A Parameterized Algorithm for Bounded-Degree Vertex Deletion

Abstract

The d-bounded-degree vertex deletion problem, to delete at most k vertices in a given graph to make the maximum degree of the remaining graph at most d, finds applications in computational biology, social network analysis and some others. It can be regarded as a special case of the (d + 2)-hitting set problem and generates the famous vertex cover problem. The d-bounded-degree vertex deletion problem is NP-hard for each fixed d ≥ 0. In terms of parameterized complexity, the problem parameterized by k is W[2]-hard for unbounded d and fixed-parameter tractable for each fixed d ≥ 0. Previously, (randomized) parameterized algorithms for this problem with running time bound O * ((d + 1) k ) are only known for d ≤ 2. In this paper, we give a uniform parameterized algorithm deterministically solving this problem in O * ((d + 1) k ) time for each d ≥ 3. Note that it is an open problem whether the d ′ -hitting set problem can be solved in O * ((d ′ − 1) k ) time for d ′ ≥ 3. Our result answers this challenging open problem affirmatively for a special case. Furthermore, our algorithm also gets a running time bound of O * (3.0645 k ) for the case that d = 2, improving the previous deterministic bound of O * (3.24 k ).The d-bounded-degree vertex deletion problem is a natural generation of the famous vertex cover problem, which is one of the best studied problems in combinatorial optimization. An application of the d-bounded-degree vertex deletion problem in computational biology is addressed by Fellows et. al. [5]: A cliquecentric approach in the analysis of genetic networks based on micro-array data can be modeled as the d-bounded-degree vertex deletion problem. The problem also plays an important role in the area of property testing [12]. Its "dual problem"-the s-plex problem was introduced in 1978 by Seidman and Foster [14] and it becomes an important problem in social network analysis now [1].

A Fast Maximum k-Plex Algorithm Parameterized by the Degeneracy Gap

“…In contrast, k-PLEX is W[1]-hard with repsect to parameter p. In the literature, Nishimura et al [2005] presented a O((d + t) t+3 t + n(d + t)) algorithm for d-BDD, followed by improvements in [Moser et al, 2012;Xiao, 2016]. For d ≥ 3, the d-BDD problem can be solved in O(|V | O(1) (d + 1) t ) [Xiao, 2016]. However, these algorithms are not practical and only of theoretical interest at the current stage.…”

An Improved Deterministic Parameterized Algorithm for Cactus Vertex Deletion

“…The gap between the running time of deterministic and randomized algorithms sometimes emerges for vertex deletion problems to "sparse" hereditary classes of graphs, such as Feedback Vertex Set. For instance, Pseudo Forest Vertex Deletion can be solved deterministically in time O * (3 k ) [2] and randomizedly in time O * (2.85 k ) [12] and Bounded Degree-2 Vertex Deletion can be solved deterministically in time O * (3.0645 k ) [19] and randomizedly in time O * (3 k ) [8]. Among others, the known gap on Cactus Vertex Deletion is remarkable: Bonnet et al [3] presented a deterministic O * (26 k )-time algorithm, while Koley et al [15] presented a randomized O * (12 k )-time algorithm.…”

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

“…A 3-path vertex cover is also known as a 1-degree-bounded deletion set. The d-degree-bounded deletion problem [11,25,26] is to delete a minimum number of vertices from a graph such that the remaining graph has degree at most d. The 3-path vertex cover problem is exactly the 1-degreebounded deletion problem. Several applications of 3-path vertex covers have been proposed in [6,16,27].…”