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Advances in Linear Programming for Bounded Width CSPs and Robust Satisfaction Problems

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This paper explores linear programming methodologies applied to Constraint Satisfaction Problems (CSPs), specifically focusing on bounded-width instances. Key contributions include solutions for robust satisfiability across various CSP types such as 2-Sat, 3-Lin(mod 2), and 3-Colorability. The authors establish significant results indicating that canonical linear programming relaxations can solve robust satisfiability problems if the width is constrained to one. This work advances the understanding of complex satisfiability problems, presenting new pathways for efficient algorithmic solutions.

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Advances in Linear Programming for Bounded Width CSPs and Robust Satisfaction Problems

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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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