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Find the shortest possible route that visits a collection of cities and returns to the origin city. Often solved using algorithms like the Held-Karp algorithm (Dynamic Programming) for exact solutions, or heuristic and approximation algorithms like Genetic Algorithms or Ant Colony Optimization for larger instances.

Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight does not exceed a given limit while the total value is as large as possible. Solved optimally by Dynamic Programming methods or approximately by Greedy algorithms.

Identify the best positions to build facilities (warehouses, stores, etc.) to minimize the cost of serving a set of clients. Solved using various heuristic algorithms such as the Greedy algorithm, Genetic Algorithms, or Particle Swarm Optimization.

Optimize a linear objective function subject to linear equality and inequality constraints. The Simplex algorithm or Interior Point Methods are commonly used for finding the optimal solution.

Like linear programming, but the solutions are restricted to integers. It is often solved with the Branch and Bound algorithm, the Cutting Plane algorithm, or by using solver libraries like IBM's CPLEX.

Determine the minimum set of edges to remove from a graph to disconnect the remaining network into two disjoint subsets. The Karger's algorithm is often used to solve this problem, especially using its randomized contraction technique.

Determine the optimal flow through a single-source, single-sink flow network that is subject to capacity constraints. The Ford-Fulkerson algorithm or the Edmonds-Karp algorithm, which is an implementation of the former using BFS, are frequently used for solving these problems.

Schedule jobs on machines to minimize the completion time of all jobs. The optimal algorithm depends on the specific variant, but common methods include Greedy algorithms, Dynamic Programming, and Integer Programming.

Find the contiguous subarray within a one-dimensional array of numbers which has the largest sum. Often solved optimally by Kadane's algorithm.