Abstract
The classes of the W-hierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives such as majority, not-all-equal, and unique. For example, a gate labelled by the majority connective outputs true if more than half of its inputs are true. For any finite set \(\mathcal{C}\) of connectives we construct the corresponding W( \(\mathcal{C}\) )-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W( \(\mathcal{C}\) )-hierarchy coincide levelwise. If \(\mathcal{C}\) only contains the majority connective then the first levels of the hierarchies coincide. We use this to show that a variant of the parameterized vertex cover problem, the majority vertex cover problem, is W[1]-complete.
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Abrahamson, K.A., Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness IV: on completeness for W[P] and PSPACE analogs. Ann. Pure Appl. Log. 73, 235–276 (1995)
Cai, L., Chen, J., Downey, R.G., Fellows, M.R.: On the parameterized complexity of short computation and factorization. Arch. Math. Log. 36, 321–337 (1997)
Chen, Y., Flum, J.: The parameterized complexity of maximality and minimality problems. Ann. Pure Appl. Log. 151, 22–61 (2007)
Chen, Y., Flum, J., Grohe, M.: Machine-based methods in parameterized complexity theory. Theor. Comput. Sci. 339, 167–199 (2005)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)
Downey, R.G., Fellows, M., Langston, M.A. (guest eds.): Special Issue on Parameterized Complexity, Part I and Part II. Comput. J. 51(1, 3) (2008)
Downey, R.G., Fellows, M.R., Taylor, U.: The parameterized complexity of relational database queries and an improved characterization of W[1]. In: Bridges, D.S., Calude, C., Gibbons, P., Reeves, S., Witten, I.H. (eds.) Combin. Complex. Log. 39, 194–213 (1996)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)
Marx, D.: Parameterized complexity of independence and domination on geometric graphs. In: Proceedings of the Second International Workshop on Parameterized and Exact Computation (IWPEC). LNCS, vol. 4169, pp. 154–165. Springer, Berlin (2006)
Melhorn, K.: Data Structures and Algorithms, Vol. 2: NP-Completeness and Graph Algorithms. EATCS Monographs on Theoretical Computer Science. Springer, Berlin (1984)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006)
Sipser, M.: Borel sets and circuit complexity. In: Proc. of the 15th Annual ACM Symposium on the Theory of Computing, pp. 61–69 (1983)
Spira, P.M.: On time hardware complexity tradeoffs for Boolean functions. In: Proceedings of the Fourth Hawaii International Symposium on System Sciences, pp. 525–527 (1971)
Wegener, I.: Relating monotone formula size and monotone depth of Boolean functions. Inf. Process. Lett. 16, 41–42 (1983)
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Research supported by the Australian Research Council, Centre in Bioinformatics, and by Fellowships to the Institute of Advanced Studies, Durham University, and to Grey College, Durham.
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Fellows, M., Flum, J., Hermelin, D. et al. W-Hierarchies Defined by Symmetric Gates. Theory Comput Syst 46, 311–339 (2010). https://doi.org/10.1007/s00224-008-9138-6
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DOI: https://doi.org/10.1007/s00224-008-9138-6