Parameterized Parallel Complexity

We consider the framework of Parameterized Complexity and we investigate the issue of which problems do admit eecient xed parameter parallel algorithms. In particular, we introduce two classes of eeciently parallelizable parameterized problems, PNC and FPP, according to the degree of eeciency we want to obtain. We sketch both some FPP-algorithms solving natural parameterized problems and a useful tool for proving membership to FPP based on the concept of treewidth. We then focus our attention on parameterized parallel intractability and prove that a necessary condition for a parameterized problem to be complete for the class of sequentially xed parameter tractable problems (with respect to reductions preserving membership to PNC) is that it is not in NC for some xed value of the parameter (unless P = NC). Finally, we give two alternative characterizations of both PNC and FPP and we use them to prove the PNC-completeness of two natural parameterized problems.