Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space

Let XNLP be the class of parameterized prob-lems such that an instance of size
$n$
with parameter
$k$
can be solved nondeterministically in time
$f$
(
$k$
) nO
⁽¹⁾
and space f (k) log(n) (for some computable function f). We give a wide variety of XNLP-complete problems, such as List Coloringand Precoloring Extensionwith pathwidth as parameter, Scheduling Of Jobs With Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidthand variants of 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 Bandwidth problem.