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When intuition is wrong Heuristics for clique and maximum clique

When intuition is wrong Heuristics for clique and maximum clique. Patrick Prosser esq. You have a set of people You have to produce the largest group of people such t hat everyone in the group knows each other How would you do that?. Solve it!.

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When intuition is wrong Heuristics for clique and maximum clique

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  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 ?

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