3,977 research outputs found
Local Optima Network Analysis for MAX-SAT
Local Optima Networks (LONs) are a valuable tool to understand fitness landscapes of optimization problems observed from the perspective of a search algorithm. Local optima of the optimization problem are linked by an edge in LONs when an operation in the search algorithm allows one of them to be reached from the other. Previous work analyzed several combinatorial optimization problems using LONs and provided a visual guide to understand why the instances are difficult or easy for the search algorithms. In this work we analyze for the first time the MAX-SAT problem. Given a Boolean formula in Conjunctive Normal Form, the goal of the MAX-SAT problem is to find an assignment maximizing the number of satistified clauses. Several random and industrial instances of MAX-SAT are analyzed using Iterated Local Search to sample the search space.Universidad de Málaga, Campus de Excelencia International Andalucía Tech.
Universidad de Stirling, Reino Unido.
Ministerio de Economía y Competitividad y FEDER (proyecto TIN2017-88213-R)
Global Landscape Structure and the Random MAX-SAT Phase Transition
We revisit the fitness landscape structure of random MAX-SAT instances, and address the question: what structural features change when we go from easy underconstrained instances to hard overconstrained ones? Some standard techniques such as autocorrelation analysis fail to explain what makes instances hard to solve for stochastic local search algorithms, indicating that deeper landscape features are required to explain the observed performance differences. We address this question by means of local optima network (LON) analysis and visualisation. Our results reveal that the number, size, and, most importantly, the connectivity pattern of local and global optima change significantly over the easy-hard transition. Our empirical results suggests that the landscape of hard MAX-SAT instances may feature sub-optimal funnels, that is, clusters of sub-optimal solutions where stochastic local search methods can get trapped
Joint Downlink Base Station Association and Power Control for Max-Min Fairness: Computation and Complexity
In a heterogeneous network (HetNet) with a large number of low power base
stations (BSs), proper user-BS association and power control is crucial to
achieving desirable system performance. In this paper, we systematically study
the joint BS association and power allocation problem for a downlink cellular
network under the max-min fairness criterion. First, we show that this problem
is NP-hard. Second, we show that the upper bound of the optimal value can be
easily computed, and propose a two-stage algorithm to find a high-quality
suboptimal solution. Simulation results show that the proposed algorithm is
near-optimal in the high-SNR regime. Third, we show that the problem under some
additional mild assumptions can be solved to global optima in polynomial time
by a semi-distributed algorithm. This result is based on a transformation of
the original problem to an assignment problem with gains , where
are the channel gains.Comment: 24 pages, 7 figures, a shorter version submitted to IEEE JSA
On the Neutrality of Flowshop Scheduling Fitness Landscapes
Solving efficiently complex problems using metaheuristics, and in particular
local searches, requires incorporating knowledge about the problem to solve. In
this paper, the permutation flowshop problem is studied. It is well known that
in such problems, several solutions may have the same fitness value. As this
neutrality property is an important one, it should be taken into account during
the design of optimization methods. Then in the context of the permutation
flowshop, a deep landscape analysis focused on the neutrality property is
driven and propositions on the way to use this neutrality to guide efficiently
the search are given.Comment: Learning and Intelligent OptimizatioN Conference (LION 5), Rome :
Italy (2011
Towards the Inferrence of Structural Similarity of Combinatorial Landscapes
One of the most common problem-solving heuristics is by analogy. For a given
problem, a solver can be viewed as a strategic walk on its fitness landscape.
Thus if a solver works for one problem instance, we expect it will also be
effective for other instances whose fitness landscapes essentially share
structural similarities with each other. However, due to the black-box nature
of combinatorial optimization, it is far from trivial to infer such similarity
in real-world scenarios. To bridge this gap, by using local optima network as a
proxy of fitness landscapes, this paper proposed to leverage graph data mining
techniques to conduct qualitative and quantitative analyses to explore the
latent topological structural information embedded in those landscapes. By
conducting large-scale empirical experiments on three classic combinatorial
optimization problems, we gain concrete evidence to support the existence of
structural similarity between landscapes of the same classes within neighboring
dimensions. We also interrogated the relationship between landscapes of
different problem classes
A quantitative analysis of estimation of distribution algorithms based on Bayesian networks
The successful application of estimation of distribution algorithms
(EDAs) to solve different kinds of problems has reinforced their candidature
as promising black-box optimization tools. However, their internal behavior
is still not completely understood and therefore it is necessary to work
in this direction in order to advance their development. This paper
presents a new methodology of analysis which provides new information
about the behavior of EDAs by quantitatively analyzing the probabilistic
models learned during the search. We particularly focus on calculating the
probabilities of the optimal solutions, the most probable solution given by
the model and the best individual of the population at each step of the
algorithm. We carry out the analysis by optimizing functions of different
nature such as Trap5, two variants of Ising spin glass and Max-SAT. By
using different structures in the probabilistic models, we also analyze the
influence of the structural model accuracy in the quantitative behavior
of EDAs. In addition, the objective function values of our analyzed key
solutions are contrasted with their probability values in order to study
the connection between function and probabilistic models. The results not
only show information about the EDA behavior, but also about the quality
of the optimization process and setup of the parameters, the relationship
between the probabilistic model and the fitness function, and even about
the problem itself. Furthermore, the results allow us to discover common
patterns of behavior in EDAs and propose new ideas in the development
of this type of algorithms
On the landscape of combinatorial optimization problems
This paper carries out a comparison of the fitness landscape for four classic optimization problems: Max-Sat, graph-coloring, traveling salesman, and quadratic assignment. We have focused on two types of properties, local average properties of the landscape, and properties of the local optima. For the local optima we give a fairly comprehensive description of the properties, including the expected time to reach a local optimum, the number of local optima at different cost levels, the distance between optima, and the expected probability of reaching the optima. Principle component analysis is used to understand the correlations between the local optima. Most of the properties that we examine have not been studied previously, particularly those concerned with properties of the local optima. We compare and contrast the behavior of the four different problems. Although the problems are very different at the low level, many of the long-range properties exhibit a remarkable degree of similarity
Dynastic Potential Crossover Operator
An optimal recombination operator for two parent solutions provides the best solution among those that take the value for each variable from one of the parents (gene transmission property). If the solutions are bit strings, the offspring of an optimal recombination operator is optimal in the smallest hyperplane containing the two parent solutions.
Exploring this hyperplane is computationally costly, in general, requiring exponential time in the worst case. However, when the variable interaction graph of the objective function is sparse, exploration can be done in polynomial time.
In this paper, we present a recombination operator, called Dynastic Potential Crossover (DPX), that runs in polynomial time and behaves like an optimal recombination operator for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with traditional crossover operators, like uniform crossover and network crossover, and with two recently defined efficient recombination operators: partition crossover and articulation points partition crossover. The empirical comparison uses NKQ Landscapes and MAX-SAT instances. DPX outperforms the other crossover operators in terms of quality of the offspring and provides better results included in a trajectory and a population-based metaheuristic, but it requires more time and memory to compute the offspring.This research is partially funded by the Universidad de M\'alaga, Consejería de Economía y Conocimiento de la Junta de Andalucía and FEDER under grant number UMA18-FEDERJA-003 (PRECOG); under grant PID 2020-116727RB-I00 (HUmove) funded by MCIN/AEI/10.13039/501100011033; and TAILOR ICT-48 Network (No 952215) funded by EU Horizon 2020 research and innovation programme. The work is also partially supported in Brazil by São Paulo Research Foundation (FAPESP), under grants 2021/09720-2 and 2019/07665-4, and National Council for Scientific and Technological Development (CNPq), under grant 305755/2018-8
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