FIXED-PARAMETER TRACTABILITY AND COMPLETENESS

For many fixed-parameter problems that are trivially soluable in polynomial-time, such as ($k$-)DOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as ($k$-)FEEDBACK VERTEX SET, exhibit fixed-parameter tractability: for each fixed $k$ the problem is soluable in time bounded by a polynomial of degree $c$, where $c$ is a constant independent of $k$. We establish the main results of a completeness program which addresses the apparent fixed-parameter intractability of many parameterized problems. In particular, we define a hierarchy of classes of parameterized problems $FPT \subseteq W[1] \subseteq W[2] \subseteq \cdots \subseteq W[SAT] \subseteq W[P]$ and identify natural complete problems for $W[t]$ for $t \geq 2$. (In other papers we have shown many problems complete for $W[1]$.) DOMINATING SET is shown to be complete for $W[2]$, and thus is not fixed-parameter tractable unless INDEPENDENT SET, CLIQUE, IRREDUNDANT SET and many other natural problems in $W[2]$ are also fixed-parameter tractable. We also give a compendium of currently known hardness results as an appendix.