Mikhail Moshkov

Michal Mankowski

This book introduces a fairly universal approach to the design and analysis of exact optimization algorithms for multi-objective combinatorial optimization problems. It proposes the circuits without repetitions representing the sets of feasible solutions along with the increasing and strictly increasing cost functions as a model for such problems. The book designs the algorithms for multi-stage and bi-criteria optimization and for counting the solutions in the framework of this model. As applications, this book studies eleven known combinatorial optimization problems: matrix chain multiplication, global sequence alignment, optimal paths in directed graphs, binary search trees, convex polygon triangulation, line breaking (text justification), one-dimensional clustering, optimal bitonic tour, segmented least squares, optimization of matchings in trees, and 0/1 knapsack problem. The results presented are useful for researchers in combinatorial optimization. This book is also useful as the basis for graduate courses.

Dynamic Programming Multi-Objective Combinatorial Optimization

Shahid Hussain

Igor Chikalov

Hassan AbouEisha

Talha Amin

Dynamic programming is an efficient technique for solving optimization problems. It is based on breaking the initial problem down into simpler ones and solving these sub-problems, beginning with the simplest ones. A conventional dynamic programming algorithm returns an optimal object from a given set of objects. This book develops extensions of dynamic programming, enabling us to (i) describe the set of objects under consideration; (ii) perform a multi-stage optimization of objects relative to different criteria; (iii) count the number of optimal objects; (iv) find the set of Pareto optimal points for bi-criteria optimization problems; and (v) to study relationships between two criteria. It considers various applications, including optimization of decision trees and decision rule systems as algorithms for problem solving, as ways for knowledge representation, and as classifiers; optimization of element partition trees for rectangular meshes, which are used in finite element methods for solving PDEs; and multi-stage optimization for such classic combinatorial optimization problems as matrix chain multiplication, binary search trees, global sequence alignment, and shortest paths. The results presented are useful for researchers in combinatorial optimization, data mining, knowledge discovery, machine learning, and finite element methods, especially those working in rough set theory, test theory, logical analysis of data, and PDE solvers. This book can be used as the basis for graduate courses.

Extensions of Dynamic Programming for Combinatorial Optimization and Data Mining

Fawaz Alsolami

Mohammad Azad

The results presented here (including the assessment of a new tool – inhibitory trees) offer valuable tools for researchers in the areas of data mining, knowledge discovery, and machine learning, especially those whose work involves decision tables with many-valued decisions. The authors consider various examples of problems and corresponding decision tables with many-valued decisions, discuss the difference between decision and inhibitory trees and rules, and develop tools for their analysis and design. Applications include the study of totally optimal (optimal in relation to a number of criteria simultaneously) decision and inhibitory trees and rules; the comparison of greedy heuristics for tree and rule construction as single-criterion and bi-criteria optimization algorithms; and the development of a restricted multi-pruning approach used in classification and knowledge representation.

Decision and Inhibitory Trees and Rules for Decision Tables with Many-valued Decisions

Beata Zielosko

Decision trees and decision rule systems are widely used in different applications as algorithms for problem solving, as predictors, and as a way for knowledge representation. Reducts play key role in the problem of attribute (feature) selection. The aims of this book are (i) the consideration of the sets of decision trees, rules and reducts; (ii) study of relationships among these objects; (iii) design of algorithms for construction of trees, rules and reducts; and (iv) obtaining bounds on their complexity. Applications for supervised machine learning, discrete optimization, analysis of acyclic programs, fault diagnosis, and pattern recognition are considered also. This is a mixture of research monograph and lecture notes. It contains many unpublished results. However, proofs are carefully selected to be understandable for students. The results considered in this book can be useful for researchers in machine learning, data mining and knowledge discovery, especially for those who are working in rough set theory, test theory and logical analysis of data. The book can be used in the creation of courses for graduate students.

Mikhail Moshkov

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