Discrete Mathematics And Optimization.pdf

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects,[1] where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

Discrete Mathematics and Optimization.pdf

Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science.

Some research literature[2] considers discrete optimization to consist of integer programming together with combinatorial optimization (which in a turn is composed of optimization problems dealing with graph structures), although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.[clarification needed]

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization. A considerable amount of it is unified by the theory of linear programming. Some examples of combinatorial optimization problems that are covered by this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable[weasel words] approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman (decision) problem,[7] this is expected unless P=NP.

Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities. This tutorial explains the fundamental concepts of Sets, Relations and Functions, Mathematical Logic, Group theory, Counting Theory, Probability, Mathematical Induction and Recurrence Relations, Graph Theory, Trees and Boolean Algebra.

We publish research articles written in English in areas such as combinatorics and graph theory, discrete and computational geometry, discrete optimization and operations research, theoretical computer science, and coding and communication theory. In particular, the following topics (2000 AMS classification numbers in parentheses):

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