Solution methods for NP-hard Discrete Optimization Problems

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Solution methods for NP-hard Discrete
Optimization Problems
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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
Least accuracy
Most accuracy
Worst time
Best time
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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
• Dont guarantee optimal solution
• Hopefully will produce fairly good solutions at
  least some of the time
• Example The nearest neighbor algorithm for TSP

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Approximation Algorithms

• Time-efficient (sometimes not as efficient as
  heuristics)
• Dont 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.

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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.

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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.