Parameterized Complexity via Combinatorial Circuits (Extended Abstract) MICHAEL FELLOWS , 1 J¨ ORG FLUM,DANNY HERMELIN,MORITZ M¨ ULLER, AND FRANCES ROSAMOND 2 ABSTRACT. The classes of the W-hierarchy are the most important classes of intractable problems in parameterized complexity. These classes were orig- inally defined via the weighted satisfiability problem for Boolean circuits. The analysis of the parameterized majority vertex cover problem and other pa- rameterized problems led us to study circuits that contain connectives such as majority, not-all-equal, and unique, instead of (or in addition to) the Boolean connectives. For example, a gate labelled by the majority connective outputs TRUE if more than half of the inputs are TRUE. For any finite set C of connec- tives we construct the corresponding W( C)-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W( C)-hierarchy co- incide levelwise. Surprisingly, if C contains only the majority connective (i.e., no booleanconnectives), then the first levels coincide. We use this to show that the majority vertex cover problem is W[1]-complete. 1 Introduction Parameterized complexity is a refinement of classical complexity theory, in which one takes into account not only the total input length n, but also other aspects of the problem codified as the parameter k. In doing so, one attempts to confine the exponential running time needed for solving many natural problems strictly to the parameter. For example, the classical V ERTEX-COVER problem can be solved in O(2 k · n) time, when parameterized by the size k of the solution sought [8] (significant improvements to this algorithm are surveyed in [9]). This running time is practical for instances with small parameter, and in general is far better than the O(n k+1 ) running time of the brute-force algorithm. More generally, a problem is said to be fixed-parameter tractable if it has an algorithm running in time f (k) · p(n), where n is the length of the input, k its parameter, f an arbitrary 1 Research supported by the Australian Research Council, Centre in Bioinformatics, by Fellowships to the Insitute of Advanced Studies, Durham University, and to Grey College, Durham, and by Aus- tralian Research Council Discovery Project DP-0773331. 2 Research supported by the Australian Research Council, Centre in Bioinformatics.