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.

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