Review of Parameterized Complexity by R. Downey and M. Fellows

One of the most pervasive critiques of Computational Complexity, coming from the applied Computer Science community, is that it is an abstract, largely irrelevant field, that fails to capture the various intuitions concerning the complexity of "real-world" instances they have gathered through experiments. One aspect of these intuitions is that while most problems of practical interest tend to be intractable from a standard complexity-theoretic point of view, in many cases such problems have natural "structural" parameters, and practically relevant instances are often associated with "small" values of these parameters.Parameterized Complexity is an extension of Complexity Theory that attempts to formalize and deal with this intuition. It has already been applied to fields such as Databases, Logic and Constraint Programming, Computational Biology, Nonmonotonic Reasoning, Natural Language Processing, Cognitive Science, and Robotics. Some of its ideas have even been discovered independently by practitioners working in these areas. We do not attempt a further presentation (the interested reader can consult two excellent surveys [DF99, DFS99]), but want to illustrate this point through a revealing example: it has long been known that the register-allocation problem for imperative programming language amounts to a coloring problem on the associated control-flow graph, thus it seems, at a first glance, hard even in an approximate sense. On the other hand Thorup [Th98] has shown that such graphs are characterized by a bounded value (at most 10) of an intrinsic parameter called treewidth, that guarantees the existence of an approximation algorithm with substantially better performance than the one available for general graphs.The present book is an attempt to a first presentation of this emerging area by two of its most active proponents.