Abstract
Many parameterized problems (such as the clique problem and the dominating set problem) ask, given an instance and a natural number k as parameter, whether there is a solution of size k . We analyze the relationship between the complexity of such a problem and the corresponding maximality (minimality) problem asking for a solution of size k maximal (minimal) with respect to set inclusion. As our results show, many maximality problems increase the parameterized complexity, while “in terms of the W-hierarchy” minimality problems do not increase the complexity. We also address the corresponding construction, listing, and counting problems.
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