Solution methods for NP-hard Discrete Optimization Problems

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

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

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.

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.

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 (considered in Math442/542) – 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. (see Chapter 4 of H&L)