  We study the classic {\sc Dominating Set} problem with respect to several
prominent parameters. Specifically, we present algorithmic results that
sidestep time complexity barriers by the incorporation of either approximation
or larger parameterization. Our results span several parameterization regimes,
including: (i,ii,iii) time/ratio-tradeoff for the parameters {\em treewidth},
{\em vertex modulator to constant treewidth} and {\em solution size}; (iv,v)
FPT-algorithms for the parameters {\em vertex cover number} and {\em feedback
edge set number}; and (vi) compression for the parameter {\em feedback edge set
number}.
