Let XNLP be the class of parameterized problems such that an instance of size $n$ with parameter $k$ can be solved nondeterministically in time $f(k)n^{O(1)}$ and space $f(k)\log(n)$ (for some computable function $f$). We give a wide variety of XNLP-complete problems, such as {\sc List Coloring} and {\sc Precoloring Extension} with pathwidth as parameter, {\sc Scheduling of Jobs with Precedence Constraints}, with both number of machines and partial order width as parameter, {\sc Bandwidth} and variants of {\sc Weighted CNF-Satisfiability} and reconfiguration problems. In particular, this implies that all these problems are $W[t]$-hard for all $t$. This also answers a long standing question on the parameterized complexity of the {\sc Bandwidth} problem.

中文翻译：

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非确定性 FPT 时间和对数空间的参数化问题

设 XNLP 是一类参数化问题，这样大小为 $n$ 且参数为 $k$ 的实例可以在时间 $f(k)n^{O(1)}$ 和空间 $f(k)\ log(n)$（对于某些可计算函数 $f$）。我们给出了各种各样的 XNLP 完全问题，例如 {\sc List Coloring} 和 {\sc Precoloring Extension} 以路径宽度为参数，{\sc Scheduling of Jobs with Precedence Constraints}，具有机器数量和偏序宽度作为参数，{\sc 带宽} 和 {\sc 加权 CNF 可满足性} 的变体和重新配置问题。特别是，这意味着所有这些问题对于所有 $t$ 都是 $W[t]$-hard。这也回答了关于 {\sc Bandwidth} 问题的参数化复杂性的一个长期存在的问题。