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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, …. - PowerPoint PPT Presentation
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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, …
CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, …
CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, …
CSPs: 2-Sat, 3-Lin(mod 2), 3-Col, …
Satisfiability Problem
Robust Satisfiability Problem Robust
In particular,
solves satisfiability.
3-Lin(mod 2): NP-hard to get .51 fraction [Hås97]
Sat problem in P
Robust-sat problem in P
All CSPs3-Col
3-Lin(G)
2-Col
Unique-Games2-Sat
Horn-k-Sat
3-Sat
Sat problem in P
All CSPs3-Col
3-Lin(G)
2-Col
Unique-Games2-Sat
Horn-k-Sat
3-Sat
Bounded Width
Thm: iff cannot gadget-reduce any 3-Lin(G) to it. [BK09]
Sat problem in P
Robust-sat problem in PBounded width
All CSPs3-Col
3-Lin(G)
2-Col
Unique-Games2-Sat
Horn-k-Sat
3-Sat
Our results
In fact, canonical LP
relaxation solves it.
In fact, canonical LP solves
robust-sat. iff width 1.
Sat problem in P
Robust-sat problem in PBounded width
All CSPs
Width 1
3-Col
3-Lin(G)
2-Col
Unique-Games2-Sat
Horn-k-Sat
3-Sat
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
E.g.: Does 2-Col have width 1?
Bob can win (play forever), though unsatisfiable.
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
E.g.: Does 2-Col have width 2?
Alice can win.
E.g.: Does 2-Col have width 2?
Any unsat. 2-Col instance has an odd cycle: Alice can win.
• ‘Arc Consistency Alg.’ solves satisfiability problem
• it has “tree duality”
• it has a “set operation”
• canonical LP relaxation solves robust-sat. problem
• satisfiability is property-testable [Yos11]
• canonical LP relaxation solves sat. problem
• it has “symmetric polymorphisms of all arities”
• it has a “measure operation”
Theorem 1
Theorem 2
• ‘Arc Consistency Alg.’ solves satisfiability problem
• it has “tree duality”
• it has a “set operation”
• canonical LP relaxation solves robust-sat. problem
• satisfiability is property-testable [Yos11]
• canonical LP relaxation solves sat. problem
• it has “symmetric polymorphisms of all arities”
• it has a “measure operation”
Theorem 1
Theorem 2
Henceforth, only worry about binary CSPs.
Theorem 1:
Proof sketch
Summary
Thanks!
