Fundamentals of Parameterized Complexity Revisited
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What is a parameterized algorithm? What is a parameterized problem and what is its complexity? Is a problem parameterized by clique-width the same as parameterized by rank-width? If so, then what is the higher-level object that these two parameters represent? Should the complexity of computing the parameter be part of the complexity of the parameterized problem? Is the concept of slices justifiable given that it only applies to decision problems whereas the parameterization paradigm extends beyond that? What is the exact relation between classical and parameterized complexity? We address these questions by presenting an alternative formalization of parameterized complexity based on the ubiquitous notion of solving a problem on a restricted set of inputs and show how it relates to the status quo.
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