- Design and Analysis of Algorithms
- Home
- Basics of Algorithms
- DAA - Introduction
- DAA - Analysis of Algorithms
- DAA - Methodology of Analysis
- Asymptotic Notations & Apriori Analysis
- Time Complexity
- Master’s Theorem
- DAA - Space Complexities
- Divide & Conquer
- DAA - Divide & Conquer
- DAA - Max-Min Problem
- DAA - Merge Sort
- DAA - Binary Search
- Strassen’s Matrix Multiplication
- Karatsuba Algorithm
- Towers of Hanoi
- Greedy Algorithms
- DAA - Greedy Method
- Travelling Salesman Problem
- Prim's Minimal Spanning Tree
- Kruskal’s Minimal Spanning Tree
- Dijkstra’s Shortest Path Algorithm
- Map Colouring Algorithm
- DAA - Fractional Knapsack
- DAA - Job Sequencing with Deadline
- DAA - Optimal Merge Pattern
- Dynamic Programming
- DAA - Dynamic Programming
- Matrix Chain Multiplication
- Floyd Warshall Algorithm
- DAA - 0-1 Knapsack
- Longest Common Subsequence
- Travelling Salesman Problem | Dynamic Programming
- Randomized Algorithms
- Randomized Algorithms
- Randomized Quick Sort
- Karger’s Minimum Cut
- Fisher-Yates Shuffle
- Approximation Algorithms
- Approximation Algorithms
- Vertex Cover Problem
- Set Cover Problem
- Travelling Salesperson Approximation Algorithm
- Graph Theory
- DAA - Spanning Tree
- DAA - Shortest Paths
- DAA - Multistage Graph
- Optimal Cost Binary Search Trees
- Heap Algorithms
- DAA - Binary Heap
- DAA - Insert Method
- DAA - Heapify Method
- DAA - Extract Method
- Sorting Techniques
- DAA - Bubble Sort
- DAA - Insertion Sort
- DAA - Selection Sort
- DAA - Shell Sort
- DAA - Heap Sort
- DAA - Bucket Sort
- DAA - Counting Sort
- DAA - Radix Sort
- Searching Techniques
- Searching Techniques Introduction
- DAA - Linear Search
- DAA - Binary Search
- DAA - Interpolation Search
- DAA - Jump Search
- DAA - Exponential Search
- DAA - Fibonacci Search
- DAA - Sublist Search
- Complexity Theory
- Deterministic vs. Nondeterministic Computations
- DAA - Max Cliques
- DAA - Vertex Cover
- DAA - P and NP Class
- DAA - Cook’s Theorem
- NP Hard & NP-Complete Classes
- DAA - Hill Climbing Algorithm
- DAA Useful Resources
- DAA - Quick Guide
- DAA - Useful Resources
- DAA - Discussion
Design and Analysis - Approximation Algorithms
Approximation algorithms are algorithms designed to solve problems that are not solvable in polynomial time for approximate solutions. These problems are known as NP complete problems. These problems are significantly effective to solve real world problems, therefore, it becomes important to solve them using a different approach.
NP complete problems can still be solved in three cases: the input could be so small that the execution time is reduced, some problems can still be classified into problems that can be solved in polynomial time, or use approximation algorithms to find near-optima solutions for the problems.
This leads to the concept of performance ratios of an approximation problem.
Performance Ratios
The main idea behind calculating the performance ratio of an approximation algorithm, which is also called as an approximation ratio, is to find how close the approximate solution is to the optimal solution.
The approximate ratio is represented using ρ(n) where n is the input size of the algorithm, C is the near-optimal solution obtained by the algorithm, C* is the optimal solution for the problem. The algorithm has an approximate ratio of ρ(n) if and only if −
$$max\left\{\frac{C}{C^{\ast} },\frac{C^{\ast }}{C} \right\}\leq \rho \left ( n \right )$$
The algorithm is then called a ρ(n)-approximation algorithm. Approximation Algorithms can be applied on two types of optimization problems: minimization problems and maximization problems. If the optimal solution of the problem is to find the maximum cost, the problem is known as the maximization problem; and if the optimal solution of the problem is to find the minimum cost, then the problem is known as a minimization problem.
For maximization problems, the approximation ratio is calculated by C*/C since 0 ≤ C ≤ C*. For minimization problems, the approximation ratio is calculated by C/C* since 0 ≤ C* ≤ C.
Assuming that the costs of approximation algorithms are all positive, the performance ratio is well defined and will not be less than 1. If the value is 1, that means the approximate algorithm generates the exact optimal solution.
Examples
Few popular examples of the approximation algorithms are −
Vertex Cover Problem
Set Cover Problem
Travelling Salesperson Problem
The Subset Sum Problem
