Solving NP-complete Problems in Polynomial Time by Using a Natural Computing Model

2016 ◽  
pp. 91-108
Author(s):  
Bogdan Aman ◽  
Gabriel Ciobanu
Keyword(s):  
Polynomial Time ◽  
Natural Computing ◽  
Computing Model ◽  
Complete Problems ◽  
Np Complete

scholarly journals Tuning Frontiers of Efficiency in Tissue P Systems with Evolutional Communication Rules

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
David Orellana-Martín ◽  
Luis Valencia-Cabrera ◽  
Bosheng Song ◽  
Linqiang Pan ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Polynomial Time ◽  
Efficient Solution ◽  
Efficient Solutions ◽  
Affirmative Answer ◽  
P Systems ◽  
Membrane Systems ◽  
Complete Problems ◽  
Np Complete ◽  
Np Problem ◽  
Computing Models

Over the last few years, a new methodology to address the P versus NP problem has been developed, based on searching for borderlines between the nonefficiency of computing models (only problems in class P can be solved in polynomial time) and the presumed efficiency (ability to solve NP-complete problems in polynomial time). These borderlines can be seen as frontiers of efficiency, which are crucial in this methodology. “Translating,” in some sense, an efficient solution in a presumably efficient model to an efficient solution in a nonefficient model would give an affirmative answer to problem P versus NP. In the framework of Membrane Computing, the key of this approach is to detect the syntactic or semantic ingredients that are needed to pass from a nonefficient class of membrane systems to a presumably efficient one. This paper deals with tissue P systems with communication rules of type symport/antiport allowing the evolution of the objects triggering the rules. In previous works, frontiers of efficiency were found in these kinds of membrane systems both with division rules and with separation rules. However, since they were not optimal, it is interesting to refine these frontiers. In this work, optimal frontiers of the efficiency are obtained in terms of the total number of objects involved in the communication rules used for that kind of membrane systems. These optimizations could be easier to translate, if possible, to efficient solutions in a nonefficient model.

Membrane Computing

2011 ◽  
pp. 1-31 ◽  
Author(s):  
Gheorghe Paun
Keyword(s):  
Membrane Computing ◽  
Living Cells ◽  
Point Of View ◽  
Natural Computing ◽  
Historical Evolution ◽  
Biological Phenomena ◽  
Complete Problems ◽  
Hard Problems ◽  
Np Complete ◽  
And Mathematics

Membrane computing is a branch of natural computing whose initial goal was to abstract computing models from the structure and the functioning of living cells. The research was initiated about five years ago (at the end of 1998), and since that time the area has been developed significantly from a mathematical point of view. The basic types of results of this research concern the computability power (in comparison with the standard Turing machines and their restrictions) and the efficiency (the possibility to solve computationally hard problems, typically NP-complete problems, in a feasible time and typically polynomial). However, membrane computing has recently become attractive also as a framework for devising models of biological phenomena, with the tendency to provide tools for modelling the cell itself, not only the local processes. This chapter surveys the basic elements of membrane computing, somewhat in its “historical” evolution: from biology to computer science and mathematics and back to biology. The presentation is informal, without any technical detail, and an invitation to membrane computing intended to acquaint the nonmathematician reader with the main directions of research of the domain, the type of central results, and the possible lines of future development, including the possible interest of the biologist looking for discrete algorithmic tools for modelling cell phenomena.

EDGE-DELETION GRAPH PROBLEMS WITH FIRST-ORDER EXPRESSIBLE SUBGRAPH PROPERTIES

1991 ◽  
Vol 02 (02) ◽  
pp. 83-99
Author(s):  
V. ARVIND ◽  
S. BISWAS
Keyword(s):  
Normal Form ◽  
Polynomial Time ◽  
Edge Deletion ◽  
Present Evidence ◽  
First Order ◽  
Graph Problems ◽  
Polynomial Time Algorithms ◽  
Complete Problems ◽  
Np Complete ◽  
New Perspective

In this paper edge-deletion problems are studied with a new perspective. In general an edge-deletion problem is of the form: Given a graph G, does it have a subgraph H obtained by deleting zero or more edges such that H satisfies a polynomial-time verifiable property? This paper restricts attention to first-order expressible properties. If the property is expressed by π, which in prenex normal form is Q(Φ) where Q is the quantifier-prefix, then we prove results on the quantifier structure that characterize the complexity of the edge-deletion problem. In particular we give polynomial-time algorithms for problems for which Q is ‘simple’ and in other cases we encode certain NP-complete problems as edge-deletion problems, essentially using the quantifier structure of π. We also present evidence that Q alone cannot capture the complexity of the edge-deletion problem.

scholarly journals Enhanced Membrane Computing Algorithm for SAT Problems Based on the Splitting Rule

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1412
Author(s):  
Hao ◽  
Liu
Keyword(s):  
Membrane Computing ◽  
Space Complexity ◽  
Natural Computing ◽  
Sat Problem ◽  
Computing Algorithm ◽  
Complete Problems ◽  
Np Complete ◽  
Splitting Rule ◽  
Np Problems ◽  
Time And Space Complexity

Boolean propositional satisfiability (SAT) problem is one of the most widely studied NP-complete problems and plays an outstanding role in many domains. Membrane computing is a branch of natural computing which has been proven to solve NP problems in polynomial time with a parallel compute mode. This paper proposes a new algorithm for SAT problem which combines the traditional membrane computing algorithm of SAT problem with a classic simplification rule, the splitting rule, which can divide a clause set into two axisymmetric subsets, deal with them respectively and simultaneously, and obtain the solution of the original clause set with the symmetry of their solutions. The new algorithm is shown to be able to reduce the space complexity by distributing clauses with the splitting rule repeatedly, and also reduce both time and space complexity by executing one-literal rule and pure-literal rule as many times as possible.

scholarly journals Solving Vertex Cover Problem by Means of Tissue P Systems with Cell Separation

2010 ◽  
Vol 5 (4) ◽  
pp. 540 ◽  
Author(s):  
Chun Lu ◽  
Xingyi Zhang
Keyword(s):  
Cell Separation ◽  
Membrane Computing ◽  
Linear Time ◽  
Vertex Cover ◽  
P Systems ◽  
Computing Model ◽  
Vertex Cover Problem ◽  
Complete Problems ◽  
Np Complete ◽  
Cover Problem

Tissue P systems is a computing model in the framework of membrane computing inspired from intercellular communication and cooperation between neurons. Many different variants of this model have been proposed. One of the most important models is known as tissue P systems with cell separation. This model has the ability of generating an exponential amount of workspace in linear time, thus it allows us to design cellular solutions to NP-complete problems in polynomial time. In this paper, we present a solution to the Vertex Cover problem via a family of such devices. This is the first solution to this problem in the framework of tissue P systems with cell separation.

Cellular Solutions to Some Numerical NP-Complete Problems

2011 ◽  
pp. 115-149 ◽  
Author(s):  
Andrés Cordón-Franco ◽  
Miguel A. Gutiérrez-Naranjo ◽  
Mario J. Pérez-Jiménez ◽  
Agustín Riscos-Núñez
Keyword(s):  
Polynomial Time ◽  
Efficient Solutions ◽  
Cellular Systems ◽  
P Systems ◽  
Knapsack Problems ◽  
Simulation Tool ◽  
Working Space ◽  
Complete Problems ◽  
Subset Sum ◽  
Np Complete

This chapter is devoted to the study of numerical NP-complete problems in the framework of cellular systems with membranes, also called P systems (Pun, 1998). The chapter presents efficient solutions to the subset sum and the knapsack problems. These solutions are obtained via families of P systems with the capability of generating an exponential working space in polynomial time. A simulation tool for P systems, written in Prolog, is also described. As an illustration of the use of this tool, the chapter includes a session in the Prolog simulator implementing an algorithm to solve one of the above problems.

scholarly journals Average polynomial time complexity of some NP-complete problems

1986 ◽  
Vol 46 ◽  
pp. 219-237 ◽  
Author(s):  
Phan Dinh Dieu ◽  
Le Cong Thanh ◽  
Le Tuan Hoa
Keyword(s):  
Polynomial Time ◽  
Time Complexity ◽  
Polynomial Time Complexity ◽  
Complete Problems ◽  
Np Complete

scholarly journals Towards algorithm-free physical equilibrium model of computing

2021 ◽  
Vol 21 (15&16) ◽  
pp. 1296-1306
Author(s):  
Seyed Mousavi
Keyword(s):  
Quantum Circuits ◽  
Physical Processes ◽  
Physical Systems ◽  
The Past ◽  
Computing Model ◽  
Computing Paradigm ◽  
Complete Problems ◽  
Proposed Model ◽  
Np Complete ◽  
Past Half

Our computers today, from sophisticated servers to small smartphones, operate based on the same computing model, which requires running a sequence of discrete instructions, specified as an algorithm. This sequential computing paradigm has not yet led to a fast algorithm for an NP-complete problem despite numerous attempts over the past half a century. Unfortunately, even after the introduction of quantum mechanics to the world of computing, we still followed a similar sequential paradigm, which has not yet helped us obtain such an algorithm either. Here a completely different model of computing is proposed to replace the sequential paradigm of algorithms with inherent parallelism of physical processes. Using the proposed model, instead of writing algorithms to solve NP-complete problems, we construct physical systems whose equilibrium states correspond to the desired solutions and let them evolve to search for the solutions. The main requirements of the model are identified and quantum circuits are proposed for its potential implementation.

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