A kernelization algorithm for d-Hitting Set

Abstract

For a given parameterized problem, π , a kernelization algorithm is a polynomial-time pre-processing procedure that transforms an arbitrary instance of π into an equivalent one whose size depends only on the input parameter(s). The resulting instance is called a problem kernel. In this paper, a kernelization algorithm for the 3-Hitting Set problem is presented along with a general kernelization for d-Hitting Set. For 3-Hitting Set, an arbitrary instance is reduced into an equivalent one that contains at most 5k 2 +k elements.This kernelization is an improvement over previously known methods that guarantee cubic-order kernels. Our method is used also to obtain quadratic kernels for several other problems. For a constant d 3, a kernelization of d-Hitting Set is achieved by a non-trivial generalization of the 3-Hitting Set method, and guarantees a kernel whose order does not exceed (2d − 1)k d−1 + k.

Parameterized Algorithms for (r,l)-Partization

“…A parameterized problem P is parameterized reducible to a parameterized problem Q if there is a FPT algorithm that transforms an instance x, k of P into an instance x , k of Q such that x, k ∈ P ⇔ x , k ∈ Q and k = g(k) for some function g. A parameterized problem P to which all problems in W [1] [28] involves a systematic exploration of the solution space of a parameterized problem in a tree-like fashion. The depths of such search trees are usually upper-bounded by values depending on the parameter.…”

“…Proof: For each fixed r, in O(n r+1 ) time, an r-Partization instance G, k can be transformed into an (r + 1)-uniform Hitting Set instance U = V (G), C = K, k = k , where K denotes the set of cliques of size (r + 1) in G. Thus, we obtain an O(k r ) size kernel using the kernelization techniques employed in [1].…”

“…Though a randomized polynomial kernelization algorithm is known for OCT by a recent result [23], the existence of a deterministic kernelization algorithm is still open. It is interesting to note that OCT on perfect graphs, being a restricted 3-Hitting Set, has a quadratic vertex kernel [1]. Determining if linear kernels exists is an interesting direction of research.…”

Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems

“…1 In this problem, the input consists of a universe U , a family F containing sets of size at most d over U , and in integer k. The objective is to determine whether there exists a set S ⊆ U of size at most k that intersects every set in F. Abu-Khzam [2] gave an improved kernel for d-Hitting Set, still with O(k d ) sets, but with O(k d−1 ) elements. The importance of the d-Hitting Set problem stems from the number of other problems that can be re-cast in terms of it.…”

“…Here, the input is a graph G and an integer k, and the task is to determine whether there exists a subset S of at most k vertices such that every connected component of G − S is a clique (such graphs are called cluster graphs). Also this problem can be formulated as a 3-Hitting Set problem where the family F contains the vertex sets of all induced P 3 's of G. An induced P 3 is a path on three vertices where the first and last vertex are non-adjacent in G. The kernel with O(k 2 ) elements for d-Hitting Set [2] can be adapted to obtain kernels with O(k 2 ) vertices for…”