A probabilistic approach to problems parameterized above or below tight bounds

We introduce a new approach for establishing fixed-parameter tractability of problems parameterized above tight lower bounds or below tight upper bounds. To illustrate the approach we consider two problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar [M. Mahajan, V. Raman, S. Sikdar, Parameterizing above or below guaranteed values, J. Comput. System Sci. 75 (2) (2009) 137-153]. We show that a generalization of one of the problems and three non-trivial special cases of the other problem admit kernels of quadratic size. As a byproduct we obtain a new probabilistic inequality that could be of independent interest. Our new inequality is dual to the Hypercontractive Inequality.