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Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: An optimization problem with discrete variables is known as a discrete optimization , in which an object such as an integer , permutation or graph must be found from a countable set .
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
An algorithm published by T. C. Hu and M.-T. Shing achieves O(n log n) computational complexity. [3] [4] [5] They showed how the matrix chain multiplication problem can be transformed (or reduced) into the problem of triangulation of a regular polygon. The polygon is oriented such that there is a horizontal bottom side, called the base, which ...
Branch and bound (BB, B&B, or BnB) is a method for solving optimization problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain the optimal solution. It is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical ...
If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems. [1] In the optimization literature this relationship is called the Bellman equation.
An optimization problem asks for finding a "best possible" solution among the set of all possible solutions to a search problem. One example is the maximum independent set problem: "Given a graph G, find an independent set of G of maximum size." Optimization problems are represented by their objective function and their constraints.
For example, one of the long-standing open questions in computer science is to determine whether there is an algorithm that outperforms the 2-approximation for the Steiner Forest problem by Agrawal et al. [3] The desire to understand hard optimization problems from the perspective of approximability is motivated by the discovery of surprising ...
Function and optimization problems are often transformed into decision problems by considering the question of whether the output is equal to or less than or equal to a given value. This allows the complexity of the corresponding decision problem to be studied; and in many cases the original function or optimization problem can be solved by ...