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

1in3

0

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Coin bias (Probability of setting a variable to true)

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maximize 3x(1-x)2

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 anassignment to the n variables which satisfiesat 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

Simple example

1in3 Problem(f):

For f <= 4/9: problem has polynomial solutionFor f = 4/9 + : problem is NP-complete,

critical transition point

easy (fluid)

hard (solid)

Example

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

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?

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).

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.

Transition Rules

• Update:

M || F || SR || N → M || F || SR || M

• if M is complete, and• unsat(M,F) < unsat(N,F).

Transition Rules

• Restart:

M || F || SR || N → { } || F || SR || N

Transition Rules

• Finale:

M || F || SR || N → M || F || SR || N

• if Φ SR or unsat(N,F) = 0.

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).

Transition Rules

Transition Rules (cont.)

Transition Manager

appmean is an approximation of the true mean