scholarly journals POLYNOMIAL SOLVABILITY OF NP-COMPLETE PROBLEMS

2014 ◽  
Author(s):  
Anatoly Panyukov
Keyword(s):  
Computational Complexity ◽  
Polynomial Algorithm ◽  
Hamiltonian Circuit ◽  
Graph Vertex ◽  
Polynomial Solvability ◽  
Complete Problems ◽  
Vertex Set ◽  
Np Complete

A polynomial algorithm for solving the ''Hamiltonian circuit'' problem is presented in the paper. Computational complexity of the algorithm is equal to $O\left(n^8\log_2^2{n}\right)$ where $n$ is the cardinality of the observed graph vertex set. Thus the polynomial solvability for ${\mathcal NP}$-complete problems is proved.

scholarly journals Representations of Edge Intersection Graphs of Paths in a Tree

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Charles Golumbic ◽  
Marina Lipshteyn ◽  
Michal Stern
Keyword(s):  
Analogous Result ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Network Applications ◽  
Host Trees ◽  
Complete Problems ◽  
Vertex Set ◽  
International Audience ◽  
Np Complete ◽  
Simple Paths

International audience Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.

Networks of Evolutionary Processors

2011 ◽  
pp. 78-114 ◽  
Author(s):  
Carlos Martin-Vide ◽  
Victor Mitrana
Keyword(s):  
Computational Complexity ◽  
Linear Time ◽  
Complexity Class ◽  
Hybrid Networks ◽  
Computational Power ◽  
Complete Problems ◽  
Computational Aspects ◽  
Np Complete ◽  
Traditional Class

The goal of this chapter is to survey, in a systematic and uniform way, the main results regarding different computational aspects of hybrid networks of evolutionary processors viewed both as generating and accepting devices, as well as solving problems with these mechanisms. We first show that generating hybrid networks of evolutionary processors are computationally complete. The same computational power is reached by accepting hybrid networks of evolutionary processors. Then, we define a computational complexity class of accepting these networks and prove that this class equals the traditional class NP. In another section, we present a few NP-complete problems and recall how they can be solved in linear time by accepting networks of evolutionary processors with linearly bounded resources (nodes, rules, symbols). Finally, we discuss some possible directions for further research.

scholarly journals Implicitly Coordinated Multi-Agent Path Finding under Destination Uncertainty: Success Guarantees and Computational Complexity

2019 ◽  
Vol 64 ◽  
pp. 497-527 ◽  
Author(s):  
Bernhard Nebel ◽  
Thomas Bolander ◽  
Thorsten Engesser ◽  
Robert Mattmüller
Keyword(s):  
Computational Complexity ◽  
Common Knowledge ◽  
Path Finding ◽  
Computational Costs ◽  
Complete Problems ◽  
Multi Agent ◽  
Np Complete

In multi-agent path finding (MAPF), it is usually assumed that planning is performed centrally and that the destinations of the agents are common knowledge. We will drop both assumptions and analyze under which conditions it can be guaranteed that the agents reach their respective destinations using implicitly coordinated plans without communication. Furthermore, we will analyze what the computational costs associated with such a coordination regime are. As it turns out, guarantees can be given assuming that the agents are of a certain type. However, the implied computational costs are quite severe. In the distributed setting, we either have to solve a sequence of NP-complete problems or have to tolerate exponentially longer executions. In the setting with destination uncertainty, bounded plan existence becomes PSPACE-complete. This clearly demonstrates the value of communicating about plans before execution starts.

Complexity, computational

2018 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Propositional Logic ◽  
Common Type ◽  
Combinatorial Games ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete ◽  
Feasible Solutions ◽  
Np Problems

The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) and space (amount of memory used). There are problems in logic, algebra and combinatorial games that are solvable in principle by a computer, but computationally intractable because the resources required by relatively small instances are practically infeasible. The theory of NP-completeness concerns a common type of problem in which a solution is easy to check but may be hard to find. Such problems belong to the class NP; the hardest ones of this type are the NP-complete problems. The problem of determining whether a formula of propositional logic is satisfiable or not is NP-complete. The class of problems with feasible solutions is commonly identified with the class P of problems solvable in polynomial time. Assuming this identification, the conjecture that some NP problems require infeasibly long times for their solution is equivalent to the conjecture that P≠NP. Although the conjecture remains open, it is widely believed that NP-complete problems are computationally intractable.

Computational Complexity and Statistical Physics

2005 ◽  
Keyword(s):  
Phase Transitions ◽  
Computational Complexity ◽  
Computer Science ◽  
Statistical Physics ◽  
Combinatorial Problems ◽  
Complete Problems ◽  
Computer Scientists ◽  
Background Material ◽  
Np Complete ◽  
Exciting Area

Computer science and physics have been closely linked since the birth of modern computing. In recent years, an interdisciplinary area has blossomed at the junction of these fields, connecting insights from statistical physics with basic computational challenges. Researchers have successfully applied techniques from the study of phase transitions to analyze NP-complete problems such as satisfiability and graph coloring. This is leading to a new understanding of the structure of these problems, and of how algorithms perform on them. Computational Complexity and Statistical Physics will serve as a standard reference and pedagogical aid to statistical physics methods in computer science, with a particular focus on phase transitions in combinatorial problems. Addressed to a broad range of readers, the book includes substantial background material along with current research by leading computer scientists, mathematicians, and physicists. It will prepare students and researchers from all of these fields to contribute to this exciting area.

Two slightly-entangled NP-complete problems

2005 ◽  
Vol 5 (6) ◽  
pp. 449-455
Author(s):  
R. Orus
Keyword(s):  
Computational Complexity ◽  
Spin System ◽  
Quantum Spin ◽  
Quantum States ◽  
Quantum Spin System ◽  
Quantum Mechanical ◽  
Mathematical Proofs ◽  
Complete Problems ◽  
Entangled Quantum States ◽  
Np Complete

We perform a mathematical analysis of the classical computational complexity of two genuine quantum-mechanical problems, which are inspired in the calculation of the expected magnetizations and the entanglement between subsystems for a quantum spin system. These problems, which we respectively call SES and SESSP, are specified in terms of pure slightly-entangled quantum states of $n$ qubits, and rigorous mathematical proofs that they belong to the NP-Complete complexity class are presented. Both SES and SESSP are, therefore, computationally equivalent to the relevant $3$-SAT problem, for which an efficient algorithm is yet to be discovered.

scholarly journals Relating Vertex and Global Graph Entropy in Randomly Generated Graphs

2018 ◽  
Author(s):  
Philip Tee ◽  
George Parisis ◽  
Luc Berthouze ◽  
Ian Wakeman
Keyword(s):  
Computational Complexity ◽  
Recent Work ◽  
Random Graphs ◽  
Real World ◽  
Close Correlation ◽  
Complete Problem ◽  
Entropy Measures ◽  
Vertex Set ◽  
Np Complete ◽  
The Greedy Algorithm

Combinatoric measures of entropy capture the complexity of a graph, but rely upon the calculation of its independent sets, or collections of non-adjacent vertices. This decomposition of the vertex set is a known NP-Complete problem and for most real world graphs is an inaccessible calculation. Recent work by Dehmer et al. and Tee et al. identified a number of alternative vertex level measures of entropy that do not suffer from this pathological computational complexity. It can be demonstrated that they are still effective at quantifying graph complexity. It is intriguing to consider whether there is a fundamental link between local and global entropy measures. In this paper, we investigate the existence of correlation between vertex level and global measures of entropy, for a narrow subset of random graphs. We use the greedy algorithm approximation for calculating the chromatic information and therefore Körner entropy. We are able to demonstrate close correlation for this subset of graphs and outline how this may arise theoretically.

scholarly journals On the Classification of NP Complete Problems and Their Duality Feature

2018 ◽  
Vol 10 (1) ◽  
pp. 67-78 ◽  
Author(s):  
Wenhong Tian ◽  
Wenxia Guo ◽  
Majun He
Keyword(s):  
Complete Problems ◽  
Np Complete

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.

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