![Example 2 : Travelling Salesman Problem
We are given an integer n > 0 and the distance between every pair of n
cities in the form of an n x n matrix [di,j], where di,j Z+ . A tour is a
close path that visits every city exactly once. The problem is to find a
tour with minimum total length.
F= {all cyclic permutations π on n cities}
C(π) → σ𝒋 𝒏 𝒅i,j
Here city j is visited after visiting city i](https://image.slidesharecdn.com/co-lec-1-210526055638/85/Combinatorial-optimization-CO-1-10-320.jpg)
Combinatorial optimization CO-1
- 1. Combinatorial Optimization CS-724 Lec-1: Date: 22-02-2021 Dr. Parikshit Saikia Assistant Professor Department of Computer Science and Engineering NIT Hamirpur (HP) India
- 2. Course Information •Theory •Day: Monday—Wednesday •Time: 3-3:35 PM •Evaluation: • Quiz + Attendance (At least 75%) + Assignment : 20% of the total marks • Mid Sem Exam: 30% of the total marks • End Sem Exam: 50% of the total marks
- 3. Combinatorial Optimization (CO) • Objective • To learn how to model problems using mathematical programs. • To learn how to formulate and solve the traditional problems in combinatorial optimization using combinatorial algorithm. • To learn the concept of linear programming and matching algorithms.
- 4. Course Outcome • Students will be able to • Identify and classify combinatorial optimization problems with real-world problems • Identify, classify and implement algorithm to solve combinatorial optimization problems • Model problems using linear programming. • Understand the inherent complexity of problems: Polynomial time, NP- completeness.
- 5. Application of CO • Algorithm Theory • Operations Research • Artificial Intelligence • Computational Complexity • Auction Theory • Software Engineering • Machine Learning • Theoretical Computer Science • …………….
- 6. • What do we optimize? * A cost function: It help us to define which solution is better (Solution A or Solution B etc) • How to find the solution having the best cost? By designing an optimization algorithm. • Can we decompose the problem into some smaller problems? By defining optimal sub-structure property • How shall we search among alternatives? By using complexity: run time, space required
- 7. Optimization Problem • An instance of an optimization problem is a pair (F, c), where F is any set, the domain of feasible points; c is the cost function s.t. c : F → R The point is to find a point f F s.t. c(f) ≤ c(y) for all y F Such a point f is called globally optimal solution.
- 8. Optimization Problems • Categorized into two types • Those with continuous variables: looking for a set of real numbers • Those with discrete variables: looking for an object from a finite, or possibly countably infinite set, typically an integer set, permutation or graphs. Optimization Problems with Discrete Variables are called Combinatorial Optimization Problems
- 9. Example 1: Minimum spanning problem Given a graph G=(V,E), w: E → Z+ An instance of MST problem is F = {all spanning trees (V, E’) with V = {1, 2, ……, n}} c: (V, E’) → σ𝒆 𝑬′ 𝒘(𝒆) Here spanning tree means connected and acyclic graph.
- 10. Example 2 : Travelling Salesman Problem We are given an integer n > 0 and the distance between every pair of n cities in the form of an n x n matrix [di,j], where di,j Z+ . A tour is a close path that visits every city exactly once. The problem is to find a tour with minimum total length. F= {all cyclic permutations π on n cities} C(π) → σ𝒋 𝒏 𝒅i,j Here city j is visited after visiting city i
- 11. Global Optima • It is the extrema (minimum or maximum) of the objective function for the entire input space • Given an instance (F, c) of an optimization problem, a feasible solution f is called globally optimal with respect to F if c(f) ≤ c(g) for all g F
- 12. Local optimization • A local optima is the extrema (minimum or maximum) of the objective function for a given region of the input space. • Given an instance (F, c) of an optimization problem and a neighborhood N, a feasible solution f is called locally optimal with respect to N if c(f) ≤ c(g) for all g N(f) Here N(f) is defined as a set of points that are close in some sense to the point f.