Looking for Reverse Transformations between NP-Complete Problems

2012 ◽  
pp. 181-206
Author(s):  
Rodolfo A.Pazos R. ◽  
Ernesto Ong C. ◽  
Héctor Fraire H. ◽  
Laura Cruz R. ◽  
José A.Martínez F.
Keyword(s):  
Polynomial Time ◽  
Bin Packing ◽  
Job Scheduling ◽  
Reverse Transformation ◽  
Time Transformation ◽  
Complete Class ◽  
Np Completeness ◽  
Decision Optimization ◽  
Complete Problems ◽  
Np Complete

The theory of NP-completeness provides a method for telling whether a decision/optimization problem is “easy” (i.e., it belongs to the P class) or “difficult” (i.e., it belongs to the NP-complete class). Many problems related to logistics have been proven to belong to the NP-complete class such as Bin Packing, job scheduling, timetabling, etc. The theory predicts that for any pair of NP-complete problems A and B there must exist a polynomial time transformation from A to B and also a reverse transformation (from B to A). However, for many pairs of NP-complete problems no reverse transformation has been reported in the literature; thus the following question arises: do reverse transformations exist for any pair of NP-complete problems? This chapter presents results on an ongoing investigation for clarifying this issue.

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.

NP-completeness of combinatorial problems with unary notation for integers

1980 ◽  
Vol 3 (3) ◽  
pp. 397-400
Author(s):  
Martti Penttonen
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Combinatorial Problems ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete

Most NP-complete problems remain NP-complete even though the notation for integers is changed to unary. The knapsack problem is an exception, it becomes provably polynomial time recognizable. However, we present a modified knapsack problem that remains NP-complete also in unary notation.

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.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

The Application of DNA Self-Assembly Model for Bin Packing Problem

2012 ◽  
Vol 3 (1) ◽  
pp. 1-15
Author(s):  
Yanfeng Wang ◽  
Xuewen Bai ◽  
Donghui Wei ◽  
Weili Lu ◽  
Guangzhao Cui
Keyword(s):  
Self Assembly ◽  
Bin Packing ◽  
Formal Model ◽  
Linear Time ◽  
Packing Problem ◽  
Assembly Model ◽  
Dna Tiles ◽  
Bin Packing Problem ◽  
Complete Problems ◽  
Np Complete

Bin Packing Problem (BPP) is a classical combinatorial optimization problem of graph theory, which has been proved to be NP-complete, and has high computational complexity. DNA self-assembly, a formal model of crystal growth, has been proposed as a mechanism for the bottom-up fabrication of autonomous DNA computing. In this paper, the authors propose a DNA self-assembly model for solving the BPP, this model consists of two units: grouping based on binary method and subtraction system. The great advantage of the model is that the number of DNA tile types used in the model is constant and it can solve any BPP within linear time. This work demonstrates the ability of DNA tiles to solve other NP-complete problems in the future.

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.

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
Sign in / Sign up

Export Citation Format

Share Document