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.

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.

scholarly journals An Average Case NP-complete Graph Colouring Problem

2018 ◽  
Vol 27 (5) ◽  
pp. 808-828 ◽  
Author(s):  
LEONID A. LEVIN ◽  
RAMARATHNAM VENKATESAN
Keyword(s):  
Random Graphs ◽  
Polynomial Time ◽  
Complete Graph ◽  
Graph Colouring ◽  
Worst Case ◽  
Average Case ◽  
Graph Problem ◽  
Complete Problems ◽  
Np Complete ◽  
Np Problems

NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proved to be easy. We show the intractability of random instances of a graph colouring problem: this graph problem is hard on average unless all NP problems under all samplable (i.e. generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities. This poses significant technical difficulties.

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.

scholarly journals Solving Combinatorial Optimization Problems on Quantum Computers

2020 ◽  
pp. 5-13
Author(s):  
Vyacheslav Korolyov ◽  
Oleksandr Khodzinskyi
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Quantum Computer ◽  
Independent Set ◽  
Cloud Services ◽  
Quantum Computers ◽  
Maximal Independent Set ◽  
Maximum Independent Set ◽  
Combinatorial Optimization Problems ◽  
D Wave

Introduction. Quantum computers provide several times faster solutions to several NP-hard combinatorial optimization problems in comparison with computing clusters. The trend of doubling the number of qubits of quantum computers every year suggests the existence of an analog of Moore's law for quantum computers, which means that soon they will also be able to get a significant acceleration of solving many applied large-scale problems. The purpose of the article is to review methods for creating algorithms of quantum computer mathematics for combinatorial optimization problems and to analyze the influence of the qubit-to-qubit coupling and connections strength on the performance of quantum data processing. Results. The article offers approaches to the classification of algorithms for solving these problems from the perspective of quantum computer mathematics. It is shown that the number and strength of connections between qubits affect the dimensionality of problems solved by algorithms of quantum computer mathematics. It is proposed to consider two approaches to calculating combinatorial optimization problems on quantum computers: universal, using quantum gates, and specialized, based on a parameterization of physical processes. Examples of constructing a half-adder for two qubits of an IBM quantum processor and an example of solving the problem of finding the maximum independent set for the IBM and D-wave quantum computers are given. Conclusions. Today, quantum computers are available online through cloud services for research and commercial use. At present, quantum processors do not have enough qubits to replace semiconductor computers in universal computing. The search for a solution to a combinatorial optimization problem is performed by achieving the minimum energy of the system of coupled qubits, on which the task is mapped, and the data are the initial conditions. Approaches to solving combinatorial optimization problems on quantum computers are considered and the results of solving the problem of finding the maximum independent set on the IBM and D-wave quantum computers are given. Keywords: quantum computer, quantum computer mathematics, qubit, maximal independent set for a graph.

A Genetic Model: Analysis and Application to MAXSAT

2000 ◽  
Vol 8 (3) ◽  
pp. 291-309 ◽  
Author(s):  
Alberto Bertoni ◽  
Marco Carpentieri ◽  
Paola Campadelli ◽  
Giuliano Grossi
Keyword(s):  
Combinatorial Optimization ◽  
Genetic Model ◽  
Optimization Problems ◽  
Case Analysis ◽  
Optimization Techniques ◽  
Worst Case Analysis ◽  
Worst Case ◽  
Average Case ◽  
Combinatorial Optimization Problems

In this paper, a genetic model based on the operations of recombination and mutation is studied and applied to combinatorial optimization problems. Results are: The equations of the deterministic dynamics in the thermodynamic limit (infinite populations) are derived and, for a sufficiently small mutation rate, the attractors are characterized; A general approximation algorithm for combinatorial optimization problems is designed. The algorithm is applied to the Max Ek-Sat problem, and the quality of the solution is analyzed. It is proved to be optimal for k≥3 with respect to the worst case analysis; for Max E3-Sat the average case performances are experimentally compared with other optimization techniques.

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