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.

翻译：我们提出以下精细粒度困难性假设：即使对于稠密NFA和固定字母表，NFA接受问题的教科书算法在亚多项式因子意义下也是最优的。通过引入着色游走问题框架，我们证明该障碍以多种变体形式出现在算法文献中。这些变体将NFA接受问题等价转化为在边着色或顶点着色图中检测具有特定颜色序列的$s$-$t$游走问题。对于稀疏NFA（等价于稀疏图中的着色游走问题），在强指数时间假说下的紧致下界已在近年被多次重新发现。我们证明，关于稠密NFA的困难性假设具有以下重要推论：- 为上下文无关语言可达性问题提供紧致下界，从而证明2NPDA完全问题类的最优性条件，解释过程间程序分析的立方瓶颈。- 对长度为$n$的字符串和总规模为$m$的词典构成的单词分割问题，给出$(n+nm^{1/3})^{1-o(1)}$的紧致下界。- 蕴含流行的OMv假设。由于NFA接受问题是静态（即非动态）问题，这为许多动态问题的困难性提供了静态原因。因此，证明NFA接受假设将解决若干重要障碍。反之，证伪NFA接受假设可能为突破上下文无关语言可达性、单词分割问题及日益增长的OMv假设下被证明困难的动态问题清单提供新途径。