wanyu/IteraTeR_v2 huggingface dataset:
Original:  On the one hand we prove that cliques minus c edges can be counted in time f(d,k) \cdot (n^{\lceil 1/2 + \sqrt{c/2 \rceil}), which generalizes the well-known O(nd^{k-1}) upper bound for counting cliques, and shows that a lower degeneracy makes it easier to find and count dense patterns.
Revised:  On the one hand we prove that cliques minus c edges can be counted in time f(d,k) \cdot \rceil}), which generalizes the well-known O(nd^{k-1}) upper bound for counting cliques, and shows that a lower degeneracy makes it easier to find and count dense patterns.
Edit intent: coherence
Original:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic }  O(n^{ ( \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}]  d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Revised:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic } O(n^{ ( \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}] d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Edit intent: clarity
Original:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic }  O(n^{ ( \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}]  d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Revised:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic } O(n^{ \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}] d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Edit intent: fluency
Original:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic }  O(n^{ ( \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}]  d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Revised:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic } O(n^{ ( \omega k/3 +2}) bound for et\v{r}il et al., 1985}] d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Edit intent: fluency
Original:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic }  O(n^{ ( \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}]  d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Revised:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic } O(n^{ ( \omega k/3 ) +2}) algorithm of et\v{r}il et al., 1985}] d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Edit intent: clarity
Original:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic }  O(n^{ ( \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}]  d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Revised:  Then we prove that any pattern on k nodes can be counted in time f(k) \cdot}%DIFDELCMD < O%%% (n^{\lfloor k/4 \rfloor+2 \, d^{k-\lfloor k/4 \rfloor-2}), beating the classic } O(n^{ ( \omega k/3 ) +2}) bound for et\v{r}il et al., 1985}] as long as d < n^{0.721} using the current matrix multiplication exponent \omega \approx 2.37.
Edit intent: clarity
Original:  On the other hand we show that algorithms with running time f(d, k) \cdot n^{o(k/\ln{k ]
Revised:  On the other hand we show that algorithms with running time f(d, k) \cdot n^{o(k/\ln{k ].
Edit intent: fluency
Original:  . Combining a simple dynamic programming with inclusion-exclusion arguments, for any given H we can count the homomorphisms from H to G in time f(d,k) \cdot O(n^{s(H)}).
Revised:  on every undirected pattern H, and leads to a dynamic programming to count the copies of H in G in time f(d,k) \cdot O(n^{s(H)}).
Edit intent: clarity
Original:  For instance {2} + \frac{\epsilon{2}} \, \rceil}), which extends the classic f(d,k) \cdot O(n) bound for cliques by Chiba and Nishizeki }[  , we can count  [ O{4}\rfloor+2}) for complete multipartite graphs plus \epsilon edges.
Revised:  For instance {2} + \frac{\epsilon{2}} \, \rceil}), which extends the classic f(d,k) \cdot O(n) bound for cliques by Chiba and Nishizeki }[ . For complete multipartite graphs , we can count [ O{4}\rfloor+2}) for complete multipartite graphs plus \epsilon edges.
Edit intent: coherence
Original:  Third, we can count } the induced or non-induced copies of any pattern faster than  the decades-old state-of-the-art O(n^{ \omega k/3 +2}) algorithm of Ne\v{s}et\v{r}il et al., 1985  ] as long as d < n^{0.721} or m < n^{1.442 Impagliazzo et al., 1998 .
Revised:  Third, we can count } the induced or non-induced copies of any pattern faster than the decades-old state-of-the-art O(n^{ \omega k/3 +2}) algorithm of Ne\v{s}et\v{r}il et al., 1985 ] for d < n^{0.721} or m < n^{1.442 Impagliazzo et al., 1998 .
Edit intent: clarity
Original:  This decomposition induces a width measure s(H) on every undirected pattern H, and leads to a dynamic programming to count the copies of H in G in  ] time f( d, k) \cdot (n ^{ s(H)\lfloor{3}}\rfloor  }) .
Revised:  This decomposition induces a width measure s(H) on every undirected pattern H, and leads to a dynamic programming to count the copies of H in G in ], with running time f( d, k) \cdot (n ^{ s(H)\lfloor{3}}\rfloor }) .
Edit intent: clarity
Original:  This decomposition induces a width measure s(H) on every undirected pattern H, and leads to a dynamic programming to count the copies of H in G in  ] time f( d, k) \cdot (n ^{ s(H)\lfloor{3}}\rfloor  }) .
Revised:  This decomposition induces a width measure s(H) on every undirected pattern H, and leads to a dynamic programming to count the copies of H in G in ] time f( k) \cdot (n ^{ s(H)\lfloor{3}}\rfloor }) .
Edit intent: clarity
Original:  This decomposition induces a width measure s(H) on every undirected pattern H, and leads to a dynamic programming to count the copies of H in G in  ] time f( d, k) \cdot (n ^{ s(H)\lfloor{3}}\rfloor  }) .
Revised:  This decomposition induces a width measure s(H) on every undirected pattern H, and leads to a dynamic programming to count the copies of H in G in ] time f( d, k) \cdot O(n ^{ s(H)\lfloor{3}}\rfloor }) .
Edit intent: clarity
Original:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le  the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot  n ^{O(\alpha(H)) where \alpha(H)is the independence number of H, and }  no algorithm can do so in f(d,k) \cdot n^{o( \alpha (H)/\ln \alpha (H))} unless ETH fails.
Revised:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot ( n ^{O(\alpha(H)) where \alpha(H)is the independence number of H, and } no algorithm can do so in f(d,k) \cdot n^{o( \alpha (H)/\ln \alpha (H))} unless ETH fails.
Edit intent: fluency
Original:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le  the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot  n ^{O(\alpha(H)) where \alpha(H)is the independence number of H, and }  no algorithm can do so in f(d,k) \cdot n^{o( \alpha (H)/\ln \alpha (H))} unless ETH fails.
Revised:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot n where \alpha(H)is the independence number of H, and } no algorithm can do so in f(d,k) \cdot n^{o( \alpha (H)/\ln \alpha (H))} unless ETH fails.
Edit intent: clarity
Original:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le  the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot  n ^{O(\alpha(H)) where \alpha(H)is the independence number of H, and }  no algorithm can do so in f(d,k) \cdot n^{o( \alpha (H)/\ln \alpha (H))} unless ETH fails.
Revised:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot n ^{O(\alpha(H)) where \alpha(H)is the independence number of H, and } no algorithm can solve the problem in time f(d,k) \cdot n^{o( \alpha (H)/\ln \alpha (H))} unless ETH fails.
Edit intent: clarity
Original:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le  the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot  n ^{O(\alpha(H)) where \alpha(H)is the independence number of H, and }  no algorithm can do so in f(d,k) \cdot n^{o( \alpha (H)/\ln \alpha (H))} unless ETH fails.
Revised:  This is tight in the following sense: if any algorithm solves the problem in time f(d ,k) \cdot n^{o(s(H)/\ln s(H))\le the induced or non-induced copies of any pattern H on k nodes (even disconnected ones) can be counted in time f(d,k) \cdot n ^{O(\alpha(H)) where \alpha(H)is the independence number of H, and } no algorithm can do so in f(d,k) \cdot n^{o( \alpha (H)/\ln \tau (H))} unless ETH fails.
Edit intent: clarity
Original:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% 9 . For complete multipartite graphs, we can count non-induced copies in f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein 13%DIFDELCMD < ]%%% ), which becomes f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Revised:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% . For complete multipartite graphs, we can count non-induced copies in f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein 13%DIFDELCMD < ]%%% ), which becomes f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Edit intent: coherence
Original:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% 9 . For complete multipartite graphs, we can count non-induced copies in f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein 13%DIFDELCMD < ]%%% ), which becomes f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Revised:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% 9 %DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein 13%DIFDELCMD < ]%%% ), which becomes f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Edit intent: coherence
Original:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% 9 . For complete multipartite graphs, we can count non-induced copies in f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein 13%DIFDELCMD < ]%%% ), which becomes f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Revised:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% 9 . For complete multipartite graphs, we can count non-induced copies in f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein %DIFDELCMD < ]%%% ), which becomes f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Edit intent: fluency
Original:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% 9 . For complete multipartite graphs, we can count non-induced copies in f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein 13%DIFDELCMD < ]%%% ), which becomes f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Revised:  \cdot O(n) bound for cliques by Chiba and Nishizeki }%DIFDELCMD < [%%% 9 . For complete multipartite graphs, we can count non-induced copies in f(d,k) \cdot%DIFDELCMD < \tilde{O}%%% (n) (essentially matching an f(d,k) \cdot O(n) bound by Eppstein 13%DIFDELCMD < ]%%% %DIFDELCMD < \tilde{O}%%% (n^{\lfloor\frac{\epsilon{4}\rfloor+2})for complete multipartite graphs plus \epsilon edges.
Edit intent: coherence
Original:  Third, } \operatorname{poly} we can count the induced or non-induced copies of any H in f(d, k) \cdot\tilde{O}(n^{ \lfloor \frac{k}{4} \rfloor + 2}), beating the decades-old state-of-the-art O(n^{\frac{\omega k Ne\v{s}et\v{r} il and Poljak 28%DIFDELCMD < ] %%% for d < n^{0.721 (\omega is the matrix multiplication exponent)}  .
Revised:  Third, } \operatorname{poly} we can count the induced copies of any H in f(d, k) \cdot\tilde{O}(n^{ \lfloor \frac{k}{4} \rfloor + 2}), beating the decades-old state-of-the-art O(n^{\frac{\omega k Ne\v{s}et\v{r} il and Poljak 28%DIFDELCMD < ] %%% for d < n^{0.721 (\omega is the matrix multiplication exponent)} .
Edit intent: clarity
Original:  Third, } \operatorname{poly} we can count the induced or non-induced copies of any H in f(d, k) \cdot\tilde{O}(n^{ \lfloor \frac{k}{4} \rfloor + 2}), beating the decades-old state-of-the-art O(n^{\frac{\omega k Ne\v{s}et\v{r} il and Poljak 28%DIFDELCMD < ] %%% for d < n^{0.721 (\omega is the matrix multiplication exponent)}  .
Revised:  Third, } \operatorname{poly} we can count the induced or non-induced copies of any H in time f( k) \cdot\tilde{O}(n^{ \lfloor \frac{k}{4} \rfloor + 2}), beating the decades-old state-of-the-art O(n^{\frac{\omega k Ne\v{s}et\v{r} il and Poljak 28%DIFDELCMD < ] %%% for d < n^{0.721 (\omega is the matrix multiplication exponent)} .
Edit intent: clarity
Original:  Third, } \operatorname{poly} we can count the induced or non-induced copies of any H in f(d, k) \cdot\tilde{O}(n^{ \lfloor \frac{k}{4} \rfloor + 2}), beating the decades-old state-of-the-art O(n^{\frac{\omega k Ne\v{s}et\v{r} il and Poljak 28%DIFDELCMD < ] %%% for d < n^{0.721 (\omega is the matrix multiplication exponent)}  .
Revised:  Third, } \operatorname{poly} we can count the induced or non-induced copies of any H in f(d, k) \cdot\tilde{O}(n^{ \lfloor \frac{k}{4} \rfloor + 2}), beating the decades-old state-of-the-art O(n^{\frac{\omega k Ne\v{s}et\v{r} il and Poljak %DIFDELCMD < ] %%% for d < n^{0.721 (\omega is the matrix multiplication exponent)} .
Edit intent: fluency
Original: A fundamental graph problem asks to compute the number of induced copies of a k-node pattern graph H in an n-node graph G. The fastest algorithm to date is still the 35-years-old algorithm by Nesetril and Poljak 31%DIFDELCMD < ]%%% , with running time f(k) \cdot O(n^{\omega\lfloork{3\rfloor + 2}) where \omega }%DIFDELCMD < \le %%% 2.373 is the matrix multiplication exponent.
Revised: A fundamental graph problem asks to compute the number of 31%DIFDELCMD < ]%%% , with running time f(k) \cdot O(n^{\omega\lfloork{3\rfloor + 2}) where \omega }%DIFDELCMD < \le %%% 2.373 is the matrix multiplication exponent.
Edit intent: clarity
Original: A fundamental graph problem asks to compute the number of induced copies of a k-node pattern graph H in an n-node graph G. The fastest algorithm to date is still the 35-years-old algorithm by Nesetril and Poljak 31%DIFDELCMD < ]%%% , with running time f(k) \cdot O(n^{\omega\lfloork{3\rfloor + 2}) where \omega }%DIFDELCMD < \le %%% 2.373 is the matrix multiplication exponent.
Revised: A fundamental graph problem asks to compute the number of induced copies of a k-node pattern graph H in an n-node graph G. The fastest algorithm to date is still the 35-years-old algorithm by Nesetril and Poljak %DIFDELCMD < ]%%% , with running time f(k) \cdot O(n^{\omega\lfloork{3\rfloor + 2}) where \omega }%DIFDELCMD < \le %%% 2.373 is the matrix multiplication exponent.
Edit intent: coherence
Original:  More precisely, after introducing a novel notion of graph width, theDAG-treewidth , we prove what follows.
Revised:  More precisely, after introducing a novel DAG-treewidth , we prove what follows.
Edit intent: clarity
Original:  If H has DAG-treewidth \tau(H) and G has degeneracy d, then the induced copies of H in G can be counted in time f(d,k) \cdot (n^{\tau(H));
Revised:  If H has DAG-treewidth \tau(H) and G has degeneracy d, then the induced copies of H in G (n^{\tau(H));
Edit intent: clarity
Original:  If H has DAG-treewidth \tau(H) and G has degeneracy d, then the induced copies of H in G can be counted in time f(d,k) \cdot (n^{\tau(H));
Revised:  If H has DAG-treewidth \tau(H) and G has degeneracy d, then the induced copies of H in G can be counted in time f(d,k) \cdot );
Edit intent: clarity
Original:  Developing bounds on \tau(H), then, we obtain natural generalisations of classic results and faster algorithms for sparse graphs}  . For example, when d=O( \log(n))
Revised:  Developing bounds on \tau(H), then, we obtain natural generalisations of classic results and faster algorithms for sparse graphs} . For example, \log(n))
Edit intent: coherence
Original:  we can count the induced copies of any  H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in }  n  \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{  .
Revised:  we can count the induced copies of any H in time (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in } n \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{ .
Edit intent: clarity
Original:  we can count the induced copies of any  H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in }  n  \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{  .
Revised:  we can count the induced copies of any H in time f(k) \cdot %DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in } n \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{ .
Edit intent: fluency
Original:  we can count the induced copies of any  H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in }  n  \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{  .
Revised:  we can count the induced copies of any H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% {4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in } n \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{ .
Edit intent: fluency
Original:  we can count the induced copies of any  H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in }  n  \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{  .
Revised:  we can count the induced copies of any H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in } n \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{ .
Edit intent: fluency
Original:  we can count the induced copies of any  H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% ) , beating the Ne\v{set\v{r}il-Poljak algorithm by essentially a cubic factor in }  n  \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{  .
Revised:  we can count the induced copies of any H in time f(k) \cdot (n\SUPERSCRIPT%DIFDELCMD < {\lfloor %%% \frac{k{4} }%DIFDELCMD < \rfloor %%% + 2 %DIFDELCMD < \MBLOCKRIGHTBRACE%%% et\v{r}il-Poljak algorithm by essentially a cubic factor in } n \log n) if G has bounded degeneracy, and in time 2^{O(k^2)} O(n\SUPERSCRIPT}{ .
Edit intent: coherence
Original:  We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model and we report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on the macroscopic concentrations.
Revised:  We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model . We report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on the macroscopic concentrations.
Edit intent: coherence
Original:  We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model and we report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on the macroscopic concentrations.
Revised:  We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model and we report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on macroscopic concentrations.
Edit intent: fluency
Original:  As a second main contribution, we study the tractability frontier of solving RDTPs by considering Hyper Temporal Networks (\HTNs), a strict generalization of \STNs grounded on hypergraphs: on one side, we prove that solving temporal problems having  either only multi-tail or ~ only multi-head hyperarc constraints and Type-2 constraints, it lies in both NP and co-NP and it admits deterministic pseudo-polynomial time algorithms;
Revised:  As a second main contribution, we study the tractability frontier of solving RDTPs by considering Hyper Temporal Networks (\HTNs), a strict generalization of \STNs grounded on hypergraphs: on one side, we prove that solving temporal problems having only Type-2 constraints and either only multi-tail or ~ only multi-head hyperarc constraints and Type-2 constraints, it lies in both NP and co-NP and it admits deterministic pseudo-polynomial time algorithms;
Edit intent: fluency