The 3- problem is also called the problem on 3-uniform hypergraphs. In this paper, we address kernelizations of the problem on 3-uniform hypergraphs. We show that this problem admits a linear kernel in three classes of 3-uniform hypergraphs. We also obtain lower and upper bounds on the kernel size for them by the parametric duality.
 The two objectives of this paper are: (1) to articulate three new general techniques for designing FPT algorithms, and (2) to apply these to obtain new FPT algorithms for Set Splitting and Vertex Cover. In the case of Set Splitting, we improve the best previous ( O ^*(72^k) ) FPT algorithm due to Dehne, Fellows and Rosamond [DFR03], to ( O ^*(8^k) ) by an approach based on greedy localization in conjunction with modeled crown reduction. In the case of Vertex Cover, we describe a new approach to 2k kernelization based on iterative compression and crown reduction, providing a potentially useful alternative to the Nemhauser-Trotter 2k kernelization. This survey reviews the basic notions of parameterized complexity, and describes some new approaches to designing FPT algorithms and problem reductions for graph problems. A kernelization algorithm for the 3-Hitting-Set problem is presented along with a general kernelization for d-Hitting-Set problems. For 3-Hitting-Set, a quadratic kernel is obtained by exploring properties of yes instances and employing what is known as crown reduction. Any 3-Hitting-Set instance is reduced into an equivalent instance that contains at most 5k2 + k elements (or vertices). This kernelization is an improvement over previously known methods that guarantee cubic-size kernels. Our method is used also to obtain a quadratic kernel for the Triangle Vertex Deletion problem. For a constant d ≥ 3, a kernelization of d-Hitting-Set is achieved by a generalization of the 3-Hitting-Set method, and guarantees a kernel whose order does not exceed (2d - 1)kd-1 + k.