Fixed-Parameter Tractability and Completeness I: Basic Results

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 intractabil-ity of many parameterized problems. In particular, we define a hierarchy of classes of parameterized problems FPT _ W[I] c_ W[2] _ _ _ _ _ _ W[SAT] c_ W[P] and identify natural complete problems for W[t] for > 2. (In other papers we have shown many problems complete for W[ ].) 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.