Parameterized complexity: A framework for systematically confronting computational intractability

In this paper we give a programmatic overview of parame-terized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how xed-parameter tractability techniques can deliver practical algorithms in two diierent ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT ker-nelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artiicial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W1]. As a corollary, this implies that k-Step Reachability for Petri Nets is complete for W1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph k-Coloring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.