1. Solution methods for NP-hard Discrete Optimization Problems

2. Three main directions to solve NP-hard discrete optimization problems:
Integer programming techniques
Approximation algorithms
Heuristics
On time-accuracy tradeoff schedule:
Integer programming
Approximation algorithms
Heuristics
Brute force
Most accuracy
Least accuracy
Worst time
Best time

3. Heuristics Based on common sense, intuition
Sometimes are based on physical, biological phenomena
(e.g., simulated annealing, genetic algorithm)
Normally very time-efficient
No rigorous mathematical analysis
Don’t guarantee optimal solution
Hopefully will produce fairly good solutions at least some of the time
Example: The nearest neighbor algorithm for TSP

4. Approximation Algorithms
Time-efficient (sometimes not as efficient as heuristics)
Don’t guarantee optimal solution
Guarantee good solution within some factor of the optimum
Rigorous mathematical analysis to prove the approximation guarantee
Often use algorithms for related problems as subroutines
Later we will consider
an approximation algorithm for TSP.

5. IP-based Solution Methods
Most discrete optimization problems can be formulated as integer programs
Guarantee optimal solution most of the time
Sometimes might be time-inefficient
Is the preferred method for most companies, especially with the advent of modern superfast computers
We will consider IP-based solution methods in details.

6. Solving Integer Programs (IP) vs solving Linear Programs (LP)
The algorithms for solving LPs are much more time-efficient than the algorithms for IPs.
LP algorithms
Simplex Method
Interior-point methods
IP algorithms use the above-mentioned LP algorithms as subroutines.
Thus, we will start by recalling the main features of Simplex Method.