Kernelization Algorithms for d-Hitting Set Problems

Fixed-Parameter Algorithms for Cluster Vertex Deletion

“…In O(nm) time, we enumerate the P 3 's of G. We then apply the lineartime kernelization by Abu-Khzam and Fernau [2], which gives us an instance with O(k 3 ) vertices. To this instance, we apply the kernelization algorithm that yields a kernel with O(k 2 ) vertices (running in O(k 9 ) time) [1]. Finally, we apply Theorem 1.…”

“…Here, we aim at finding a vertex set of minimum weight such that its deletion transforms a given graph into a cluster graph. 1 Weighted Cluster Vertex Deletion Instance: An undirected graph G = (V, E), a vertex weight function ω : V → [1, ∞), and a nonnegative number k. Question: Is there a vertex set X ⊆ V with v∈X ω(v) ≤ k such that deleting all vertices in X from G results in a cluster graph (i. e., a graph where every connected component forms a complete graph)?…”

A Problem Kernelization for Graph Packing

“…One dimension of an instance of a parameterized problem is the input size n, and the other is the parameter k. A parameterized problem is fixed-parameter tractable if it can be solved in f (k) · n O (1) time, where f is a computable function depending only on the parameter k, not on the input size n. Problem kernelization is a core tool to develop parameterized algorithms [16,17,23]. A kernelization is often described with a set of data reduction rules that are applied to the instance I with parameter k of a problem and that change that instance into an smaller instance I ′ with parameter k ′ ≤ k in polynomial time, such that (I, k) is a yes-instance if and only if (I ′ , k ′ ) is a yes-instance.…”

“…The algorithm compute witness, given in Figure 2, computes a witness W with respect to H\R, where R is a set of "unnecessary" copies of H in H, that is, if there exists a size-k H-packing, then there is a size-k H-packing that does not use any element of R. The identification of unnecessary copies of H is derived from a combination of ideas for data reduction rules for Hitting Set [1] and generalized matching and set cover problems [10]. The basic idea is that if there are many copies of H in W that intersect in the same vertex subset S, then some of them do not need to be considered for a maximum H-packing in G and can therefore be removed from the graph.…”

Backdoor Sets of Quantified Boolean Formulas

“…Currently, the fastest known fixed-parameter algorithm for the former problem is due to Chen, Kanj, and Xia [7] and runs in time O(1.2738 k + kn). The fastest known fixed-parameter algorithm for the latter problem is due to Niedermeier and Rossmanith [21] and runs in time O(2.270 k + n); an alternative algorithm is due to Abu-Khzam [1]. If no hitting set of size at most k is found, then we know by Lemmas 3 and 4 that F has no strong C-backdoor set of size at most k, and we can reject the instance.…”