Longest Run Subsequence is a problem introduced recently in the context of the scaffolding phase of genome assembly (Schrinner et al.,WABI 2020). The problem asks for a maximum length subsequence of a given string that contains at most one run for each symbol (a run is a maximum substring of consecutive identical symbols). The problem has been shown to be NP-hard and to be fixed-parameter tractable when the parameter is the size of the alphabet on which the input string is defined. In this paper we further investigate the complexity of the problem and we show that it is fixed-parameter tractable when it is parameterized by the number of runs in a solution, a smaller parameter. Moreover, we investigate the kernelization complexity of Longest Run Subsequence and we prove that it does not admit a polynomial kernel when parameterized by the size of the alphabet or by the number of runs. Finally, we consider the restriction of Longest Run Subsequence when each symbol has at most two occurrences in the input string and we show that it is APX-hard.

中文翻译：

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最长运行子序列问题：进一步复杂的结果

最长运行子序列是最近在基因组组装的支架阶段背景下引入的一个问题（Schrinner 等人，WABI 2020）。该问题要求给定字符串的最大长度子序列，其中每个符号最多包含一次运行（一次运行是连续相同符号的最大子串）。当参数是定义输入字符串的字母表的大小时，该问题已被证明是 NP-hard 并且是固定参数可处理的。在本文中，我们进一步研究了问题的复杂性，并表明当它通过解决方案中的运行次数（一个较小的参数）进行参数化时，它是固定参数易处理的。而且，我们研究了 Longest Run Subsequence 的核化复杂度，并证明当以字母表的大小或运行次数为参数时，它不接受多项式核。最后，当每个符号在输入字符串中最多出现两次时，我们考虑最长运行子序列的限制，并且我们证明它是 APX 难的。