# Approximation Algorithms

## Overview

Many real-world algorithmic problems cannot be solved efficiently using traditional algorithmic tools, for example because the problems are NP-hard. The goal of this course is to become familiar with important algorithmic concepts and techniques needed to effectively deal with such problems. These techniques apply when we don't require the optimal solution to certain problems, but an approximation that is close to the optimal solution. We will see how to efficiently find such approximations.

Prerequisites:

In order to successfully take this course, you should already have a basic knowledge of algorithms and mathematics. Here's a short list of what you are supposed to know:

- O-notation, Ω-notation, Θ-notation; how to analyze algorithms

- Basic calculus: manipulating summations, solving recurrences, working with logarithms, etc.

- Basic probability theory: events, probability distributions, random variables, expected values etc.

- Basic data structures: linked lists, stacks, queues, heaps

- (Balanced) binary search trees

- Basic sorting algorithms, for example MergeSort, InsertionSort, QuickSort

- Graph terminology, representations of graphs (adjacency lists and adjacency matrix), basic graph algorithms (BFS, DFS, topological sort, shortest paths)

The material for this course is based on the course notes that can be found under the resources tab. We will not cover everything from the course notes. The course notes are there both for students who did not fully understand the lectures as well as for students who would like to dive deeper into the topics.

The video lectures contain a few very minor mistakes. A list of these mistakes can be found under resources (in the document called "Errata"). If you think you found an error, report a problem by clicking the square flag at the bottom of the lecture or quiz where you found the error.

Prerequisites:

In order to successfully take this course, you should already have a basic knowledge of algorithms and mathematics. Here's a short list of what you are supposed to know:

- O-notation, Ω-notation, Θ-notation; how to analyze algorithms

- Basic calculus: manipulating summations, solving recurrences, working with logarithms, etc.

- Basic probability theory: events, probability distributions, random variables, expected values etc.

- Basic data structures: linked lists, stacks, queues, heaps

- (Balanced) binary search trees

- Basic sorting algorithms, for example MergeSort, InsertionSort, QuickSort

- Graph terminology, representations of graphs (adjacency lists and adjacency matrix), basic graph algorithms (BFS, DFS, topological sort, shortest paths)

The material for this course is based on the course notes that can be found under the resources tab. We will not cover everything from the course notes. The course notes are there both for students who did not fully understand the lectures as well as for students who would like to dive deeper into the topics.

The video lectures contain a few very minor mistakes. A list of these mistakes can be found under resources (in the document called "Errata"). If you think you found an error, report a problem by clicking the square flag at the bottom of the lecture or quiz where you found the error.

## Syllabus

Introduction to Approximation algorithms

-In the module the motivation for studying approximation algorithms will be given. We will discuss what optimization problems are, and what the difference between heuristics and approximation algorithms is. Finally, we will introduce the concept of approximation ratio, which plays a central role in the analysis of the quality of approximation algorithms.

The Load Balancing problem

-In this module we will study various approximation algorithms for the load balancing problem. This problems asks to distribute a given set of jobs, each with a certain processing time, over a number of machine. The goal is to do this such that all jobs are finished as soon as possible. We will analyze the quality of the computed solutions computed using the concept of rho-approximation, which we saw in the previous lecture. In this analysis we will see that lower bounds on the optimal solution play a crucial role in the analysis (or, for maximization problems: upper bounds).

LP Relaxation

-In this module we will introduce the technique of LP relaxation to design approximation algorithms, and explain how to analyze the approximation ratio of an algorithm based in LP relaxation. We will do this using the (weighted) Vertex Cover problem as an example. Before we explain the technique of LP relaxation, however, we first give a simple 2-approximation algorithm for the unweighted Vertex Cover problem.

Polynomial-time approximation schemes

-In this module we will introduce the concept of Polynomial-Time Approximation Scheme (PTAS), which are algorithms that can get arbitrarily close to an optimal solution. We describe a general technique to design PTASs, and apply it to the famous Knapsack problem. Finally we will see how to analyze PTASs that are designed with the general technique.

-In the module the motivation for studying approximation algorithms will be given. We will discuss what optimization problems are, and what the difference between heuristics and approximation algorithms is. Finally, we will introduce the concept of approximation ratio, which plays a central role in the analysis of the quality of approximation algorithms.

The Load Balancing problem

-In this module we will study various approximation algorithms for the load balancing problem. This problems asks to distribute a given set of jobs, each with a certain processing time, over a number of machine. The goal is to do this such that all jobs are finished as soon as possible. We will analyze the quality of the computed solutions computed using the concept of rho-approximation, which we saw in the previous lecture. In this analysis we will see that lower bounds on the optimal solution play a crucial role in the analysis (or, for maximization problems: upper bounds).

LP Relaxation

-In this module we will introduce the technique of LP relaxation to design approximation algorithms, and explain how to analyze the approximation ratio of an algorithm based in LP relaxation. We will do this using the (weighted) Vertex Cover problem as an example. Before we explain the technique of LP relaxation, however, we first give a simple 2-approximation algorithm for the unweighted Vertex Cover problem.

Polynomial-time approximation schemes

-In this module we will introduce the concept of Polynomial-Time Approximation Scheme (PTAS), which are algorithms that can get arbitrarily close to an optimal solution. We describe a general technique to design PTASs, and apply it to the famous Knapsack problem. Finally we will see how to analyze PTASs that are designed with the general technique.