1 / 67

When intuition is wrong Heuristics for clique and maximum clique

This work explores the challenge of identifying the largest group of people in a network where every member knows each other, known as the maximum clique problem. Using random graphs G(n, p), we analyze the presence of cliques of size k or more, a problem classified as NP-complete. We generate 100 instances with varying parameters, apply heuristics based on maximum and minimum degree choices, and examine their effectiveness and efficiency. We delve into the complexities and phase transitions associated with clique problems, seeking to optimize heuristics for better decision-making in combinatorial contexts.

tien
Télécharger la présentation

When intuition is wrong Heuristics for clique and maximum clique

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. When intuition is wrong Heuristics for clique and maximum clique Patrick Prosser esq.

  2. You have a set of people You have to produce the largest group of people such that everyone in the group knows each other How would you do that?

  3. Solve it! To a man with a hammer everything looks like a nail

  4. lovely lovely java

  5. Model 3

  6. Model 3 colouring lower bound

  7. GT[19] clique(n,p,k) Given a random graph G(n,p) is there a clique of size k or more?

  8. GT[19] clique(n,p,k) Given a random graph G(n,p) is there a clique of size k or more? A “decision problem”, NP-complete

  9. GT[19] clique(n,p,k) Given a random graph G(n,p) is there a clique of size k or more? A “decision problem”, NP-complete • What we did: • Generate 100 instances of G(50,0.9) • Vary k from 1 to 50 • apply Model 1 with max-degree heuristic • measure search cost of clique(G(50,0.9), k) • determine if sat or unsat • Analyse results (5,000 points)

  10. Search effort (decisions)

  11. Search effort (decisions) max mean scatter med Vary that

  12. Complexity peak coincide with crossover point

  13. vary p

  14. vary n

  15. clique(50,0.9,k) for four heuristics

  16. Search is on a log scale!

  17. What are those heuristics?!

  18. H00: choose max degree and reject H01: choose max degree and accept H10: choose min degree and reject H11: choose min degree and accept

  19. H00: choose max degree and reject H01: choose max degree and accept H10: choose min degree and reject H11: choose min degree and accept H1*

  20. WHY? H1*

  21. SoCS-TV

  22. Normal service will soon be resumed SoCS-TV

  23. … but just to build tension, here’s what happens when we compare models (normal service will soon be resumed)

  24. … and we are back SoCS-TV

  25. What’s all that “phase transition” malarkey?

  26. Constrainedness (circa 1996)

  27. kappa for clique

  28. kappa for clique kappa seems to work … it fits the empirical results reasonably well

  29. minimise kappa

  30. minimise kappa When you make a decision make sure it drives you into the easy soluble region … capiche? easy soluble

  31. minimise kappa

  32. minimise kappa

  33. minimise kappa

  34. minimise kappa

  35. minimise kappa

  36. I wonder if kappa can predict heuristic behaviour?

  37. If a heuristic is good for the decision problem, will it be good for optimisation?

  38. optimisation

  39. What about the DIMACS benchmarks?

  40. DIMACS

  41. DIMACS

  42. DIMACS ?

More Related