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Overview

Overview. MAX-CSP Input = List of Constraint Constraint = Relation + List of Variable Relation = int. // Relation number Variable = int. Example Input: One relation (22) of rank 3 22 1 2 3 22 1 2 4 22 1 3 4. 22 = 1in3. maximize 3x(1-x) 2. Simple example.

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Overview

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  1. Overview • MAX-CSP • Input = List of Constraint • Constraint = Relation + List of Variable • Relation = int. // Relation number • Variable = int. • Example Input: One relation (22) of rank 3 • 22 1 2 3 • 22 1 2 4 • 22 1 3 4 22 = 1in3

  2. maximize 3x(1-x)2

  3. Simple example 1in3 Problem(f): Given a constraint system (a bag of constraints) expressed in n variables which may assume only the values 0 or 1, find an assignment to the n variables which satisfies at least the fraction f of the constraints. Example: Constraints are of the form 1in3. 22 1 2 3 22 1 2 4 22 1 3 4 22 2 3 4

  4. Simple example 1in3 Problem(f): For f <= 4/9: problem has polynomial solution For f = 4/9 + e: problem is NP-complete, e>0. 1 hard (solid) 4/9 = critical transition point easy (fluid) 0

  5. Example x1 + x2 + x3 = 1 x1 + x2 + + x4 = 1 can satisfy 6/7 x1 + x3 + x4 = 1 x1 + x3 + x4 = 1 x1 + x2 + + x5 = 1 x1 + x3 + x5 = 1 x2 + x3 + x5 =1

  6. Forget about computation ... • Focus on purely mathematical question first • Algorithmic solution will follow • Mathematical question: Given a constraint system S, which fraction of the constraints can always be satisfied by some (0,1) assignment? In which constraint systems is it impossible to satisfy many constraints? What is the worst-case?

  7. Transition Rules • Unit-Propagation (UP): M || F || SR || N → Mk || F || SR || N • if k is undefined in M, and • unsat (M¬k,SR) > 0 or unsat(M¬k,F) ≥ unsat(N,F).

  8. Transition Rules • Decide (D): M || F || SR || N → Mkd || F || SR || N • if k is undefined in M, and • v(k) occurs in some constraint of F.

  9. Transition Rules • Update: M || F || SR || N → M || F || SR || M • if M is complete, and • unsat(M,F) < unsat(N,F).

  10. Transition Rules • Restart: M || F || SR || N → { } || F || SR || N

  11. Transition Rules • Finale: M || F || SR || N → M || F || SR || N • if Φ SR or unsat(N,F) = 0.

  12. Transition Rules • Semi-Superresolution (SSR): NewSR = V (¬k), where k Md M || F || SR || N → M || F || SR, NewSR || N • if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

  13. Transition Rules

  14. Transition Rules (cont.)

  15. Transition Manager

  16. appmean is an approximation of the true mean

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