A New Polynomial-Time Solvable Graph Class for the Minimum Biclique Edge Cover Problem

2020 ◽  
Vol E103.A (10) ◽  
pp. 1202-1205
Author(s):  
Hideaki OTSUKI
Keyword(s):  
Polynomial Time ◽  
Edge Cover ◽  
Graph Class ◽  
Solvable Graph ◽  
Cover Problem

On the Complexity of the Upper r-Tolerant Edge Cover Problem

2020 ◽  
pp. 32-47
Author(s):  
Ararat Harutyunyan ◽  
Mehdi Khosravian Ghadikolaei ◽  
Nikolaos Melissinos ◽  
Jérôme Monnot ◽  
Aris Pagourtzis
Keyword(s):  
Edge Cover ◽  
Cover Problem

scholarly journals Hardness of covering all the triangles

2018 ◽  
Author(s):  
Thinh D. Nguyen
Keyword(s):  
Combinatorial Optimization ◽  
Polynomial Time ◽  
Optimization Problem ◽  
Vertex Cover ◽  
Research Area ◽  
Combinatorial Optimization Problem ◽  
Edge Cover ◽  
Hard Edge ◽  
Problem Solvability

Vertex Cover and Edge Cover are two classical examples that are often used to show the contrast of problem solvability. While Vertex Cover is hard, Edge Cover can be solved in polynomial time. We claim that the former remains intractable even if the objects to be covered are triangles instead of edges. Therefore, one more combinatorial optimization problem, namely Covering Triangles, is added to the decades-old list of the problems in this research area.

A polynomial time algorithm for the 2-Poset Cover Problem

2021 ◽  
Vol 169 ◽  
pp. 106106
Author(s):  
Ivy Ordanel ◽  
Proceso Fernandez ◽  
Henry Adorna
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Cover Problem

scholarly journals Recognizing the P_4-structure of claw-free graphs and a larger graph class

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Luitpold Babel ◽  
Andreas Brandstädt ◽  
Van Bang Le
Keyword(s):  
Large Class ◽  
Polynomial Time ◽  
Uniform Hypergraph ◽  
Free Graph ◽  
Constructive Algorithm ◽  
Time Criterion ◽  
Graph Class ◽  
Vertex Set ◽  
International Audience

International audience The P_4-structure of a graph G is a hypergraph \textbfH on the same vertex set such that four vertices form a hyperedge in \textbfH whenever they induce a P_4 in G. We present a constructive algorithm which tests in polynomial time whether a given 4-uniform hypergraph is the P_4-structure of a claw-free graph and of (banner,chair,dart)-free graphs. The algorithm relies on new structural results for (banner,chair,dart)-free graphs which are based on the concept of p-connectedness. As a byproduct, we obtain a polynomial time criterion for perfectness for a large class of graphs properly containing claw-free graphs.

scholarly journals K3 Edge Cover Problem in a Wide Sense

2020 ◽  
Vol 28 (0) ◽  
pp. 849-858
Author(s):  
Kyohei Chiba ◽  
Rémy Belmonte ◽  
Hiro Ito ◽  
Michael Lampis ◽  
Atsuki Nagao ◽  
...  
Keyword(s):  
Wide Sense ◽  
Edge Cover ◽  
Cover Problem

Constant-approximation for minimum weight partial sensor cover

2020 ◽  
pp. 2150047
Author(s):  
Siwen Liu ◽  
Hongmin W. Du
Keyword(s):  
Time Constant ◽  
Polynomial Time ◽  
Wireless Sensors ◽  
Minimum Weight ◽  
Total Weight ◽  
Sensor Cover ◽  
Nonnegative Weight ◽  
Cover Problem

Consider a set of homogeneous wireless sensors, [Formula: see text] with nonnegative weight [Formula: see text] for each sensor [Formula: see text]. Let [Formula: see text] be a set of target points. Given a integer [Formula: see text], we study the minimum weight partial sensor cover problem, that is, find the minimum total weight subset of sensors covering at least [Formula: see text] target points in [Formula: see text]. In this paper, we show the existence of polynomial-time constant-approximation for this problem.

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.

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