scholarly journals Optimization Problems over Unit-Distance Representations of Graphs

10.37236/3128 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Marcel Kenji De Carli Silva ◽  
Levent Tunçel
Keyword(s):  
Optimization Problems ◽  
Positive Semidefinite ◽  
Weighted Version ◽  
Unit Distance ◽  
Graph Homomorphisms ◽  
Hard Problems ◽  
Theta Number ◽  
The Relationship ◽  
Lovász Theta Number ◽  
Np Hard Problems

We study the relationship between unit-distance representations and the Lovász theta number of graphs, originally established by Lovász. We derive and prove min-max theorems. This framework allows us to derive a weighted version of the hypersphere number of a graph and a related min-max theorem. Then, we connect to sandwich theorems via graph homomorphisms. We present and study a generalization of the hypersphere number of a graph and the related optimization problems. The generalized problem involves finding the smallest ellipsoid of a given shape which contains a unit-distance representation of the graph. Arbitrary positive semidefinite forms describing the ellipsoids yield NP-hard problems.

scholarly journals Coupling Elephant Herding with Ordinal Optimization for Solving the Stochastic Inequality Constrained Optimization Problems

2020 ◽  
Vol 10 (6) ◽  
pp. 2075 ◽  
Author(s):  
Shih-Cheng Horng ◽  
Shieh-Shing Lin
Keyword(s):  
Optimization Problems ◽  
Solution Space ◽  
Inequality Constraints ◽  
Optimization Methods ◽  
Ordinal Optimization ◽  
Np Hard ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Hard Problems ◽  
Np Hard Problems

The stochastic inequality constrained optimization problems (SICOPs) consider the problems of optimizing an objective function involving stochastic inequality constraints. The SICOPs belong to a category of NP-hard problems in terms of computational complexity. The ordinal optimization (OO) method offers an efficient framework for solving NP-hard problems. Even though the OO method is helpful to solve NP-hard problems, the stochastic inequality constraints will drastically reduce the efficiency and competitiveness. In this paper, a heuristic method coupling elephant herding optimization (EHO) with ordinal optimization (OO), abbreviated as EHOO, is presented to solve the SICOPs with large solution space. The EHOO approach has three parts, which are metamodel construction, diversification and intensification. First, the regularized minimal-energy tensor-product splines is adopted as a metamodel to approximately evaluate fitness of a solution. Next, an improved elephant herding optimization is developed to find N significant solutions from the entire solution space. Finally, an accelerated optimal computing budget allocation is utilized to select a superb solution from the N significant solutions. The EHOO approach is tested on a one-period multi-skill call center for minimizing the staffing cost, which is formulated as a SICOP. Simulation results obtained by the EHOO are compared with three optimization methods. Experimental results demonstrate that the EHOO approach obtains a superb solution of higher quality as well as a higher computational efficiency than three optimization methods.

scholarly journals An overview on polynomial approximation of NP-hard problems

2009 ◽  
Vol 19 (1) ◽  
pp. 3-40 ◽  
Author(s):  
Vangelis Paschos
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Approximation ◽  
Optimization Problems ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Computational Time ◽  
Np Hard ◽  
Hard Problems ◽  
Feasible Solutions ◽  
Np Hard Problems

The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.

scholarly journals Embedding Ordinal Optimization into Tree–Seed Algorithm for Solving the Probabilistic Constrained Simulation Optimization Problems

2018 ◽  
Vol 8 (11) ◽  
pp. 2153 ◽  
Author(s):  
Shih-Cheng Horng ◽  
Shieh-Shing Lin
Keyword(s):  
Optimization Problems ◽  
Simulation Optimization ◽  
Inequality Constraint ◽  
Search Space ◽  
Ordinal Optimization ◽  
Np Hard ◽  
Heuristic Approaches ◽  
Probabilistic Inequality ◽  
Hard Problems ◽  
Np Hard Problems

Probabilistic constrained simulation optimization problems (PCSOP) are concerned with allocating limited resources to achieve a stochastic objective function subject to a probabilistic inequality constraint. The PCSOP are NP-hard problems whose goal is to find optimal solutions using simulation in a large search space. An efficient “Ordinal Optimization (OO)” theory has been utilized to solve NP-hard problems for determining an outstanding solution in a reasonable amount of time. OO theory to solve NP-hard problems is an effective method, but the probabilistic inequality constraint will greatly decrease the effectiveness and efficiency. In this work, a method that embeds ordinal optimization (OO) into tree–seed algorithm (TSA) (OOTSA) is firstly proposed for solving the PCSOP. The OOTSA method consists of three modules: surrogate model, exploration and exploitation. Then, the proposed OOTSA approach is applied to minimize the expected lead time of semi-finished products in a pull-type production system, which is formulated as a PCSOP that comprises a well-defined search space. Test results obtained by the OOTSA are compared with the results obtained by three heuristic approaches. Simulation results demonstrate that the OOTSA method yields an outstanding solution of much higher computing efficiency with much higher quality than three heuristic approaches.

scholarly journals A Novel Metaheuristic for Travelling Salesman Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Vahid Zharfi ◽  
Abolfazl Mirzazadeh
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Travelling Salesman Problem ◽  
Network Routing ◽  
Np Hard ◽  
Travelling Salesman ◽  
Combinatorial Optimization Problems ◽  
Hard Problems ◽  
Insight Into ◽  
Np Hard Problems

One of the well-known combinatorial optimization problems is travelling salesman problem (TSP). This problem is in the fields of logistics, transportation, and distribution. TSP is among the NP-hard problems, and many different metaheuristics are used to solve this problem in an acceptable time especially when the number of cities is high. In this paper, a new meta-heuristic is proposed to solve TSP which is based on new insight into network routing problems.

scholarly journals Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

2021 ◽  
Author(s):  
Stefano Massei
Keyword(s):  
Computer Science ◽  
Recent Work ◽  
Linear Algebra ◽  
Optimization Problems ◽  
Approximation Error ◽  
Positive Semidefinite ◽  
Numerical Linear Algebra ◽  
Maximum Volume ◽  
Deterministic Algorithms ◽  
Principal Submatrix

AbstractVarious applications in numerical linear algebra and computer science are related to selecting the $$r\times r$$ r × r submatrix of maximum volume contained in a given matrix $$A\in \mathbb R^{n\times n}$$ A ∈ R n × n . We propose a new greedy algorithm of cost $$\mathcal O(n)$$ O ( n ) , for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $$(r+1)$$ ( r + 1 ) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost $$\mathcal O(n^3)$$ O ( n 3 ) .

scholarly journals Nanolaser-based emulators of spin Hamiltonians

Nanophotonics ◽  
2020 ◽  
Vol 9 (13) ◽  
pp. 4193-4198 ◽  
Author(s):  
Midya Parto ◽  
William E. Hayenga ◽  
Alireza Marandi ◽  
Demetrios N. Christodoulides ◽  
Mercedeh Khajavikhan
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Exchange Interactions ◽  
Np Hard ◽  
Spin Models ◽  
Geometric Frustration ◽  
Spin Hamiltonians ◽  
Hard Problems ◽  
On Chip ◽  
Classical Spin Models

AbstractFinding the solution to a large category of optimization problems, known as the NP-hard class, requires an exponentially increasing solution time using conventional computers. Lately, there has been intense efforts to develop alternative computational methods capable of addressing such tasks. In this regard, spin Hamiltonians, which originally arose in describing exchange interactions in magnetic materials, have recently been pursued as a powerful computational tool. Along these lines, it has been shown that solving NP-hard problems can be effectively mapped into finding the ground state of certain types of classical spin models. Here, we show that arrays of metallic nanolasers provide an ultra-compact, on-chip platform capable of implementing spin models, including the classical Ising and XY Hamiltonians. Various regimes of behavior including ferromagnetic, antiferromagnetic, as well as geometric frustration are observed in these structures. Our work paves the way towards nanoscale spin-emulators that enable efficient modeling of large-scale complex networks.

scholarly journals A characterization of the weighted version of McEliece–Rodemich–Rumsey–Schrijver number based on convex quadratic programming

2015 ◽  
Vol 07 (04) ◽  
pp. 1550050
Author(s):  
Carlos J. Luz
Keyword(s):  
Quadratic Programming ◽  
Discrete Math ◽  
Convex Quadratic Programming ◽  
Stability Number ◽  
Weighted Version ◽  
Similar Characterization ◽  
Number Formula ◽  
Weighted Stability ◽  
Theta Number

For any graph [Formula: see text] Luz and Schrijver [A convex quadratic characterization of the Lovász theta number, SIAM J. Discrete Math. 19(2) (2005) 382–387] introduced a characterization of the Lovász number [Formula: see text] based on convex quadratic programming. A similar characterization is now established for the weighted version of the number [Formula: see text] independently introduced by McEliece, Rodemich, and Rumsey [The Lovász bound and some generalizations, J. Combin. Inform. Syst. Sci. 3 (1978) 134–152] and Schrijver [A Comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory 25(4) (1979) 425–429]. Also, a class of graphs for which the weighted version of [Formula: see text] coincides with the weighted stability number is characterized.

scholarly journals In some curved spaces, one can solve NP-hard problems in polynomial time

2009 ◽  
Vol 158 (5) ◽  
pp. 727-740 ◽  
Author(s):  
V. Kreinovich ◽  
M. Margenstern
Keyword(s):  
Polynomial Time ◽  
Np Hard ◽  
Curved Spaces ◽  
Hard Problems ◽  
Np Hard Problems

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.

Approximating NP-hard Problems

2006 ◽  
pp. 353-363
Keyword(s):  
Np Hard ◽  
Hard Problems ◽  
Np Hard Problems
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