scholarly journals Hamilton Connectivity of Convex Polytopes with Applications to Their Detour Index

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Sakander Hayat ◽  
Asad Khan ◽  
Suliman Khan ◽  
Jia-Bao Liu
Keyword(s):  
Computer Science ◽  
Connected Graph ◽  
Electrical Engineering ◽  
Convex Polytopes ◽  
Hamiltonian Path ◽  
Infinite Family ◽  
Complete Problem ◽  
Analytical Formulas ◽  
Np Complete ◽  
Hamilton Connected

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

scholarly journals On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Sakander Hayat ◽  
Muhammad Yasir Hayat Malik ◽  
Ali Ahmad ◽  
Suliman Khan ◽  
Faisal Yousafzai ◽  
...  
Keyword(s):  
Convex Polytopes ◽  
Hamiltonian Path ◽  
Convex Polytope ◽  
Arbitrary Graph ◽  
Structural Graph ◽  
Complete Problems ◽  
Finite Set ◽  
Np Complete ◽  
Hamilton Connected ◽  
Set Of Points

A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

scholarly journals NLocalSAT: Boosting Local Search with Solution Prediction

2020 ◽  
Author(s):  
Wenjie Zhang ◽  
Zeyu Sun ◽  
Qihao Zhu ◽  
Ge Li ◽  
Shaowei Cai ◽  
...  
Keyword(s):  
Neural Network ◽  
Local Search ◽  
Prediction Model ◽  
Computer Science ◽  
Boolean Satisfiability ◽  
Stochastic Local Search ◽  
Boolean Satisfiability Problem ◽  
Complete Problem ◽  
Random Manner ◽  
Np Complete

The Boolean satisfiability problem (SAT) is a famous NP-complete problem in computer science. An effective way for solving a satisfiable SAT problem is the stochastic local search (SLS). However, in this method, the initialization is assigned in a random manner, which impacts the effectiveness of SLS solvers. To address this problem, we propose NLocalSAT. NLocalSAT combines SLS with a solution prediction model, which boosts SLS by changing initialization assignments with a neural network. We evaluated NLocalSAT on five SLS solvers (CCAnr, Sparrow, CPSparrow, YalSAT, and probSAT) with instances in the random track of SAT Competition 2018. The experimental results show that solvers with NLocalSAT achieve 27% ~ 62% improvement over the original SLS solvers.

scholarly journals Matroids, Complexity and Computation

2021 ◽  
Author(s):  
◽  
Michael Snook
Keyword(s):  
Finite Fields ◽  
Infinite Family ◽  
Np Hard ◽  
Edge Deletion ◽  
Fixed Field ◽  
Complete Problem ◽  
Np Complete

<p>The node deletion problem on graphs is: given a graph and integer k, can we delete no more than k vertices to obtain a graph that satisfies some property π. Yannakakis showed that this problem is NP-complete for an infinite family of well- defined properties. The edge deletion problem and matroid deletion problem are similar problems where given a graph or matroid respectively, we are asked if we can delete no more than k edges/elements to obtain a graph/matroid that satisfies a property π. We show that these problems are NP-hard for similar well-defined infinite families of properties.  In 1991 Vertigan showed that it is #P-complete to count the number of bases of a representable matroid over any fixed field. However no publication has been produced. We consider this problem and show that it is #P-complete to count the number of bases of matroids representable over any infinite fixed field or finite fields of a fixed characteristic.  There are many different ways of describing a matroid. Not all of these are polynomially equivalent. That is, given one description of a matroid, we cannot create another description for the same matroid in time polynomial in the size of the first description. Due to this, the complexity of matroid problems can vary greatly depending on the method of description used. Given one description a problem might be in P while another description gives an NP-complete problem. Based on these interactions between descriptions, we create and study the hierarchy of all matroid descriptions and generalize this to all descriptions of countable objects.</p>

scholarly journals Polynomial time solution to NP-complete problem Hamiltonian cycle (P=NP Proof)

2021 ◽  
Author(s):  
Yasaman KalantarMotamedi
Keyword(s):  
Computer Science ◽  
Polynomial Time ◽  
Hamiltonian Cycle ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Rigorous Proof ◽  
Complete Problem ◽  
Time Solution ◽  
Np Complete

P vs NP is one of the open and most important mathematics/computer science questions that has not been answered since it was raised in 1971 despite its importance and a quest for a solution since 2000. P vs NP is a class of problems that no polynomial time algorithm exists for any. If any of the problems in the class gets solved in polynomial time, all can be solved as the problems are translatable to each other. One of the famous problems of this kind is Hamiltonian cycle. Here we propose a polynomial time algorithm with rigorous proof that it always finds a solution if there exists one. It is expected that this solution would address all problems in the class and have a major impact in diverse fields including computer science, engineering, biology, and cryptography.

scholarly journals The Property of Hamiltonian Connectedness in Toeplitz Graphs

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Ayesha Shabbir ◽  
Muhammad Faisal Nadeem ◽  
Tudor Zamfirescu
Keyword(s):  
Hamiltonian Path ◽  
Complete Problem ◽  
Hamiltonian Connected ◽  
The Family ◽  
Np Complete

A spanning path in a graph G is called a Hamiltonian path. To determine which graphs possess such paths is an NP-complete problem. A graph G is called Hamiltonian-connected if any two vertices of G are connected by a Hamiltonian path. We consider here the family of Toeplitz graphs. About them, it is known only for n=3 that Tnp,q is Hamiltonian-connected, while some particular cases of Tnp,q,r for p=1 and q=2,3,4 have also been investigated regarding Hamiltonian connectedness. Here, we prove that the nonbipartite Toeplitz graph Tn1,q,r is Hamiltonian-connected for all 1<q<r<n and n≥5r−2.

A Polynomial Algorithm for Constructing a Large Bipartite Subgraph, with an Application to a Satisfiability Problem

1982 ◽  
Vol 34 (3) ◽  
pp. 519-524 ◽  
Author(s):  
Svatopluk Poljak ◽  
Daniel Turzík
Keyword(s):  
Connected Graph ◽  
Polynomial Algorithm ◽  
Short Proof ◽  
Satisfiability Problem ◽  
Boolean Formula ◽  
Complete Problem ◽  
Bipartite Subgraph ◽  
Np Complete ◽  
Image Position

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,for a connected graph G of n vertices and m edges, whereand ﹛x﹜ denotes the smallest integer ≧ x.We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

FRACTAL STRUCTURE ON k-SAT

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 227-232 ◽  
Author(s):  
QIN WANG ◽  
LIFENG XI
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Computer Science ◽  
Fractal Structure ◽  
Reasonable Condition ◽  
Complete Problem ◽  
Np Complete ◽  
Boolean Expressions

The k-SAT (k ≥ 3) is a typical NP-complete problem in computer science. To visualize k-SAT, we embed the Boolean expressions in k-CNF, with variables in an infinite list, into cube [0, 1]k of Euclidean space. In this paper, we find fractal structure of the visualized image of satisfiable expressions, we also prove that the image has full Hausdorff dimension k under some reasonable condition, by constructing some Moran subset. The results show this image is quite different from that with respect to a finite list of Boolean variables as in Refs. 1 and 2.

scholarly journals Matroids, Complexity and Computation

2021 ◽  
Author(s):  
◽  
Michael Snook
Keyword(s):  
Finite Fields ◽  
Infinite Family ◽  
Np Hard ◽  
Edge Deletion ◽  
Fixed Field ◽  
Complete Problem ◽  
Np Complete

<p>The node deletion problem on graphs is: given a graph and integer k, can we delete no more than k vertices to obtain a graph that satisfies some property π. Yannakakis showed that this problem is NP-complete for an infinite family of well- defined properties. The edge deletion problem and matroid deletion problem are similar problems where given a graph or matroid respectively, we are asked if we can delete no more than k edges/elements to obtain a graph/matroid that satisfies a property π. We show that these problems are NP-hard for similar well-defined infinite families of properties.  In 1991 Vertigan showed that it is #P-complete to count the number of bases of a representable matroid over any fixed field. However no publication has been produced. We consider this problem and show that it is #P-complete to count the number of bases of matroids representable over any infinite fixed field or finite fields of a fixed characteristic.  There are many different ways of describing a matroid. Not all of these are polynomially equivalent. That is, given one description of a matroid, we cannot create another description for the same matroid in time polynomial in the size of the first description. Due to this, the complexity of matroid problems can vary greatly depending on the method of description used. Given one description a problem might be in P while another description gives an NP-complete problem. Based on these interactions between descriptions, we create and study the hierarchy of all matroid descriptions and generalize this to all descriptions of countable objects.</p>

Application of DNA Nanoparticle Conjugation on the Hamiltonian Path Problem

2021 ◽  
Vol 16 (3) ◽  
pp. 501-505
Author(s):  
Jingjing Ma
Keyword(s):  
Graph Theory ◽  
Self Assembly ◽  
Dna Computing ◽  
Hamiltonian Path ◽  
Au Nanoparticle ◽  
Assembly Model ◽  
Hamiltonian Path Problem ◽  
Complete Problem ◽  
Np Complete ◽  
Nanoparticle Conjugation

A DNA computing algorithm is proposed in this paper. The algorithm uses the assembly of DNA/Au nanoparticle conjugation to solve an NP-complete problem in the Graph theory, the Hamiltonian Path problem. According to the algorithm, I designed the special DNA/Au nanoparticle conjugations which assembled based on a specific graph, then, a series of experimental techniques are utilized to get the final result. This biochemical algorithm can reduce the complexity of the Hamiltonian Path problem greatly, which will provide a practical way to the best use of DNA self-assembly model.

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