## An analysis of the W*-hierarchy.(English)Zbl 1122.03040

The well-known W-hierarchy was introduced by Downey and Fellows as a hierarchy of fixed-parameter intractable problems. The W*-hierarchy, introduced by Downey, Fellows, and Taylor, is a variant of the W-hierarchy. In this paper, it is shown that $$\text{W}[t]\subseteq \text{W}^*[t]\subseteq \text{W}[2t-2]$$ for each $$t\geq 2$$. Thus, the union over the levels of the W-hierarchy and over the W*-hierarchy, respectively, are the same. It was known before that $$\text{W}[1]\subseteq \text{W}^*[1]$$ and $$\text{W}[2]\subseteq \text{W}^*[2]$$. The other main result of the paper is a new logical characterization of the W*-hierarchy in terms of “Fagin-definable problems”. As a by-product, Chen, Flum and Grohe obtain an improvement of their earlier characterization of the W*-hierarchy in terms of model-checking problems. Furthermore, they obtain new complete problems for the classes W[3] and W*[3].

### MSC:

 03D15 Complexity of computation (including implicit computational complexity) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)

### Keywords:

W-hierarchy; complete problems; W[3]; W*[3]
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### References:

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