scholarly journals The k ‐path vertex cover: General bounds and chordal graphs

Networks ◽  
2021 ◽  
Author(s):  
Csilla Bujtás ◽  
Marko Jakovac ◽  
Zsolt Tuza
Keyword(s):  
Vertex Cover ◽  
Chordal Graphs

scholarly journals Layered Graphs: A Class that Admits Polynomial Time Solutions for Some Hard Problems

2018 ◽  
Author(s):  
Bhadrachalam Chitturi ◽  
Srijith Balachander ◽  
Sandeep Satheesh ◽  
Krithic Puthiyoppil
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Vertex Cover ◽  
Independent Set ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Graph Class ◽  
Hard Problems ◽  
Connected Vertex Cover ◽  
Circular Arc Graphs

The independent set, IS, on a graph G = ( V , E ) is V * ⊆ V such that no two vertices in V * have an edge between them. The MIS problem on G seeks to identify an IS with maximum cardinality, i.e. MIS. V * ⊆ V is a vertex cover, i.e. VC of G = ( V , E ) if every e ∈ E is incident upon at least one vertex in V * . V * ⊆ V is dominating set, DS, of G = ( V , E ) if ∀ v ∈ V either v ∈ V * or ∃ u ∈ V * and ( u , v ) ∈ E . The MVC problem on G seeks to identify a vertex cover with minimum cardinality, i.e. MVC. Likewise, MCV seeks a connected vertex cover, i.e. VC which forms one component in G, with minimum cardinality, i.e. MCV. A connected DS, CDS, is a DS that forms a connected component in G. The problems MDS and MCD seek to identify a DS and a connected DS i.e. CDS respectively with minimum cardinalities. MIS, MVC, MDS, MCV and MCD on a general graph are known to be NP-complete. Polynomial time algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs for some of these problems. We introduce a novel graph class, layered graph, where each layer refers to a subgraph containing at most some k vertices. Inter layer edges are restricted to the vertices in adjacent layers. We show that if k = Θ ( log ∣ V ∣ ) then MIS, MVC and MDS can be computed in polynomial time and if k = O ( ( log ∣ V ∣ ) α ) , where α < 1 , then MCV and MCD can be computed in polynomial time. If k = Θ ( ( log ∣ V ∣ ) 1 + ϵ ) , for ϵ > 0 , then MIS, MVC and MDS require quasi-polynomial time. If k = Θ ( log ∣ V ∣ ) then MCV, MCD require quasi-polynomial time. Layered graphs do have constraints such as bipartiteness, planarity and acyclicity.

Complexity of the Maximum k-Path Vertex Cover Problem

2020 ◽  
Vol E103.A (10) ◽  
pp. 1193-1201
Author(s):  
Eiji MIYANO ◽  
Toshiki SAITOH ◽  
Ryuhei UEHARA ◽  
Tsuyoshi YAGITA ◽  
Tom C. van der ZANDEN
Keyword(s):  
Vertex Cover ◽  
Vertex Cover Problem ◽  
Cover Problem

scholarly journals Approximation algorithm for minimum connected 3-path vertex cover

2020 ◽  
Vol 287 ◽  
pp. 77-84
Author(s):  
Pengcheng Liu ◽  
Zhao Zhang ◽  
Xianyue Li ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Vertex Cover

scholarly journals From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1036
Author(s):  
Abel Cabrera Martínez ◽  
Alejandro Estrada-Moreno ◽  
Juan Alberto Rodríguez-Velázquez
Keyword(s):  
Vertex Cover ◽  
Domination Number ◽  
Theoretical Computer Science ◽  
Discrete Mathematics ◽  
Special Issue ◽  
Total Domination Number ◽  
Theoretical Computer ◽  
Graph Parameters ◽  
Semitotal Domination ◽  
Domination In Graphs

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x∈V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X⊆V(G) is defined to be N(X)=⋃x∈XN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)∖X. Now, for every set X⊆V(G) and every vertex x∈X, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={y∈V(G)∖X:N(y)∩X={x}}. Let Xw={x∈X:Pe(x,X)≠⌀}. The strong differential of X is defined to be ∂s(X)=|Ne(X)|−|Xw|, while the quasi-total strong differential of G is defined to be ∂s*(G)=max{∂s(X):X⊆V(G)andXw⊆N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard.

scholarly journals Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

AbstractIn computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

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