Explore solution methods for NP-hard discrete optimization problems, including Integer Programming techniques, Approximation Algorithms, and Heuristics. Learn about the trade-offs between time and accuracy, and the different approaches such as IP-based solution methods. Discover heuristics, approximation algorithms, and the comparison between solving Integer Programs (IP) and Linear Programs (LP).
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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
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 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.
