Combinatorial optimization in the context of Planar graph

Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Automated planning and scheduling, sometimes denoted as simply AI planning, is a branch of artificial intelligence that concerns the realization of strategies or action sequences, typically for execution by intelligent agents, autonomous robots and unmanned vehicles. Unlike classical control and classification problems, the solutions are complex and must be discovered and optimized in multidimensional space. Planning is also related to decision theory.

In known environments with available models, planning can be done offline. Solutions can be found and evaluated prior to execution. In dynamically unknown environments, the strategy often needs to be revised online. Models and policies must be adapted. Solutions usually resort to iterative trial and error processes commonly seen in artificial intelligence. These include dynamic programming, reinforcement learning and combinatorial optimization. Languages used to describe planning and scheduling are often called action languages.

The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?" The problem first appeared, as the truck dispatching problem, in a paper by George Dantzig and John Ramser in 1959, in which it was applied to petrol deliveries. Often, the context is that of delivering goods located at a central depot to customers who have placed orders for such goods. However, variants of the problem consider, e.g, collection of solid waste and the transport of the elderly and the sick to and from health-care facilities. The standard objective of the VRP is to minimise the total route cost. Other objectives, such as minimising the number of vehicles used or travelled distance are also considered.

The VRP generalises the travelling salesman problem (TSP), which is equivalent to requiring a single route to visit all locations. As the TSP is NP-hard, the VRP is also NP-hard.

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.

The travelling purchaser problem, the vehicle routing problem and the ring star problem are three generalizations of TSP.

The knapsack problem is the following problem in combinatorial optimization:

It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively.

A genetic operator is an operator used in evolutionary algorithms (EA) to guide the algorithm towards a solution to a given problem. There are three main types of operators (mutation, crossover and selection), which must work in conjunction with one another in order for the algorithm to be successful. Genetic operators are used to create and maintain genetic diversity (mutation operator), combine existing solutions (also known as chromosomes) into new solutions (crossover) and select between solutions (selection).

The classic representatives of evolutionary algorithms include genetic algorithms, evolution strategies, genetic programming and evolutionary programming. In his book discussing the use of genetic programming for the optimization of complex problems, computer scientist John Koza has also identified an 'inversion' or 'permutation' operator; however, the effectiveness of this operator has never been conclusively demonstrated and this operator is rarely discussed in the field of genetic programming. For combinatorial problems, however, these and other operators tailored to permutations are frequently used by other EAs.

In computer science and mathematical optimization, a metaheuristic is a higher-level procedure or heuristic designed to find, generate, tune, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimization problem or a machine learning problem, especially with incomplete or imperfect information or limited computation capacity. Metaheuristics sample a subset of solutions which is otherwise too large to be completely enumerated or otherwise explored. Metaheuristics may make relatively few assumptions about the optimization problem being solved and so may be usable for a variety of problems. Their use is always of interest when exact or other (approximate) methods are not available or are not expedient, either because the calculation time is too long or because, for example, the solution provided is too imprecise.

Compared to optimization algorithms and iterative methods, metaheuristics do not guarantee that a globally optimal solution can be found on some class of problems. Many metaheuristics implement some form of stochastic optimization, so that the solution found is dependent on the set of random variables generated. In combinatorial optimization, there are many problems that belong to the class of NP-complete problems and thus can no longer be solved exactly in an acceptable time from a relatively low degree of complexity. Metaheuristics then often provide good solutions with less computational effort than approximation methods, iterative methods, or simple heuristics. This also applies in the field of continuous or mixed-integer optimization. As such, metaheuristics are useful approaches for optimization problems. Several books and survey papers have been published on the subject. Literature review on metaheuristic optimization, suggested that it was Fred Glover who coined the word metaheuristics.

Éva Tardos (born 1 October 1957) is a Hungarian mathematician and the Jacob Gould Schurman Professor of Computer Science at Cornell University, known for her work in theoretical computer science. For her work, she has received the Fulkerson Prize (1988), the Dantzig Prize (2006), and the IEEE John von Neumann Medal (2019).

Tardos's research interest is algorithms and algorithmic game theory. Her work focuses on the design and analysis of efficient methods for combinatorial optimization problems on graphs or networks. She has done some work on network flow algorithms like approximation algorithms for network flows, cut, and clustering problems. Her recent work focuses on algorithmic game theory and simple auctions.