This module will cover integer linear programming and its use in solving NP-hard (combinatorial optimization) problems. We will cover some examples of what integer linear programming is by formulating problems such as Knapsack, Vertex Cover and Graph Coloring. Next, we will study the concept of integrality gap and look at the special case of integrality gap for vertex cover problems. We will conclude with a tutorial on formulating and solving integer linear programs using the python library Pulp.

We will introduce approximation algorithms for solving NP-hard problems. These algorithms are fast (often greedy algorithms) that may not produce an optimal solution but guarantees that its solution is not "too far away" from the best possible. We will present some of these algorithms starting from a basic introduction to the concepts involved followed by a series of approximation algorithms for scheduling problems, vertex cover problem and the maximum satisfiability problem.

We will present the travelling salesperson problem (TSP): a very important and widely applicable combinatorial optimization problem, its NP-hardness and the hardness of approximating a general TSP with a constant factor. We present integer linear programming formulation and a simple yet elegant dynamic programming algorithm. We will present a 3/2 factor approximation algorithm by Christofides and discuss some heuristic approaches for solving TSPs. We will conclude by presenting approximation schemes for the knapsack problem.