There is a close connection between sub-exponential time solvability (SUBEPT) and fixed parameter tractability (FPT). The link between them is provided in the following paper.

An isomorphism between subexponential and parameterized complexity theory, Yijia Chen and Martin Grohe, 2006.

In brief, they introduced a notion called miniaturization mapping, which maps a parameterized problem $(P,\nu)$ into another parameterized problem $(Q,\kappa)$. By viewing a normal problem as a problem parameterized by the input size, we have the following connection. (See theorem 16 in the paper)

Theorem. $(P,\nu)$ is in SUBEPT iff $(Q,\kappa)$ is in FPT.

Be careful of the definitions here. Normally we view $k$-clique problem as parameterized in $k$, so there is no sub-exponential time algorithm for it assuming Exponential time hypothesis. But here we let the problem be parameterized by the input size $O(m+n)$, thus the problem can be solved in $2^{O(\sqrt{m}\log m)}$, which is a sub-exponential time algorithm. And the theorem tells us that the $k$-clique problem is fixed parameter tractable under the some twist of the parameter $k$, which is reasonable.

In general, problems in SUBEPT under SERF-reductions (sub-exponential reduction families) can be transformed into problems in FPT under FPT-reductions. (Theorem 20 in the paper) Furthermore, the connections are even stronger since they provided an isomorphism theorem between a whole hierarchy of problems in exponential time complexity theory and parameterized complexity theory. (Theorem 25 and 47) Though the isomorphism is not complete (there are some missing links between them), it is still nice to have a clear picture about these problems, and we can study sub-exponential time algorithms via parameterized complexity.

See the survey by Jacobo Torán, Jörg Flum and Martin Grohe for more information.

There is a close connection between sub-exponential time solvability (SUBEPT) and fixed parameter tractability (FPT). The link between them is provided in the following paper.

An isomorphism between subexponential and parameterized complexity theory, Yijia Chen and Martin Grohe, 2006.

In brief, they introduced a notion called miniaturization mapping, which maps a parameterized problem $(P,\nu)$ into another parameterized problem $(Q,\kappa)$. By viewing a normal problem as a problem parameterized by the input size, we have the following connection. (See theorem 16 in the paper)

Theorem. $(P,\nu)$ is in SUBEPT iff $(Q,\kappa)$ is in FPT.

Be careful of the definitions here. Normally we view $k$-clique problem as parameterized in $k$, so there is no sub-exponential time algorithm for it assuming Exponential time hypothesis. But here we let the problem be parameterized by the input size $O(m+n)$, thus the problem can be solved in $2^{O(\sqrt{m}\log m)}$, which is a sub-exponential time algorithm. And the theorem tells us that the $k$-clique problem is fixed parameter tractable under the some twist of the parameter $k$, which is reasonable.

In general, problems in SUBEPT under SERF-reductions (sub-exponential reduction families) can be transformed into problems in FPT under FPT-reductions. (Theorem 20 in the paper) Furthermore, the connections are even stronger since they provided an isomorphism theorem between a whole hierarchy of problems in exponential time complexity theory and parameterized complexity theory. (Theorem 25 and 47) Though the isomorphism is not complete (there are some missing links between them), it is still nice to have a clear picture about these problems, and we can study sub-exponential time algorithms via parameterized complexity.

See the survey by Jörg Flum and Martin Grohe, together with Jacobo Torán, the editor of the complexity column, for more information.

There is a close connection between sub-exponential time solvability (SUBEPT) and fixed parameter tractability (FPT). The link between them is provided in this paper by Yijia Chen and Martin Grohe.

In brief, they introduced a notion called miniaturization mapping, which maps a parameterized problem $(P,\nu)$ into another parameterized problem $(Q,\kappa)$. By viewing a normal problem as a problem parameterized by the input size, we have the following connection. (See theorem 16 in the paper)

Theorem. $(P,\nu)$ is in $\mathsf{SUBEPT}$ iff $(Q,\kappa)$ is in $\mathsf{FPT}$.

Be careful of the definitions here. Normally we view $k$-clique problem as parameterized in $k$, so there is no sub-exponential time algorithm for it assuming Exponential time hypothesis. But here we let the problem be parameterized by the input size $O(m+n)$, thus the problem can be solved in $2^{O(\sqrt{m}\log m)}$, which is a sub-exponential time algorithm. And the theorem tells us that the $k$-clique problem is fixed parameter tractable under the normal parameter $k$, which is reasonable.

In general, problems in $\mathsf{SUBEPT}$ under SERF-reductions (sub-exponential reduction families) can be transformed into problems in $\mathsf{FPT}$ under FPT-reductions. (Theorem 20 in the paper) Furthermore, the connections are even stronger since they provided an isomorphism theorem between a whole hierarchy of problems in exponential time complexity theory and parameterized complexity theory. (Theorem 25 and 47) Though the isomorphism is not complete (there are some missing links between them), it is still nice to have a clear picture about these problems, and we can study sub-exponential time algorithms via parameterized complexity.

There is a close connection between sub-exponential time solvability (SUBEPT) and fixed parameter tractability (FPT). The link between them is provided in the following paper.

An isomorphism between subexponential and parameterized complexity theory, Yijia Chen and Martin Grohe, 2006.

In brief, they introduced a notion called miniaturization mapping, which maps a parameterized problem $(P,\nu)$ into another parameterized problem $(Q,\kappa)$. By viewing a normal problem as a problem parameterized by the input size, we have the following connection. (See theorem 16 in the paper)

Theorem. $(P,\nu)$ is in SUBEPT iff $(Q,\kappa)$ is in FPT.

Be careful of the definitions here. Normally we view $k$-clique problem as parameterized in $k$, so there is no sub-exponential time algorithm for it assuming Exponential time hypothesis. But here we let the problem be parameterized by the input size $O(m+n)$, thus the problem can be solved in $2^{O(\sqrt{m}\log m)}$, which is a sub-exponential time algorithm. And the theorem tells us that the $k$-clique problem is fixed parameter tractable under the some twist of the parameter $k$, which is reasonable.

In general, problems in SUBEPT under SERF-reductions (sub-exponential reduction families) can be transformed into problems in FPT under FPT-reductions. (Theorem 20 in the paper) Furthermore, the connections are even stronger since they provided an isomorphism theorem between a whole hierarchy of problems in exponential time complexity theory and parameterized complexity theory. (Theorem 25 and 47) Though the isomorphism is not complete (there are some missing links between them), it is still nice to have a clear picture about these problems, and we can study sub-exponential time algorithms via parameterized complexity.

See the survey by Jacobo Torán, Jörg Flum and Martin Grohe for more information.