  In the Shortest Common Superstring (SCS) problem, one is given a collection
of strings, and needs to find a shortest string containing each of them as a
substring. SCS admits $2\frac{11}{23}$-approximation in polynomial time (Mucha,
SODA'13). While this algorithm and its analysis are technically involved, the
30 years old Greedy Conjecture claims that the trivial and efficient Greedy
Algorithm gives a 2-approximation for SCS.
  We develop a graph-theoretic framework for studying approximation algorithms
for SCS. The framework is reminiscent of the classical 2-approximation for
Traveling Salesman: take two copies of an optimal solution, apply a trivial
edge-collapsing procedure, and get an approximate solution. In this framework,
we observe two surprising properties of SCS solutions, and we conjecture that
they hold for all input instances. The first conjecture, that we call
Collapsing Superstring conjecture, claims that there is an elementary way to
transform any solution repeated twice into the same graph $G$. This conjecture
would give an elementary 2-approximate algorithm for SCS. The second conjecture
claims that not only the resulting graph $G$ is the same for all solutions, but
that $G$ can be computed by an elementary greedy procedure called Greedy
Hierarchical Algorithm.
  While the second conjecture clearly implies the first one, perhaps
surprisingly we prove their equivalence. We support these equivalent
conjectures by giving a proof for the special case where all input strings have
length at most 3. We prove that the standard Greedy Conjecture implies Greedy
Hierarchical Conjecture, while the latter is sufficient for an efficient greedy
2-approximate approximation of SCS. Except for its (conjectured) good
approximation ratio, the Greedy Hierarchical Algorithm provably finds a
3.5-approximation.
