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 List Coloring and Precoloring Extension with pathwidth as parameter, Scheduling of Jobs with Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidth and variants of Weighted CNF-Satisfiability. 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 Bandwidth problem.

翻译：令 XNLP 表示一类参数化问题，其规模为 n、参数为 k 的实例可在时间 $f(k)n^{O(1)}$ 和空间 $f(k)\log(n)$（对于某个可计算函数 f）内非确定性地求解。我们给出了广泛的 XNLP 完备问题，例如以路径宽度为参数的列表着色和预着色扩展、具有优先约束的作业调度（以机器数量和偏序宽度为参数）、带宽以及加权 CNF 可满足性的变体。特别地，这意味着所有这些问题对于所有 t 都是 W[t]-困难的。这也回答了关于带宽问题参数化复杂性的一个长期未决问题。