scholarly journals The Power of Human–Algorithm Collaboration in Solving Combinatorial Optimization Problems

Algorithms ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 253
Author(s):  
Tapani Toivonen ◽  
Markku Tukiainen
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Maximum Clique ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Problems ◽  
Combinatorial Optimization Problems ◽  
Multiplicative Factor ◽  
Intractable Problems ◽  
Polynomial Factor

Many combinatorial optimization problems are often considered intractable to solve exactly or by approximation. An example of such a problem is maximum clique, which—under standard assumptions in complexity theory—cannot be solved in sub-exponential time or be approximated within the polynomial factor efficiently. However, we show that if a polynomial time algorithm can query informative Gaussian priors from an expert poly(n) times, then a class of combinatorial optimization problems can be solved efficiently up to a multiplicative factor ϵ, where ϵ is arbitrary constant. In this paper, we present proof of our claims and show numerical results to support them. Our methods can cast new light on how to approach optimization problems in domains where even the approximation of the problem is not feasible. Furthermore, the results can help researchers to understand the structures of these problems (or whether these problems have any structure at all!). While the proposed methods can be used to approximate combinatorial problems in NPO, we note that the scope of the problems solvable might well include problems that are provable intractable (problems in EXPTIME).

scholarly journals Reducing the Size of Combinatorial Optimization Problems Using the Operator Vaccine by Fuzzy Selector with Adaptive Heuristics

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Oscar Montiel ◽  
Francisco Javier Díaz Delgadillo
Keyword(s):  
Combinatorial Optimization ◽  
Fuzzy Inference ◽  
Optimization Problems ◽  
Travelling Salesman Problem ◽  
Combinatorial Problems ◽  
Reduction Step ◽  
Problem Size ◽  
Inference System ◽  
Combinatorial Optimization Problems ◽  
Adaptive Heuristics

Nowadays, solving optimally combinatorial problems is an open problem. Determining the best arrangement of elements proves being a very complex task that becomes critical when the problem size increases. Researchers have proposed various algorithms for solving Combinatorial Optimization Problems (COPs) that take into account the scalability; however, issues are still presented with larger COPs concerning hardware limitations such as memory and CPU speed. It has been shown that the Reduce-Optimize-Expand (ROE) method can solve COPs faster with the same resources; in this methodology, the reduction step is the most important procedure since inappropriate reductions, applied to the problem, will produce suboptimal results on the subsequent stages. In this work, an algorithm to improve the reduction step is proposed. It is based on a fuzzy inference system to classify portions of the problem and remove them, allowing COPs solving algorithms to utilize better the hardware resources by dealing with smaller problem sizes, and the use of metadata and adaptive heuristics. The Travelling Salesman Problem has been used as a case of study; instances that range from 343 to 3056 cities were used to prove that the fuzzy logic approach produces a higher percentage of successful reductions.

scholarly journals Combinatorial optimization by weight annealing in memristive hopfield networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Z. Fahimi ◽  
M. R. Mahmoodi ◽  
H. Nili ◽  
Valentin Polishchuk ◽  
D. B. Strukov
Keyword(s):  
Neural Network ◽  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Main Idea ◽  
Hopfield Neural Network ◽  
Combinatorial Problems ◽  
Hopfield Network ◽  
Global Minima ◽  
Mixed Signal ◽  
Combinatorial Optimization Problems

AbstractThe increasing utility of specialized circuits and growing applications of optimization call for the development of efficient hardware accelerator for solving optimization problems. Hopfield neural network is a promising approach for solving combinatorial optimization problems due to the recent demonstrations of efficient mixed-signal implementation based on emerging non-volatile memory devices. Such mixed-signal accelerators also enable very efficient implementation of various annealing techniques, which are essential for finding optimal solutions. Here we propose a “weight annealing” approach, whose main idea is to ease convergence to the global minima by keeping the network close to its ground state. This is achieved by initially setting all synaptic weights to zero, thus ensuring a quick transition of the Hopfield network to its trivial global minima state and then gradually introducing weights during the annealing process. The extensive numerical simulations show that our approach leads to a better, on average, solutions for several representative combinatorial problems compared to prior Hopfield neural network solvers with chaotic or stochastic annealing. As a proof of concept, a 13-node graph partitioning problem and a 7-node maximum-weight independent set problem are solved experimentally using mixed-signal circuits based on, correspondingly, a 20 × 20 analog-grade TiO2 memristive crossbar and a 12 × 10 eFlash memory array.

scholarly journals On the Problem of a Linear Function Localization on Permutations

2020 ◽  
pp. 14-18
Author(s):  
G.A. Donets ◽  
V.I. Biletskyi
Keyword(s):  
Combinatorial Optimization ◽  
Linear Function ◽  
Optimization Problems ◽  
Target Function ◽  
Combinatorial Problems ◽  
Second Step ◽  
Localization Method ◽  
Combinatorial Optimization Problems ◽  
New Methods ◽  
Is Evaluation

Combinatorial optimization problems and methods of their solution have been a subject of numerous studies, since a large number of practical problems are described by combinatorial optimization models. Many studies consider approaches to and describe methods of solution for combinatorial optimization problems with linear or fractionally linear target functions on combinatorial sets such as permutations and arrangements. Studies consider solving combinatorial problems by means of well-known methods, as well as developing new methods and algorithms of searching a solution. We describe a method of solving a problem of a linear target function localization on a permutation set. The task is to find those locally admissible permutations on the permutation set, for which the linear function possesses a given value. In a general case, this problem may have no solutions at all. In the article, we propose a newly developed method that allows us to obtain a solution of such a problem (in the case that such solution exists) by the goal-oriented seeking for locally admissible permutations with a minimal enumeration that is much less than the number of all possible variants. Searching for the solution comes down to generating various permutations and evaluating them. Evaluation of each permutation includes two steps. The first step consists of function decreasing by transposing the numbers in the first n – 3 positions, and the second step is evaluation of the permutations for the remaining three numbers. Then we analyze the correlation (which is called balance) to define whether the considered permutation is the solution or not. In our article, we illustrate the localization method by solving the problem for n = 5. Keywords: localization, linear function, permutation, transposition, balance, position.

scholarly journals Pendekatan Pengembangan Metaheuristik dalam optimisasi kombinatorial

2019 ◽  
Vol 1 ◽  
pp. 1193
Author(s):  
Muhammad Iqbal ◽  
Muhammad Zarlis ◽  
T Tulus ◽  
Herman Mawengkang
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Combinatorial Problems ◽  
Computer Applications ◽  
Computer Design ◽  
Combinatorial Optimization Problems ◽  
Starting Point ◽  
Finite Set ◽  
Propagation Methods ◽  
Feasible Solutions

Combinatorial problems are problems that have a finite set of feasible solutions. Although in principle the solution to this problem can be obtained with complete enumeration, in complex problems it takes time that cannot be practically accepted, combinatorial analysis is a mathematical study of the arrangement, grouping, ordering, or selection of discrete objects, usually limited in number (finite in number) , many combinatorial optimization problems have been generated by research in computer design, computational theory, and by computer applications on numerical problems that require new methods, new approaches, and new mathematical insights, on combinatorial optimization, the function g (x) is defined as mapping g: {0,1} n → {0,1}, obtaining feasible regions Most methods for solving combinatorial optimization problems propose feasible regions, namely areas that are constrained by problem constraints after relaxation such as counts or binary variables, Field slices generally use the metaheuristic method proposed by para Other researchers, for example Genetic algorithms, simulated annealing, taboo search, plant propagation, methods do not use feasible regional concepts, but proposing the starting point for abstract completion is a brief summary of the paper to help readers quickly ascertain the research objectives and match the research needs.

A New Differential Evolution Based Metaheuristic for Discrete Optimization

2010 ◽  
Vol 1 (2) ◽  
pp. 15-32 ◽  
Author(s):  
Ricardo Sérgio Prado ◽  
Rodrigo César Pedrosa Silva ◽  
Frederico Gadelha Guimarães ◽  
Oriane Magela Neto
Keyword(s):  
Combinatorial Optimization ◽  
Differential Evolution ◽  
Discrete Optimization ◽  
Optimization Problems ◽  
Travelling Salesman Problem ◽  
Search Space ◽  
Combinatorial Problems ◽  
Combinatorial Optimization Problems ◽  
De Algorithm ◽  
The Difference

The Differential Evolution (DE) algorithm is an important and powerful evolutionary optimizer in the context of continuous numerical optimization. Recently, some authors have proposed adaptations of its differential mutation mechanism to deal with combinatorial optimization, in particular permutation-based integer combinatorial problems. In this paper, the authors propose a novel and general DE-based metaheuristic that preserves its interesting search mechanism for discrete domains by defining the difference between two candidate solutions as a list of movements in the search space. In this way, the authors produce a more meaningful and general differential mutation for the context of combinatorial optimization problems. The movements in the list can then be applied to other candidate solutions in the population as required by the differential mutation operator. This paper presents results on instances of the Travelling Salesman Problem (TSP) and the N-Queen Problem (NQP) that suggest the adequacy of the proposed approach for adapting the differential mutation to discrete optimization.

GUIDING CONSTRUCTIVE SEARCH WITH STATISTICAL INSTANCE-BASE LEARNING

2002 ◽  
Vol 11 (02) ◽  
pp. 247-266 ◽  
Author(s):  
ORESTIS TELELIS ◽  
PANAGIOTIS STAMATOPOULOS
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Statistical Information ◽  
Combinatorial Problems ◽  
Set Partitioning ◽  
Greedy Heuristics ◽  
High Quality ◽  
Search Spaces ◽  
Combinatorial Optimization Problems ◽  
Real World Applications

Several real world applications involve solving combinatorial optimization problems. Commonly, existing heuristic approaches are designed to address specific difficulties of the underlying problem and are applicable only within its framework. We suspect, however, that search spaces of combinatorial problems are rich in intuitive statistical and numerical information, which could be exploited heuristically in a generic manner, towards achievement of optimized solutions. Our work presents such a heuristic methodology, which can be adequately configured for several types of optimization problems. Experimental results are discussed, concerning two widely used problem models, namely the Set Partitioning and the Kanpsack problems. It is shown that, by gathering statistical information upon previously found solutions to the problems, the heuristic is able to incrementally adapt its behaviour and reach high quality solutions, exceeding the ones obtained by commonly used greedy heuristics.

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