# What is combinatorial optimization?

by Stephen M. Walker II, Co-Founder / CEO

## What is combinatorial optimization?

Combinatorial optimization is a subfield of mathematical optimization that focuses on finding the optimal solution from a finite set of objects. The set of feasible solutions is discrete or can be reduced to a discrete set.

In more formal terms, a combinatorial optimization problem is defined as a quadruple $(I, f, m, g)$, where:

- $I$ is a set of instances
- For an instance $x \in I$, $f(x)$ is the finite set of feasible solutions
- For an instance $x$ and a feasible solution $y$ of $x$, $m(x, y)$ denotes the measure of $y$, which is usually a positive real number
- $g$ is the goal function, which is either $\min$ or $\max$. The goal is to find an optimal solution $y$ for some instance $x$ such that $m(x, y) = g{m(x, y') | y' \in f(x)}$.

Combinatorial optimization has wide-ranging applications in various fields, including logistics, supply chain optimization, network design, job allocation, water distribution networks, and earth science problems. It also plays a crucial role in computer science domains like artificial intelligence, machine learning, auction theory, and software engineering.

Solving combinatorial optimization problems can be challenging due to the large search space and the discrete nature of the solutions. Various methods are used to tackle these problems, including polynomial-time algorithms for certain special classes of problems, specialized algorithms that quickly rule out large parts of the search space, and approximation algorithms. In some cases, problems can be solved exactly using techniques like Branch and Bound, but in other cases, randomized search algorithms like simulated annealing, genetic algorithms, and tabu search are employed.

Examples of combinatorial optimization problems include the Traveling Salesman Problem, Bin-Packing, Integer Linear Programming, and the Knapsack problem. These problems often arise in real-world scenarios, such as resource allocation, logistics operations, and transportation route optimization.

## Combinatorial Optimization in AI: Problems, Algorithms, and Heuristics

Combinatorial optimization in AI encompasses a variety of problems where the goal is to find the best solution from a vast set of possibilities. Common problems include the knapsack problem, the traveling salesman problem, and the minimum spanning tree problem. To tackle these, several algorithms are employed, each with unique advantages. Genetic algorithms are useful for large, unknown search spaces, while simulated annealing and tabu search are effective for landscapes with multiple local optima and specific constraints, respectively.

Choosing the right algorithm is crucial as it significantly affects system performance. It's often beneficial to experiment with multiple algorithms to determine which yields the best results for a given problem. In addition to these algorithms, heuristics like the greedy algorithm, hill climbing, and simulated annealing are employed to find satisfactory solutions efficiently. These heuristics prioritize immediate, incremental improvements without exhaustive future consequence analysis.

Despite the power of combinatorial optimization, challenges such as vast or complex search spaces and dynamic solution sets can impede finding optimal solutions. To enhance algorithm performance, one can optimize data structures, employ heuristics and meta-heuristics for search guidance, or leverage machine learning techniques to predict effective solutions.