Calculating the probability of an individual solution being selected under lexicase selection is an important problem in attempts to develop a deeper theoretical understanding of lexicase selection, a state-of-the art parent selection algorithm in evolutionary computation. Discovering a fast solution to this problem would also have implications for efforts to develop practical improvements to lexicase selection. Here, I prove that this problem, which I name lex-prob, is NP-Hard. I achieve this proof by reducing SAT, a well-known NP-Complete problem, to lex-prob in polynomial time. This reduction involves an intermediate step in which a popular variant of lexicase selection, epsilon-lexicase selection, is reduced to standard lexicase selection. This proof has important practical implications for anyone needing a fast way of calculating the probabilities of individual solutions being selected under lexicase selection. Doing so in polynomial time would be incredibly challenging, if not all-together impossible. Thus, finding approximation algorithms or practical optimizations for speeding up the brute-force solution is likely more worthwhile. This result also has deeper theoretical implications about the relationship between epsilon-lexicase selection and lexicase selection and the relationship between lex-prob and other NP-Hard problems.

翻译：在尝试深入理解词法选择（进化计算中一种先进的父代选择算法）的理论基础时，计算单个解在词法选择下被选中的概率是一个重要问题。找到该问题的快速解法，还将对词法选择的实际改进工作产生影响。这里，我证明该问题（我称之为lex-prob）是NP难的。我通过将著名的NP完全问题SAT在多项式时间内归约为lex-prob来完成这一证明。该归约过程包含一个中间步骤，即将词法选择的流行变体ε-词法选择归约为标准词法选择。这一证明对任何需要快速计算词法选择下单个解被选中概率的人具有重要的实际意义：在多项式时间内求解该问题将极其困难，甚至完全不可能。因此，寻找近似算法或加速暴力求解法的实用优化方法可能更有价值。该结果还对ε-词法选择与标准词法选择之间的关系，以及lex-prob与其他NP难问题之间的关系具有更深层的理论意义。