  We pose the fine-grained hardness hypothesis that the textbook algorithm for
the NFA Acceptance problem is optimal up to subpolynomial factors, even for
dense NFAs and fixed alphabets.
  We show that this barrier appears in many variations throughout the
algorithmic literature by introducing a framework of Colored Walk problems.
These yield fine-grained equivalent formulations of the NFA Acceptance problem
as problems concerning detection of an $s$-$t$-walk with a prescribed color
sequence in a given edge- or node-colored graph. For NFA Acceptance on sparse
NFAs (or equivalently, Colored Walk in sparse graphs), a tight lower bound
under the Strong Exponential Time Hypothesis has been rediscovered several
times in recent years. We show that our hardness hypothesis, which concerns
dense NFAs, has several interesting implications:
  - It gives a tight lower bound for Context-Free Language Reachability. This
proves conditional optimality for the class of 2NPDA-complete problems,
explaining the cubic bottleneck of interprocedural program analysis.
  - It gives a tight $(n+nm^{1/3})^{1-o(1)}$ lower bound for the Word Break
problem on strings of length $n$ and dictionaries of total size $m$.
  - It implies the popular OMv hypothesis. Since the NFA acceptance problem is
a static (i.e., non-dynamic) problem, this provides a static reason for the
hardness of many dynamic problems.
  Thus, a proof of the NFA Acceptance hypothesis would resolve several
interesting barriers. Conversely, a refutation of the NFA Acceptance hypothesis
may lead the way to attacking the current barriers observed for Context-Free
Language Reachability, the Word Break problem and the growing list of dynamic
problems proven hard under the OMv hypothesis.
