Complement, Complexity, and Symmetric Representation

2015 ◽  
Vol 26 (05) ◽  
pp. 557-581 ◽  
Author(s):  
Thomas E. O'Neil
Keyword(s):  
Time Complexity ◽  
Linear Time ◽  
Vertex Cover ◽  
Independent Set ◽  
Symmetric Representation ◽  
Exponential Time ◽  
Complexity Of Algorithms ◽  
Space Efficiency ◽  
Complete Problems ◽  
Dense Sets

A representation for a set is defined to be symmetric if the space required for the representation of the set is the same as the space required for representation of the set's complement. The use of symmetric representation is shown to be important when studying the time complexity of algorithms. A symmetric data structure called a flip list is defined, and it is employed for the Clique, Independent Set, and Vertex Cover problems in a case study. The classic reductions among these problems require the complement of either a graph's edge set or a subset of its vertices. Flip lists can be complemented in constant time with no increase in space. When a flip list is used to represent the edge set of a graph, Clique, Independent Set, and Vertex Cover are shown to have identical (and strongly exponential) time complexity when the classical complexity parameter of input length is used. On the other hand, when a flip list is used to represent a set of numbers as input for the Partition problem, an algorithm can be built that retains strongly sub-exponential time complexity. This provides new evidence with respect to which NP- complete problems should be classified as sub-exponential. Symmetric representation has the advantage of space efficiency, at most linear-time and space complement operations, and symmetry in representing sparse and dense sets. These features can have a significant impact on complexity studies.

scholarly journals A Linear Time Complexity of Breadth-First Search UsingPSystem with Membrane Division

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Einallah Salehi ◽  
Siti Mariyam Shamsuddin ◽  
Kourosh Nemati
Keyword(s):  
Time Complexity ◽  
Membrane Computing ◽  
Linear Time ◽  
Real Life ◽  
Brute Force ◽  
Breadth First Search ◽  
Exponential Time ◽  
Complete Problems ◽  
Division Rule ◽  
Blind Search

One of the known methods for solving the problems with exponential time complexity such as NP-complete problems is using the brute force algorithms. Recently, a new parallel computational framework called Membrane Computing is introduced which can be applied in brute force algorithms. The usual way to find a solution for the problems with exponential time complexity with Membrane Computing techniques is byPSystem with active membrane using division rule. It makes an exponential workspace and solves the problems with exponential complexity in a polynomial (even linear) time. On the other hand, searching is currently one of the most used methods for finding solution for problems in real life, that the blind search algorithms are accurate, but their time complexity is exponential such as breadth-first search (BFS) algorithm. In this paper, we proposed a new approach for implementation of BFS by usingPsystem with division rule technique for first time. The theorem shows time complexity of BSF in this framework on randomly binary trees reduced fromO(2d)toO(d).

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

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.

scholarly journals Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Yaoxin Li ◽  
Jing Liu ◽  
Guozheng Lin ◽  
Yueyuan Hou ◽  
Muyun Mou ◽  
...  
Keyword(s):  
Objective Function ◽  
Statistical Physics ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Vertex Cover ◽  
Independent Set ◽  
Computational Social Science ◽  
Maximization Problem ◽  
Maximum Independent Set ◽  
Gradient Descent Algorithm

AbstractIn computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

An Area Preserving Transformation Algorithm for Press Forming Blank Development

1993 ◽  
Author(s):  
Nirmal K. Nair ◽  
James H. Oliver
Keyword(s):  
Time Complexity ◽  
Linear Time ◽  
Geometric Interpretation ◽  
Surface Geometry ◽  
Forming Process ◽  
Surface Design ◽  
Design Assessment ◽  
Press Forming ◽  
Blank Shape ◽  
Area Preserving

Abstract An efficient algorithm is presented to determine the blank shape necessary to manufacture a surface by press forming. The technique is independent of material properties and instead uses surface geometry and an area conservation constraint to generate a geometrically feasible blank shape. The algorithm is formulated as an approximate geometric interpretation of the reversal of the forming process. The primary applications for this technique are in preliminary surface design, assessment of manufacturability, and location of binder wrap. Since the algorithm exhibits linear time complexity, it is amenable to implementation as an interactive design aid. The algorithm is applied to two example surfaces and the results are discussed.

scholarly journals Entropy-Penalized Semidefinite Programming

2019 ◽  
Author(s):  
Mikhail Krechetov ◽  
Jakub Marecek ◽  
Yury Maximov ◽  
Martin Takac
Keyword(s):  
Machine Learning ◽  
Time Complexity ◽  
Optimization Problems ◽  
Linear Time ◽  
Broad Class ◽  
Low Rank ◽  
Learning Problems ◽  
Unified Framework ◽  
Gradient Computation ◽  
Machine Learning Applications

Low-rank methods for semi-definite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schatten-norm penalties, which are difficult to implement in practice due to high computational efforts. In this paper, we propose Entropy-Penalized Semi-Definite Programming (EP-SDP), which provides a unified framework for a broad class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit an efficient numerical algorithm, having (almost) linear time complexity of the gradient computation; this makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems.

Further Study on Reverse 1-Center Problem on Trees

2020 ◽  
Vol 37 (06) ◽  
pp. 2050034
Author(s):  
Ali Reza Sepasian ◽  
Javad Tayyebi
Keyword(s):  
Dynamic Programming ◽  
Time Complexity ◽  
Numerical Experiments ◽  
Linear Time ◽  
Minimum Cost ◽  
Fixed Number ◽  
The Other ◽  
Center Problem ◽  
Objective Value ◽  
Linear Cost

This paper studies two types of reverse 1-center problems under uniform linear cost function where edge lengths are allowed to reduce. In the first type, the aim is that the objective value is bounded by a prescribed fixed value [Formula: see text] at minimum cost. The aim of the other is to improve the objective value as much as possible within a given budget. An algorithm based on dynamic programming is proposed to solve the first problem in linear time. Then, this algorithm is applied as a subroutine to design an algorithm to solve the second type of the problem in [Formula: see text] time in which [Formula: see text] is a fixed number dependent on the problem parameters. Under the similarity assumption, this algorithm has a better complexity than the Nguyen algorithm (2013) with quadratic-time complexity. Some numerical experiments are conducted to validate this fact in practice.

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