w)}n^{O(1)} on  graphs of treewidth w, but it was only recently that this running time was improved to 2^{O(w)}n^{O(1)}, that is, to single-exponential parameterized by treewidth.
Revised: It has long been known that Feedback Vertex Set can be solved in time 2^{O(w\log w)}n^{O(1)} on n-vertex graphs of treewidth w, but it was only recently that this running time was improved to 2^{O(w)}n^{O(1)}, that is, to single-exponential parameterized by treewidth.
Edit intent: clarity
Original:  Formally, for a class of graphs P , the Bounded P-Block Vertex Deletion problem asks, given a graph  G on n vertices and positive integers k and d, whether G contains a set  S of at most k vertices such that each block of G-S has at most d vertices and is in P.
Revised:  Formally, for a class \mathcal{P , the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G-S has at most d vertices and is in P.
Edit intent: clarity
Original:  Formally, for a class of graphs P , the Bounded P-Block Vertex Deletion problem asks, given a graph  G on n vertices and positive integers k and d, whether G contains a set  S of at most k vertices such that each block of G-S has at most d vertices and is in P.
Revised:  Formally, for a class of graphs P , the Bounded P-Block Vertex Deletion problem asks, given a graph ~ G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G-S has at most d vertices and is in P.
Edit intent: fluency
Original:  Formally, for a class of graphs P , the Bounded P-Block Vertex Deletion problem asks, given a graph  G on n vertices and positive integers k and d, whether G contains a set  S of at most k vertices such that each block of G-S has at most d vertices and is in P.
Revised:  Formally, for a class of graphs P , the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set ~ S of at most k vertices such that each block of G-S has at most d vertices and is in P.
Edit intent: fluency
Original:  (1) if P consists only of chordal graphs, then the problem can be solved in time 2^{O(wd^2)} n^{\mathcal{O}(1)}, (2) if \mathcal{P} contains a graph with an induced cycle of length \ell\ge 4, then the problem is not solvable in time 2^{o(w\log w)} n^{\mathcal{O}(1)} even for fixed d=\ell, unless the ETH fails.
Revised:  if P consists only of chordal graphs, then the problem can be solved in time 2^{O(wd^2)} n^{\mathcal{O}(1)}, (2) if \mathcal{P} contains a graph with an induced cycle of length \ell\ge 4, then the problem is not solvable in time 2^{o(w\log w)} n^{\mathcal{O}(1)} even for fixed d=\ell, unless the ETH fails.
Edit intent: fluency
Original:  (1) if P consists only of chordal graphs, then the problem can be solved in time 2^{O(wd^2)} n^{\mathcal{O}(1)}, (2) if \mathcal{P} contains a graph with an induced cycle of length \ell\ge 4, then the problem is not solvable in time 2^{o(w\log w)} n^{\mathcal{O}(1)} even for fixed d=\ell, unless the ETH fails.
Revised:  (1) if P consists only of chordal graphs, then the problem can be solved in time 2^{O(wd^2)} n^{\mathcal{O}(1)}, and if \mathcal{P} contains a graph with an induced cycle of length \ell\ge 4, then the problem is not solvable in time 2^{o(w\log w)} n^{\mathcal{O}(1)} even for fixed d=\ell, unless the ETH fails.
Edit intent: coherence
Original:  We also study a similar problem, called Bounded \mathcal{P}-Component Vertex Deletion, where the target graphs have connected components of small size instead of having blocks of small size, and we show analogous results.
Revised:  We also study a similar problem, called Bounded \mathcal{P}-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we show analogous results.
Edit intent: coherence
Original:  We also study a similar problem, called Bounded \mathcal{P}-Component Vertex Deletion, where the target graphs have connected components of small size instead of having blocks of small size, and we show analogous results.
Revised:  We also study a similar problem, called Bounded \mathcal{P}-Component Vertex Deletion, where the target graphs have connected components of small size instead of having blocks of small size, and we present analogous results.
Edit intent: clarity
Original:  For this problem, we further show that if d is a part of the input and \mathcal{P} contains all chordal graphs, then it cannot be solved in time f(w)n^{o(w)} for some function f, unless  ETH fails.
Revised:  For this problem, we also show that if d is a part of the input and \mathcal{P} contains all chordal graphs, then it cannot be solved in time f(w)n^{o(w)} for some function f, unless ETH fails.
Edit intent: clarity
Original:  For this problem, we further show that if d is a part of the input and \mathcal{P} contains all chordal graphs, then it cannot be solved in time f(w)n^{o(w)} for some function f, unless  ETH fails.
Revised:  For this problem, we further show that if d is part of the input and \mathcal{P} contains all chordal graphs, then it cannot be solved in time f(w)n^{o(w)} for some function f, unless ETH fails.
Edit intent: fluency
Original:  For this problem, we further show that if d is a part of the input and \mathcal{P} contains all chordal graphs, then it cannot be solved in time f(w)n^{o(w)} for some function f, unless  ETH fails.
Revised:  For this problem, we further show that if d is a part of the input and \mathcal{P} contains all chordal graphs, then it cannot be solved in time f(w)n^{o(w)} for some function f, unless the ETH fails.
Edit intent: coherence
Original: The problem of ranking a set of items is fundamental in today's data-driven world.
Revised: The problem of ranking a set of items is a fundamental algorithmic task in today's data-driven world.
Edit intent: clarity
Original:  However, recent events have pointed to the fact that algorithmic bias in rankings, which results in decreased fairness or diversity in the type of content presented, can promote stereotypes and propagate injustices .
Revised:  However, recent events and studies show that bias exists in the output of such applications. This results in unfairness or decreased diversity in the type of content presented, can promote stereotypes and propagate injustices .
Edit intent: clarity
Original:  However, recent events have pointed to the fact that algorithmic bias in rankings, which results in decreased fairness or diversity in the type of content presented, can promote stereotypes and propagate injustices .
Revised:  However, recent events have pointed to the fact that algorithmic bias in rankings, which results in decreased fairness or diversity in the presentation of the content and can exacerbate stereotypes and manipulate perceptions .
Edit intent: clarity
Original:  Given a collection of items along with 1) the value of placing an item at a particular position, 2) the collection of possibly non-disjoint attributes (e.g., gender , race ) of each item and 3) a collection of fairness constraints that upper bound the number of items with each attribute that are allowed to appear in the top positions of the ranking , the goal is to output a ranking that maximizes value while respecting the constraints.
Revised:  Given a collection of items along with 1) the value of placing an item at a particular position, 2) the collection of possibly non-disjoint attributes (e.g., gender , race ) of each item and 3) a collection of fairness constraints that bound the number of items with each attribute that are allowed to appear in the top positions of the ranking , the goal is to output a ranking that maximizes value while respecting the constraints.
Edit intent: clarity
Original:  Given a collection of items along with 1) the value of placing an item at a particular position, 2) the collection of possibly non-disjoint attributes (e.g., gender , race ) of each item and 3) a collection of fairness constraints that upper bound the number of items with each attribute that are allowed to appear in the top positions of the ranking , the goal is to output a ranking that maximizes value while respecting the constraints.
Revised:  Given a collection of items along with 1) the value of placing an item at a particular position, 2) the collection of possibly non-disjoint attributes (e.g., gender , race ) of each item and 3) a collection of fairness constraints that upper bound the number of items with each attribute that are allowed to appear in the top positions of the ranking . The goal is to output a ranking that maximizes value while respecting the constraints.
Edit intent: fluency
Original:  Our main technical contributions are exact and approximation algorithms and hardness results that , together, come close to settling the approximability of the constrained ranking maximization problem .
Revised:  Our main technical contributions are exact and approximation algorithms and hardness results that , together, come close to settling the complexity of this constrained ranking maximization problem .
Edit intent: clarity
Original:  However, recent events and studiesshow thatbias exists in the output of such applications.
Revised:  Recent studies, however, have shown that, left unchecked, the output of such applications.
Edit intent: clarity
Original:  In our model, we are given a collection of items along with 1) the value of placing an item at a particular position  , 2) the collection of possibly non-disjoint attributes (e.g., genderand ethnicity or genre and price point depending on the context ) of each item and 3) a collection of fairness constraints that bound the number of items with each attribute that are allowed to appear in the top  positions of the ranking .
Revised:  In our model, we are given a collection of items along with 1) the value of placing an item in a particular position , 2) the collection of possibly non-disjoint attributes (e.g., genderand ethnicity or genre and price point depending on the context ) of each item and 3) a collection of fairness constraints that bound the number of items with each attribute that are allowed to appear in the top positions of the ranking .
Edit intent: fluency
Original:  In our model, we are given a collection of items along with 1) the value of placing an item at a particular position  , 2) the collection of possibly non-disjoint attributes (e.g., genderand ethnicity or genre and price point depending on the context ) of each item and 3) a collection of fairness constraints that bound the number of items with each attribute that are allowed to appear in the top  positions of the ranking .
Revised:  In our model, we are given a collection of items along with 1) the value of placing an item at a particular position in the ranking , 2) the collection of possibly non-disjoint attributes (e.g., genderand ethnicity or genre and price point depending on the context ) of each item and 3) a collection of fairness constraints that bound the number of items with each attribute that are allowed to appear in the top positions of the ranking .
Edit intent: clarity
Original:  In our model, we are given a collection of items along with 1) the value of placing an item at a particular position  , 2) the collection of possibly non-disjoint attributes (e.g., genderand ethnicity or genre and price point depending on the context ) of each item and 3) a collection of fairness constraints that bound the number of items with each attribute that are allowed to appear in the top  positions of the ranking .
Revised:  In our model, we are given a collection of items along with 1) the value of placing an item at a particular position , 2) the collection of sensitive attributes (such as gender, race, political opinion ) of each item and 3) a collection of fairness constraints that bound the number of items with each attribute that are allowed to appear in the top positions of the ranking .
Edit intent: clarity
Original:  We present algorithms along with complementary hardness results which , together, come close to settling the complexity of this constrained ranking maximization problem  .
Revised:  We present algorithms along with complementary hardness results that , together, come close to settling the complexity of this constrained ranking maximization problem .
Edit intent: fluency
Original:  We present algorithms along with complementary hardness results which , together, come close to settling the complexity of this constrained ranking maximization problem  .
Revised:  We present algorithms along with complementary hardness results which , together, come close to settling the approximability of this constrained ranking maximization problem .
Edit intent: clarity
Original:  The art gallery problem has stimulated a myriad of research in geometry and in algorithms.
Revised:  The art gallery problem has stimulated extensive research in geometry and in algorithms.
Edit intent: clarity
Original: We investigate several computational problems related to  stochastic convex hull (SCH).
Revised: We investigate several computational problems related to the stochastic convex hull (SCH).
Edit intent: fluency