On miniaturized problems in parameterized complexity theory

We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the appropriate logical formalism, we show that the miniaturization of a definable problem in W[t] lies in W[t], too. In particular, the miniaturization of the dominating set problem is in W[2]. Furthermore, we investigate the relation between f(k)ċno(k) time and subexponential time algorithms for the dominating set problem and for the clique problem.

[1]  Jörg Flum,et al.  Parametrized Complexity and Subexponential Time (Column: Computational Complexity) , 2004, Bull. EATCS.

[2]  Jörg Flum,et al.  Bounded Fixed-Parameter Tractability and log2n Nondeterministic Bits , 2004, ICALP.

[3]  Michael R. Fellows,et al.  Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems , 2003, CATS.

[4]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[5]  Liming Cai,et al.  Subexponential Parameterized Algorithms Collapse the W-Hierarchy , 2001, ICALP.

[6]  Russell Impagliazzo,et al.  On the Complexity of k-SAT , 2001, J. Comput. Syst. Sci..

[7]  Ge Xia,et al.  Linear FPT reductions and computational lower bounds , 2004, STOC '04.

[8]  Iyad A. Kanj,et al.  Tight lower bounds for certain parameterized NP-hard problems , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[9]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[10]  Rodney G. Downey,et al.  Parameterized complexity for the skeptic , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[11]  Russell Impagliazzo,et al.  Which Problems Have Strongly Exponential Complexity? , 2001, J. Comput. Syst. Sci..

[12]  Michael R. Fellows,et al.  New Directions and New Challenges in Algorithm Design and Complexity, Parameterized , 2003, WADS.

[13]  Jörg Flum,et al.  Fixed-Parameter Tractability, Definability, and Model-Checking , 1999, SIAM J. Comput..

[14]  Michael R. Fellows,et al.  Descriptive Complexity and the W Hierarchy , 2022 .

[15]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..