scholarly journals Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

2015 ◽  
Vol 54 (2) ◽  
pp. 368-389 ◽  
Author(s):  
Oswin Aichholzer ◽  
Wolfgang Mulzer ◽  
Alexander Pilz
Keyword(s):  
Simple Polygon ◽  
Np Complete ◽  
Flip Distance

scholarly journals Reconstruction of Weakly Simple Polygons from Their Edges

2018 ◽  
Vol 28 (02) ◽  
pp. 161-180
Author(s):  
Hugo A. Akitaya ◽  
Csaba D. Tóth
Keyword(s):  
Simple Polygon ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Line Segments ◽  
Text Input ◽  
Simple Polygons ◽  
Minimum Number ◽  
Np Complete ◽  
Arbitrary Polygon ◽  
Minimum Total Length

We address the problem of reconstructing a polygon from the multiset of its edges. Given [Formula: see text] line segments in the plane, find a polygon with [Formula: see text] vertices whose edges are these segments, or report that none exists. It is easy to solve the problem in [Formula: see text] time if we seek an arbitrary polygon or a simple polygon. We show that the problem is NP-complete for weakly simple polygons, that is, a polygon whose vertices can be perturbed by at most [Formula: see text], for any [Formula: see text], to obtain a simple polygon. We give [Formula: see text]-time algorithms for reconstructing weakly simple polygons: when all segments are collinear or the segment endpoints are in general position. These results extend to the variant in which the segments are directed. We study related problems for the case that the union of the [Formula: see text] input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.

scholarly journals Discovering the maximum k-clique on social networks using bat optimization algorithm

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Akram Khodadadi ◽  
Shahram Saeidi
Keyword(s):  
Genetic Algorithm ◽  
Social Networks ◽  
Social Network ◽  
Coding Theory ◽  
Evaluation Criteria ◽  
Convergence Speed ◽  
Optimization Approach ◽  
Computational Results ◽  
Complete Subgraph ◽  
Np Complete

AbstractThe k-clique problem is identifying the largest complete subgraph of size k on a network, and it has many applications in Social Network Analysis (SNA), coding theory, geometry, etc. Due to the NP-Complete nature of the problem, the meta-heuristic approaches have raised the interest of the researchers and some algorithms are developed. In this paper, a new algorithm based on the Bat optimization approach is developed for finding the maximum k-clique on a social network to increase the convergence speed and evaluation criteria such as Precision, Recall, and F1-score. The proposed algorithm is simulated in Matlab® software over Dolphin social network and DIMACS dataset for k = 3, 4, 5. The computational results show that the convergence speed on the former dataset is increased in comparison with the Genetic Algorithm (GA) and Ant Colony Optimization (ACO) approaches. Besides, the evaluation criteria are also modified on the latter dataset and the F1-score is obtained as 100% for k = 5.

scholarly journals Popular Matching in Roommates Setting Is NP-hard

2021 ◽  
Vol 13 (2) ◽  
pp. 1-20
Author(s):  
Sushmita Gupta ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
Meirav Zehavi
Keyword(s):  
Computational Complexity ◽  
Np Hard ◽  
Np Complete ◽  
Open Question ◽  
Popular Matching

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

scholarly journals An improved FPT algorithm for the flip distance problem

2021 ◽  
pp. 104708
Author(s):  
Qilong Feng ◽  
Shaohua Li ◽  
Xiangzhong Meng ◽  
Jianxin Wang
Keyword(s):  
Fpt Algorithm ◽  
Distance Problem ◽  
Flip Distance

scholarly journals Shellability is NP-complete

2019 ◽  
Vol 66 (3) ◽  
pp. 1-18
Author(s):  
Xavier Goaoc ◽  
Pavel Paták ◽  
Zuzana Patáková ◽  
Martin Tancer ◽  
Uli Wagner
Keyword(s):  
Np Complete

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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