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Linear programming, width-1 CSPs, and robust satisfaction

Linear programming, width-1 CSPs, and robust satisfaction. Ryan O’Donnell Carnegie Mellon University. joint with Gabor Kun (IAS), Suguru Tamaki (Kyoto), Yuichi Yoshida (Kyoto), Yuan Zhou (CMU). CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, ….

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Linear programming, width-1 CSPs, and robust satisfaction

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  1. Linear programming, width-1 CSPs, and robust satisfaction Ryan O’Donnell Carnegie Mellon University joint with Gabor Kun (IAS), Suguru Tamaki (Kyoto), Yuichi Yoshida (Kyoto), Yuan Zhou (CMU)

  2. CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, …

  3. CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, …

  4. CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, …

  5. CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, …

  6. Satisfiability Problem

  7. Robust Satisfiability Problem Robust In particular, solves satisfiability.

  8.  3-Lin(mod 2): NP-hard to get .51 fraction [Hås97]    

  9. All CSPs 3-Col 3-Sat Sat problem in P 3-Lin(G) Robust-sat problem in P Unique-Games 2-Sat 2-Col Horn-k-Sat

  10. All CSPs 3-Col 3-Sat Sat problem in P 3-Lin(G) Bounded Width Unique-Games 2-Sat 2-Col Horn-k-Sat

  11. Thm: iff cannotgadget-reduce any 3-Lin(G) to it. [BK09]

  12. All CSPs 3-Col 3-Sat Sat problem in P 3-Lin(G) Bounded width Robust-sat problem in P Unique-Games 2-Sat 2-Col Horn-k-Sat

  13. Our results In fact, canonical LP relaxation solves it. In fact, canonical LP solves robust-sat. iff width 1.

  14. All CSPs 3-Col 3-Sat Sat problem in P 3-Lin(G) Bounded width Robust-sat problem in P Unique-Games 2-Sat 2-Col Width 1 Horn-k-Sat

  15. E.g.: Does 2-Col have width 1?

  16. E.g.: Does 2-Col have width 1?

  17. E.g.: Does 2-Col have width 1?

  18. E.g.: Does 2-Col have width 1?

  19. E.g.: Does 2-Col have width 1?

  20. E.g.: Does 2-Col have width 1?

  21. E.g.: Does 2-Col have width 1?

  22. E.g.: Does 2-Col have width 1?

  23. E.g.: Does 2-Col have width 1?

  24. E.g.: Does 2-Col have width 1?

  25. E.g.: Does 2-Col have width 1?

  26. E.g.: Does 2-Col have width 1? Bob can win (play forever), though unsatisfiable.

  27. E.g.: Does 2-Col have width 2?

  28. E.g.: Does 2-Col have width 2?

  29. E.g.: Does 2-Col have width 2?

  30. E.g.: Does 2-Col have width 2?

  31. E.g.: Does 2-Col have width 2?

  32. E.g.: Does 2-Col have width 2?

  33. E.g.: Does 2-Col have width 2?

  34. E.g.: Does 2-Col have width 2?

  35. E.g.: Does 2-Col have width 2?

  36. E.g.: Does 2-Col have width 2?

  37. E.g.: Does 2-Col have width 2?

  38. E.g.: Does 2-Col have width 2?

  39. E.g.: Does 2-Col have width 2?

  40. E.g.: Does 2-Col have width 2?

  41. E.g.: Does 2-Col have width 2?

  42. E.g.: Does 2-Col have width 2?

  43. E.g.: Does 2-Col have width 2?

  44. E.g.: Does 2-Col have width 2?

  45. E.g.: Does 2-Col have width 2?

  46. E.g.: Does 2-Col have width 2? Alice can win.

  47. E.g.: Does 2-Col have width 2? Any unsat. 2-Col instance has an odd cycle: Alice can win.

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