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.

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 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 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.

Networks of Evolutionary Processors

2011 ◽  
pp. 78-114 ◽  
Author(s):  
Carlos Martin-Vide ◽  
Victor Mitrana
Keyword(s):  
Computational Complexity ◽  
Linear Time ◽  
Complexity Class ◽  
Hybrid Networks ◽  
Computational Power ◽  
Complete Problems ◽  
Computational Aspects ◽  
Np Complete ◽  
Traditional Class

The goal of this chapter is to survey, in a systematic and uniform way, the main results regarding different computational aspects of hybrid networks of evolutionary processors viewed both as generating and accepting devices, as well as solving problems with these mechanisms. We first show that generating hybrid networks of evolutionary processors are computationally complete. The same computational power is reached by accepting hybrid networks of evolutionary processors. Then, we define a computational complexity class of accepting these networks and prove that this class equals the traditional class NP. In another section, we present a few NP-complete problems and recall how they can be solved in linear time by accepting networks of evolutionary processors with linearly bounded resources (nodes, rules, symbols). Finally, we discuss some possible directions for further research.

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.

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.

Application of formal languages in polynomial transformations of instances between NP-complete problems

2013 ◽  
Vol 14 (8) ◽  
pp. 623-633
Author(s):  
Jorge A. Ruiz-Vanoye ◽  
Joaquín Pérez-Ortega ◽  
Rodolfo A. Pazos Rangel ◽  
Ocotlán Díaz-Parra ◽  
Héctor J. Fraire-Huacuja ◽  
...  
Keyword(s):  
Formal Languages ◽  
Complete Problems ◽  
Polynomial Transformations ◽  
Np Complete

Logical Characterizations of Bounded Query Classes I: Logspace Oracle Machines

1993 ◽  
Vol 18 (1) ◽  
pp. 65-92
Author(s):  
Iain A. Stewart
Keyword(s):  
Truth Table ◽  
Complexity Class ◽  
Polynomial Hierarchy ◽  
Complexity Classes ◽  
Turing Machines ◽  
Complete Problems

We consider three sub-logics of the logic (±HP)*[FOs] and show that these sub-logics capture the complexity classes obtained by considering logspace deterministic oracle Turing machines with oracles in NP where the number of oracle calls is unrestricted and constant, respectively; that is, the classes LNP and LNP[O(1)]. We conclude that if certain logics are of the same expressibility then the Polynomial Hierarchy collapses. We also exhibit some new complete problems for the complexity class LNP via projection translations (the first to be discovered: projection translations are extremely weak logical reductions between problems) and characterize the complexity class LNP[O(1)] as the closure of NP under a new, extremely strict truth-table reduction (which we introduce in this paper).

NP vs QMA_\log(2)

2010 ◽  
Vol 10 (1&2) ◽  
pp. 141-151
Author(s):  
S. Beigi
Keyword(s):  
Quantum Algorithms ◽  
Np Hard ◽  
Complete Problems ◽  
Hard Problems ◽  
Np Complete ◽  
Np Hard Problems

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve NP-complete problems given a "short" quantum proof; more precisely, NP\subseteq QMA_{\log}(2) where QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion NP\subseteq QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 1/24n^6. Moreover, Aaronson et al. have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if QMA_{\log}(2) with a constant gap contains NP. In this paper, we show that 3-SAT admits a QMA_{\log}(2) protocol with the gap 1/n^{3+\epsilon}} for every constant \epsilon>0.

Sign in / Sign up

Export Citation Format

Share Document