A pattern $\alpha$ is a string of variables and terminal letters. We say that $\alpha$ matches a word $w$, consisting only of terminal letters, if $w$ can be obtained by replacing the variables of $\alpha$ by terminal words. The matching problem, i.e., deciding whether a given pattern matches a given word, was heavily investigated: it is NP-complete in general, but can be solved efficiently for classes of patterns with restricted structure. In this paper, we approach this problem in a generalized setting, by considering approximate pattern matching under Hamming distance. More precisely, we are interested in what is the minimum Hamming distance between $w$ and any word $u$ obtained by replacing the variables of $\alpha$ by terminal words. Firstly, we address the class of regular patterns (in which no variable occurs twice) and propose efficient algorithms for this problem, as well as matching conditional lower bounds. We show that the problem can still be solved efficiently if we allow repeated variables, but restrict the way the different variables can be interleaved according to a locality parameter. However, as soon as we allow a variable to occur more than once and its occurrences can be interleaved arbitrarily with those of other variables, even if none of them occurs more than once, the problem becomes intractable.

中文翻译：

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汉明距离下的变量匹配模式

模式 $\alpha$ 是一串变量和终端字母。我们说 $\alpha$ 匹配一个单词 $w$，它只由终结字母组成，如果 $w$ 可以通过用终结词替换 $\alpha$ 的变量来获得。匹配问题，即决定给定模式是否与给定单词匹配，被大量研究：它通常是 NP 完全的，但可以有效地解决具有受限结构的模式类别。在本文中，我们通过考虑汉明距离下的近似模式匹配，在广义设置中解决这个问题。更准确地说，我们感兴趣的是 $w$ 与通过将 $\alpha$ 的变量替换为终结词而获得的任何单词 $u$ 之间的最小汉明距离是多少。首先，我们解决了常规模式类（其中没有变量出现两次）并为这个问题提出了有效的算法，以及匹配条件下界。我们表明，如果我们允许重复变量，问题仍然可以有效解决，但根据局部参数限制不同变量的交错方式。然而，一旦我们允许一个变量多次出现并且它的出现可以与其他变量的那些任意交错，即使它们都没有出现一次以上，问题就变得棘手了。