  In the Set Cover problem, the input is a ground set of $n$ elements and a
collection of $m$ sets, and the goal is to find the smallest sub-collection of
sets whose union is the entire ground set. The fastest algorithm known runs in
time $O(mn2^n)$ [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo)
[Cygan et al., TALG 2016] asserts that for every fixed $\varepsilon>0$, no
algorithm can solve Set Cover in time $2^{(1-\varepsilon)n}poly(m)$, even if
set sizes are bounded by $\Delta=\Delta(\varepsilon)$. We show strong
connections between this problem and kTree, a special case of Subgraph
Isomorphism where the input is an $n$-node graph $G$ and a $k$-node tree $T$,
and the goal is to determine whether $G$ has a subgraph isomorphic to $T$.
  First, we propose a weaker conjecture Log-SeCoCo, that allows input sets of
size $\Delta=O(1/\varepsilon \cdot\log n)$, and show that an algorithm breaking
Log-SeCoCo would imply a faster algorithm than the currently known $2^n
poly(n)$-time algorithm [Koutis and Williams, TALG 2016] for Directed nTree,
which is kTree with $k=n$ and arbitrary directions to the edges of $G$ and $T$.
This would also improve the running time for Directed Hamiltonicity, for which
no algorithm significantly faster than $2^n poly(n)$ is known despite extensive
research.
  Second, we prove that if Set Cover cannot be solved significantly faster than
$2^npoly(m)$ (an assumption even weaker than Log-SeCoCo), then kTree cannot be
computed significantly faster than $2^kpoly(n)$, the running time of the Koutis
and Williams' algorithm. Applying the same techniques to the p-Partial Cover
problem, a parameterized version of Set Cover that requires covering at least
$p$ elements, we obtain a new algorithm with running time $(2+\varepsilon)^p
(m+n)^{O(1/\varepsilon)}$ for arbitrary $\varepsilon>0$, which improves
previous work and is nearly optimal assuming say Log-SeCoCo.
