scholarly journals Identification Conditions for the Solvability of NP-complete Problems for the Class of Pre-fractal Graphs

2021 ◽  
Vol 28 (2) ◽  
pp. 126-135
Author(s):  
Aleksandr Vasil'evich Tymoshenko ◽  
Rasul Ahmatovich Kochkarov ◽  
Azret Ahmatovich Kochkarov
Keyword(s):  
Communication Networks ◽  
Hamiltonian Cycle ◽  
Optimization Problems ◽  
Independent Set ◽  
Network Systems ◽  
Logistics Networks ◽  
Complete Problems ◽  
Discrete Problems ◽  
Np Complete ◽  
Discrete Optimization Problems

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.

Different adiabatic quantum optimization algorithms

2011 ◽  
Vol 11 (7&8) ◽  
pp. 638-648
Author(s):  
Vicky Choi
Keyword(s):  
Optimization Problems ◽  
Quantum Computer ◽  
Independent Set ◽  
Worst Case ◽  
Average Case ◽  
Negative Results ◽  
Complete Problems ◽  
Np Complete ◽  
Quantum Optimization ◽  
Exact Cover

One of the most important questions in studying quantum computation is: whether a quantum computer can solve NP-complete problems more efficiently than a classical computer? In 2000, Farhi, et al. (Science, 292(5516):472--476, 2001) proposed the adiabatic quantum optimization (AQO), a paradigm that directly attacks NP-hard optimization problems. How powerful is AQO? Early on, van-Dam and Vazirani claimed that AQO failed (i.e. would take exponential time) for a family of 3SAT instances they constructed. More recently, Altshuler, et al. (Proc Natl Acad Sci USA, 107(28): 12446--12450, 2010) claimed that AQO failed also for random instances of the NP-complete Exact Cover problem. In this paper, we make clear that all these negative results are only for a specific AQO algorithm. We do so by demonstrating different AQO algorithms for the same problem for which their arguments no longer hold. Whether AQO fails or succeeds for solving the NP-complete problems (either the worst case or the average case) requires further investigation. Our AQO algorithms for Exact Cover and 3SAT are based on the polynomial reductions to the NP-complete Maximum-weight Independent Set (MIS) problem.

A Hybrid Approach of Heuristic and Neural Network for Packing Problems

1994 ◽  
Author(s):  
Zuo Dai ◽  
Jianzhong Cha
Keyword(s):  
Neural Network ◽  
Large Scale ◽  
Optimization Problems ◽  
Travelling Salesman Problem ◽  
Hybrid Approach ◽  
Two Dimensions ◽  
Packing Problems ◽  
Combinatorial Optimization Problems ◽  
Complete Problems ◽  
Np Complete

Abstract Artificial Neural Networks, particularly the Hopfield-Tank network, have been effectively applied to the solution of a variety of tasks formulated as large scale combinatorial optimization problems, such as Travelling Salesman Problem and N Queens Problem [1]. The problem of optimally packing a set of geometries into a space with finite dimensions arises frequently in many applications and is far difficult than general NP-complete problems listed in [2]. Until now within accepted time limit, it can only be solved with heuristic methods for very simple cases (e.g. 2D layout). In this paper we propose a heuristic-based Hopfield neural network designed to solve the rectangular packing problems in two dimensions, which is still NP-complete [3]. By comparing the adequacy and efficiency of the results with that obtained by several other exact and heuristic approaches, it has been concluded that the proposed method has great potential in solving 2D packing problems.

Solving the Independent-set Problem in a DNA-Based Supercomputer Model

2005 ◽  
Vol 15 (04) ◽  
pp. 469-479 ◽  
Author(s):  
WENG-LONG CHANG ◽  
MINYI GUO ◽  
JESSE WU
Keyword(s):  
Genetic Algorithm ◽  
Parallel Computing ◽  
Deoxyribonucleic Acid ◽  
Independent Set ◽  
Parallel Genetic Algorithm ◽  
Independent Set Problem ◽  
Complete Problems ◽  
Np Complete

In this paper, it is demonstrated how the DNA (DeoxyriboNucleic Acid) operations presented by Adleman and Lipton can be used to develop the parallel genetic algorithm that solves the independent-set problem. The advantage of the genetic algorithm is the huge parallelism inherent in DNA based computing. Furthermore, this work represents obvious evidence for the ability of DNA based parallel computing to solve NP-complete problems.

scholarly journals Recognizing Maximal Unfrozen Graphs with respect to Independent Sets is CO-NP-complete

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Nesrine Abbas ◽  
Joseph Culberson ◽  
Lorna Stewart
Keyword(s):  
Independent Set ◽  
Independent Sets ◽  
Complete Problems ◽  
International Audience ◽  
Np Complete

International audience A graph is unfrozen with respect to k independent set if it has an independent set of size k after the addition of any edge. The problem of recognizing such graphs is known to be NP-complete. A graph is maximal if the addition of one edge means it is no longer unfrozen. We designate the problem of recognizing maximal unfrozen graphs as MAX(U(k-SET)) and show that this problem is CO-NP-complete. This partially fills a gap in known complexity cases of maximal NP-complete problems, and raises some interesting open conjectures discussed in the conclusion.

On a family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation

2019 ◽  
Vol 29 (1) ◽  
pp. 7-14
Author(s):  
Aleksandr N. Maksimenko
Keyword(s):  
Travelling Salesman Problem ◽  
Independent Set ◽  
Packing Problem ◽  
Set Partitioning ◽  
Partial Ordering ◽  
Set Covering ◽  
Covering Problem ◽  
Maximum Independent Set ◽  
Complete Problems ◽  
Np Complete

Abstract In 1995 T. Matsui considered a special family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation. In 2012 the author demonstrated that all polytopes of this family appear as faces of polytopes associated with the following NP-complete problems: the travelling salesman problem, the 3-satisfiability problem, the knapsack problem, the set covering problem, the partial ordering problem, the cube subgraph problem, and some others. Here it is shown that none of the polytopes of the aforementioned special family (with the exception of the one-dimensional segment) can appear as a face in a polytope associated with the problem of the maximum independent set, the set packing problem, the set partitioning problem, and the problem of 3-assignments.

scholarly journals THE 2-LOCAL HAMILTONIAN PROBLEM ENCOMPASSES NP

2003 ◽  
Vol 01 (03) ◽  
pp. 349-357 ◽  
Author(s):  
PAWEL WOCJAN ◽  
THOMAS BETH
Keyword(s):  
Independent Set ◽  
Complexity Class ◽  
Complete Problems ◽  
Quantum Analog ◽  
Np Complete ◽  
Quantum Complexity

We show that the NP-complete problems max cut and independent set can be formulated as the 2-local Hamiltonian problem as defined by Kitaev. The 5-local Hamiltonian problem was the first problem to be shown to be complete for the quantum complexity class QMA — the quantum analog of NP. Subsequently, it was shown that 3-locality is already sufficient for QMA-completeness. It is still not known whether the 2-local Hamiltonian problem is QMA-complete. Therefore it is interesting to determine what problems can be reduced to the 2-local Hamiltonian problem. Kitaev showed that 3-SAT can be formulated as a 3-local Hamiltonian problem. We extend his result by showing that 2-locality is sufficient in order to encompass NP.

Sign in / Sign up

Export Citation Format

Share Document