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Using Python to Solve Computationally Hard Problems Using Python to - - PowerPoint PPT Presentation

Oct 31, 2023 393 likes •873 views

Using Python to Solve Computationally Hard Problems Using Python to Solve Computationally Hard Problems Rachael Madsen Optimal Design Software LLC BS in Mathematics Software Engineer & Architect Python programmer

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Using Python to Solve Computationally Hard Problems

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Using Python to Solve Computationally Hard Problems

Rachael Madsen

Optimal Design Software LLC

– BS in Mathematics – Software Engineer & Architect – Python programmer

rachael@optimaldesignsoftware.com

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NP-Complete Problems

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NP-Complete Problems

An individual solution can be checked for correctness in polynomial time, but.. A solution cannot be derived in polynomial time

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NP-Complete Problems Nodes 1 5 10 64 Polynomial (𝒐𝟑) 1 25 100 4096 Exponential (𝟑𝒐) 2 32 1024 1.8 x 10 19 Factorial (n!) 1 120 3.6 x 10 6 1.3 x 10 89

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NP-Complete Problems

  • Sequencing tasks or objects

for any optimization

  • Database design & normalization
  • Games
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The Traveling Salesman Problem

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Traveling Salesman Problem

Given a list of cities and their pairwise distances, find the shortest possible route that visits each city exactly once and returns to the origin city.

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Algorithms

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Algorithms

Changing data may change which algorithm is best

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Algorithms

Searching available software implementations may yield the best results

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Considerations

Algorithms

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Ease of Implementation

Algorithms > Considerations

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Quality of Solution

Algorithms > Considerations

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You are looking for a solution, not the solution

Algorithms > Considerations > Quality of Solution

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Algorithms > Considerations > Quality of Solution

𝜌 = 1 16𝑙 4 8𝑙 + 1 − 2 8𝑙 + 4 − 1 8𝑙 + 5 − 1 8𝑙 + 6

∞ 𝑙=0

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Will this algorithm generate valleys?

Algorithms > Considerations > Quality of Solution

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Complexity

Algorithms > Considerations

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  • Simple

–50 lines of code

  • Complex

–100,000+ lines of code

Algorithms > Considerations > Complexity

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Parallelization

Algorithms > Considerations

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Processing Power

Algorithms > Considerations

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Starting From Scratch

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Starting From Scratch (it’s a bad idea)

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Python Options

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Python Options

Where to Look

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Python Options > Where to Look

  • Google code
  • Git
  • Forums
  • Technical books
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Python Options

Licenses

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Python Options

Evaluation

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Python Options > Evaluation

Operations Research Tools

Developed at Google

http://code.google.com/p/or-tools/source/ browse/trunk/python/tsp.py?r=303

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Runtime

  • Lines of code: 132
  • Hardware availability

Python Options > Evaluation > or-tools

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Python Options > Evaluation > or-tools

Quality of Solution

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Python Options > Evaluation

Github user trevlovett

Python Ant Colony TSP Solver

https://github.com/trevlovett/ Python-Ant-Colony-TSP-Solver

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Runtime

  • Lines of code: 132
  • Hardware availability

Python Options > Evaluation > Ant Colony TSP Solver

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Python Options > Evaluation > Ant Colony TSP Solver

Quality of Solution

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Optimization

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  • Confirm implementation of

algorithm as written

  • Check for most efficient coding

practices

Optimization

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  • Use scientific package such as SciPy

to rewrite code for speed

  • Break out select code into C routine

Optimization

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  • Parallelize

–GPU –CPU

Optimization

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Beware of fencepost errors and differences in hardware

Optimization

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Web Resources

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  • The Stony Brook Code Repository, Concorde TSP Solver

http://www.cs.sunysb.edu/~algorith/implement/concorde/im plement.shtml

  • TSP, github user denlunev, https://github.com/denlunev/TSP
  • Operations Research Tools, Google,

http://code.google.com/p/or- tools/source/browse/trunk/python/tsp.py?r=303

  • Ant Colony TSP Solver, github user trevlovett

https://github.com/trevlovett/Python-Ant-Colony-TSP-Solver

Web Resources

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Bibliography

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  • Introduction to Graph Theory, 2nd Edition, Douglas Brent

West, Prentice Hall, 2000

  • The Traveling Salesman Problem: A Computational Study,

Applegate, Bixby, Chvátal & Cook, Princeton U. Press, 2007

  • The Traveling Salesman Problem: A Guided Tour of

Combinatorial Optimization, Lawler, Lenstra, Kan & Shmoys, Wiley, 1985

  • The Traveling Salesman Problem and its Variations, Gutin &

Punnen, Springer, 2002

Bibliography

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  • Combinatorial Optimization: Theory and Algorithms, 2nd

Edition, Korte & Vygen, Springer, 2002

  • Combinatorial Optimization, Cook, Cunningham, Pulleyblank

& Schrijver, Wiley-Interscience, 1997

  • Local Search in Combinatorial Optimization, Aarts & Lenstra,

Wiley, 1997

  • Combinatorial Algorithms: Generation, Enumeration, and

Search, Kreher & Stinson,CRC Press, 1998

Bibliography

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  • Counting: The Art of Enumerative Combinatorics, Martin,

Springer, 2001

  • Counting and Configurations, Herman, Kucera & Simsa,

Springer, 2003

  • Abstract Algebra, 3rd Edition, Dummit & Foote, Wiley, 2003

Bibliography

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Thank You!