On a generalization of Nemhauser and Trotter's local optimization theorem

Xiao, Mingyu

doi:10.1016/j.jcss.2016.08.003

Cited by 50 publications

(12 citation statements)

References 26 publications

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“…The k-plex problem with any fixed positive integer k is an NP-complete problem [2], it reduces to the well-known maximum clique problem (MC) when k = 1, one of Karp's 21 NP-complete problems [15]. The k-plex problem has a number of applications in information retrieval, code theory, signal transmission, social networks, classification theory amongst others [8,16,20,26].…”

Section: Introductionmentioning

confidence: 99%

“…Due in part to the wide variety of real-world applications, increased research effort is being devoted to solving this problem. Over the past few years, several exact algorithms have been proposed for finding the maximum k-plex of a given graph [2,17,19,24,26]. These methods can find optimal solutions for graphs with around a thousand vertices in a reasonable amount of computing time (within 3 hours).…”

Section: Introductionmentioning

confidence: 99%

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Effective reinforcement learning based local search for the maximum k-plex problem

Jin,

Drake,

Benlic

et al. 2019

Preprint

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The maximum k-plex problem is a computationally complex problem, which emerged from graph-theoretic social network studies. This paper presents an effective hybrid local search for solving the maximum k-plex problem that combines the recently proposed breakout local search algorithm with a reinforcement learning strategy. The proposed approach includes distinguishing features such as: a unified neighborhood search based on the swapping operator, a distance-and-quality reward for actions and a new parameter control mechanism based on reinforcement learning. Extensive experiments for the maximum k-plex problem (k = 2, 3, 4, 5) on 80 benchmark instances from the second DIMACS Challenge demonstrate that the proposed approach can match the best-known results from the literature in all but four problem instances. In addition, the proposed algorithm is able to find 32 new best solutions.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Effective reinforcement learning based local search for the maximum k-plex problem

Jin,

Drake,

Benlic

et al. 2019

Preprint

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“…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].…”

Section: Introductionmentioning

confidence: 99%

“…A kernelization algorithm is a polynomial-time algorithm which, for an input graph with a parameter (G, k) either concludes that G has no 3-path vertex cover of size k or returns an equivalent instance (G ′ , k ′ ), called a kernel, such that k ′ ≤ k and the size of G ′ is bounded by a function of k. Kernelization for the d-degreebounded deletion problem has been studied in the literature [11,25]. For d = 1, Fellows et al's algorithm [11] implies a kernel of 15k vertices for the 3-path vertex cover problem, and Xiao's algorithm [25] implies a kernel of 13k vertices. There is another closed related problem, called the 3-path packing problem.…”

Section: Introductionmentioning

confidence: 99%

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

Xiao

Kou

2017

Lecture Notes in Computer Science

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Abstract. A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most k. In this paper, we give a kernel of 5k vertices and an O * (1.7485 k )-time polynomial-space algorithm for this problem, both new results improve previous known bounds.

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“…A linear vertex kernel for the case that d = 2 was developed in [4]. Recently, a refined generation of the NT-theorem was proved [17], which can get a linear vertex kernel for each fixed d ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

A Parameterized Algorithm for Bounded-Degree Vertex Deletion

Xiao

2016

Lecture Notes in Computer Science

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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].

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On a generalization of Nemhauser and Trotter's local optimization theorem

Cited by 50 publications

References 26 publications

Effective reinforcement learning based local search for the maximum k-plex problem

Effective reinforcement learning based local search for the maximum k-plex problem

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A Parameterized Algorithm for Bounded-Degree Vertex Deletion

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