The Connected Vertex Cover problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al. [STOC 2017], have shown an α-approximate kernel for the problem for every α > 1, in the framework of approximate or lossy kernelization. We exhibit lossy kernels and FPT algorithms for Connected Vertex Cover for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size.Our first result is a lossy kernel for Connected Vertex Cover parameterized by the size k of a split deletion set. A split graph is a graph whose vertex set can be partitioned into a clique and an independent set and a split deletion set is a set of vertices whose deletion results in a split graph. Let n denote the number of vertices in the input graph. We show that
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V (G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λP : X → Q for X ⊆ V (G), decide whether λP can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O * (2 k )-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n − k colors for every v ∈ V (G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k 2 ) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k 2 ) colors.
A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP ⊆ coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t ≥ 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k c , where c could be a function of k, we obtain a kernel of size O(2 c 2 k 7+c log 3 k) whenever c ∈ o( √ k). Thus the kernel size varies from being sub-exponential when c ∈ o( √ k), to quasipolynomial when c ∈ o(log k), ∈ R + , and polynomial when c ∈ O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t ∈ O(k ), for any 0 ≤ < 1.
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