Combinatorial optimization

Combinatorial optimization is a part of mathematical optimization where we try to find the best solution from a list of choices. These choices are separate and clear, so we can count them one by one. This area helps solve special problems, like finding the shortest route a salesman could take to visit many cities and return home — known as the travelling salesman problem. Since checking every route would take too long, scientists use smart methods to find good answers quickly.

These problems connect to many important areas, such as operations research, which looks at making processes better, and algorithm theory, which studies steps to solve problems. Combinatorial optimization is useful in many modern technologies, from artificial intelligence that helps computers learn, to designing software and creating efficient electronic circuits. It helps us solve tricky puzzles in many different fields.

Applications

Combinatorial optimization helps solve many real-world problems. It can be used in logistics and supply chain optimization, like planning the best routes for airplanes or deciding where taxis should go to pick up passengers.

It is also useful for assigning jobs to people in the best way possible, designing water distribution networks, and solving problems in Earth science.

Methods

Combinatorial optimization is about finding the best solution from a set of choices. Some problems can be solved using linear programming. This helps with tasks like finding the shortest paths or spanning trees.

For harder problems, researchers look at simpler cases. They also study ways to solve typical examples and find solutions that are almost the best. Common techniques include branch-and-bound and dynamic programming. These might not always find the perfect solution quickly. Each problem has a matching question, like whether a path between two points can be found using a specific number of steps.

NP optimization problem

An NP-optimization problem is a special type of combinatorial optimization problem. The size of any solution must fit within limits that grow slowly compared to the input size. We can check if a solution works and find its value quickly.

These problems are grouped into subclasses based on how close we can get to the best solution in a reasonable time. Some problems, like the Knapsack problem, can be solved almost perfectly. Others, like the TSP (travelling salesman problem), are harder, and we can only get close to the best answer.

Specific problems

Further information: Category:Combinatorial optimization

This is a list of some famous problems in combinatorial optimization. You can help by adding more items with references to reliable sources.

Some well-known problems include:

• Assignment problem
• Bin packing problem
• Chinese postman problem
• Closure problem
• Constraint satisfaction problem
• Cutting stock problem
• Dominating set problem
• Integer programming
• Job shop scheduling
• Knapsack problem
• Metric k-center / vertex k-center problem
• Minimum relevant variables in linear system
• Minimum spanning tree
• Nurse scheduling problem
• Ring star problem
• Set cover problem
• Talent scheduling
• Traveling salesman problem
• Vehicle rescheduling problem
• Vehicle routing problem
• Weapon target assignment problem