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The Evergreen Project: How To Learn From Mistakes Caused by Blurry Vision in MAX-CSP Solving. Karl J. Lieberherr Northeastern University Boston. joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart. Where we are. Introduction Look-forward Look-backward Packed truth tables - PowerPoint PPT Presentation
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PhD March 2007 1
The Evergreen Project:How To Learn From Mistakes Caused by Blurry Vision in
MAX-CSP Solving
Karl J. Lieberherr
Northeastern University
Boston
joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart
PhD March 2007 2
Where we are
• Introduction
• Look-forward
• Look-backward
• Packed truth tables
• SPOT: how to use the look-ahead polynomials (look-forward) together with superresolution (look-backward).
PhD March 2007 3
Problem Snapshot• SAT: classic problem in complexity theory• SAT & MAX-SAT Solvers: working on CNFs (a
multi-set of disjunctions).
• CSP: constraint satisfaction problem– Each constraint uses a Boolean relation.– e.g. a Boolean relation 1in3(x y z) is satisfied iff
exactly one of its parameters is true.
• CSP & MAX-CSP Solvers: working on CSP instances (a multi-set of constraints).
PhD March 2007 4
Introduction
• Boolean MAX-CSP(G) for rank d, G = set of relations of rank d– Input
• Input = Bag of Constraint• Constraint = Relation + Set of Variable• Relation = int. // Relation number < 2 ^ (2 ^ d) in G• Variable = int
– Output• (0,1) assignment to variables which maximizes the number of
satisfied constraints.
• Example Input: G = {22} of rank 3– 22:1 2 3 0 – 22:1 2 4 0 – 22:1 3 4 0 1in3 has number 22
M = {1 !2 !3 !4} satisfies all
PhD March 2007 5
Variation
MAX-CSP(G,f): Given a MAX-CSP(G) instance 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: G = {22} of rank 3MAX-CSP({22},f):
22:1 2 3 0 22:1 2 4 0 22:1 3 4 022: 2 3 4 0
PhD March 2007 6
Our Approach
• Superresolution & P-Optimality Based MAX-CSP Solver
• Highlights– Look Forward (in Abstract Representation)– Look Backward (in Transition System)– Packed Truth Tables (in Intermediate Representation)
PhD March 2007 7
Where we are
• Introduction
• Look-forward
• Look-backward
• Packed truth tables
• SPOT: how to use the look-ahead polynomials together with superresolution.
PhD March 2007 8
Look Forward
• Why?– To make informed decisions
• How?– Abstract representation based on look-ahead
polynomials
PhD March 2007 9
Look-ahead Polynomial(Intuition)
• The look-ahead polynomial computes the expected fraction of satisfied constraints among all random assignments that are produced with bias p.
PhD March 2007 10
Consider an instance: 40 variables,1000 constraints (1in3)
1, … ,40
22: 6 7 9 0
22: 12 27 38 0
Abstract representation:reduce the instance tolook-ahead poly. 3p(1-p)2
PhD March 2007 11
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)
Fra
ctio
n o
f co
nst
rain
ts t
hat
ar
e g
uar
ante
ed t
o b
e sa
tisf
ied
3p(1-p)2 for MAX-CSP({22})
PhD March 2007 12
Look-ahead Polynomial(Definition)
• F is a MAX-CSP(G) instance.
• N is an arbitrary assignment.
• The look-ahead polynomial laF,N(p) computes the expected fraction of satisfied constraints of F when each variable in N is flipped with probability p.
PhD March 2007 13
The general case MAX-CSP(G)
G = {R1, … }, tR(F) = fraction of constraints in F that use R.
x = p
PhD March 2007 14
PhD March 2007 15
Look-ahead Polynomial in Action
• Focus on purely mathematical question first
• Algorithmic solution will follow
• Mathematical question: Given a MAX-CSP(G) instance. For which fractions f is there always an assignment satisfying fraction f of the constraints? In which constraint systems is it impossible to satisfy many constraints?
PhD March 2007 16
Remember?
MAX-CSP(G,f): Given a MAX-CSP(G) instance 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: G = {22} of rank 3MAX-CSP({22},f):
22:1 2 3 0 22:1 2 4 0 22:1 3 4 022: 2 3 4 0
PhD March 2007 17
Simple example
MAX-CSP({22},f):
For f <= u: problem has always a solutionFor f = u + : problem has not always a solution,
u critical transition point
always (fluid)
not always (solid)
PhD March 2007 18
The Magic Number
• u = 4/9
PhD March 2007 19
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)
Fra
ctio
n o
f co
nst
rain
ts t
hat
ar
e g
uar
ante
ed t
o b
e sa
tisf
ied
3p(1-p)2 for MAX-CSP({22})
PhD March 2007 20
Produce the Magic Number
• Use an optimally biased coin– 1/3 in this case
• In general: min max problem
PhD March 2007 21
General Dichotomy Theorem
MAX-CSP(G,f): For each finite set G of relationsthere exists an algebraic number tG
For f <= tG: MAX-CSP(G,f) has polynomial solutionFor f = tG+ : MAX-CSP(G,f) is NP-complete,
tG critical transition point
easy (fluid)
hard (solid)
due to Lieberherr/Specker
polynomial solution:Use optimally biased coin.Derandomize.P-Optimal.
PhD March 2007 22
Observations
• The look-ahead polynomial look-forward approach has not been used in state-of-the-art MAX-SAT and Boolean MAX-CSP solvers.
• Often a fair coin is used. The optimally biased coin is often significantly better.
PhD March 2007 23
PhD March 2007 24N0 ={!v1,!v2,!v3,!v4}
PhD March 2007 25N0‘ ={v1,!v2,!v3,!v4}
PhD March 2007 26
SAT Rank 2 example9 constraints
14 : 1 2 014 : 3 4 014 : 5 6 0 7 : 1 3 0 7 : 1 5 0 7 : 3 5 0 7 : 2 4 0 7 : 2 6 0 7 : 4 6 0
14: 1 2 = or(1 2) 7: 1 3 = or(!1 !3)
What is the look-aheadpolynomial?
PhD March 2007 27appmean = lookahead is an approximation of the true mean
Blurry vision
What do we learn from the abstract representation?• set 1/3 of the variables to true (maximize).• the best assignment will satisfy at least 7/9 constraints.• very useful but the vision is blurred in the “middle”.
excellent peripheral vision
PhD March 2007 28
Where we are
• Introduction
• Look-forward
• Look-back
• Packed truth tables
• SPOT: how to use the look-ahead polynomials
PhD March 2007 29
Look Backward
• Why?– to avoid past mistakes
• How?– Transition system based on superresolution
PhD March 2007 30
Observation
• Optimally biased coin technique based on look-ahead polynomials is “best-possible”.
• If we could improve it by a trillionth in polynomial time, then P=NP.
• We improve it now by learning new constraints that will influence the polynomial.
PhD March 2007 31
Clause Learning• Let’s go beyond what an optimally biased
coin guarantees!• Goal: satisfy the maximum number of
constraints. • Approach: Superresolution.
– When to apply: number of constraints guaranteed to be unsatisfied doesn’t decrease
• A mistake is made.
– Who to blame: the decision literals• They are the culprits.
– How to penalize: add the disjunctions of their negations as a superresolvent
• The gang of culprits is watched.
PhD March 2007 32
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).
PhD March 2007 33
Algorithm plan
• start with an arbitrary assignment N.
• while (proof incomplete) {– try to improve N by creating new assignment
from scratch using optimally biased coin to flip the assignments;
• success: Update N;• failure: learn a new constraint that will prevent
same mistake and will “improve” the polynomial. }
PhD March 2007 34
PhD March 2007 35
PhD March 2007 36
Properties of TS
• TS finds the maximum in a finite number of steps.
• It creates a proof that we indeed found the maximum.
PhD March 2007 37
Optimized Semi-Superresolution
• Not all decision literals may be responsible for the “mistake”.
• Want to find a minimal superresolvent so that deleting one literal would destroy the superresolvent property.
• Can be implemented by a traversal back the implication graph that is built as part of unit propagation.
PhD March 2007 38
Where we are
• Introduction
• Look-forward
• Look-back
• Packed Truth Tables
• SPOT: how to use the look-ahead polynomials
PhD March 2007 39
Requirements for Packed Truth Tables
• The look-ahead polynomial can be computed efficiently. Requires efficient truth table analysis.
• Reduction of an instance must be efficient.
• Efficiently compute the forced variables.
• Each relation has a unique representation.
PhD March 2007 40
Packed Truth Tables
22 254
PhD March 2007 41
RelationI: implemented by bitwise operations
int isForced(int variablePosition)boolean isIrrelevant(int variablePosition)int nMap(int variablePosition)int numberOfRelevantVariables()int q(int s) int reduce(int variablePosition, int value)int rename(int permutationSemantics,
int... permutation)
PhD March 2007 42
Where we are
• Introduction
• Look-forward
• Look-back
• Packed truth tables
• SPOT: how to use the look-ahead polynomials with superresolution
PhD March 2007 43
Using the look-ahead polynomials
• Value Ordering– Decide: how to set the variable
• Variable Ordering– Which variable to set next
PhD March 2007 44
There is hope that the look-ahead polynomials are useful
PhD March 2007 45
What is new?
• New: Packed Truth Tables
• New: Superresolution for MAX-CSP
• New: Integration of look-ahead polynomials with superresolution
• Old: Superresolution for SAT (1977)
• Old: Look-ahead polynomials (1983)
PhD March 2007 46
Future work
• Exploring best combination of look-forward and look-back techniques.
• Find all maximum-assignments or estimate their number.
• Robustness of maximum assignments.
• Are our MAX-CSP solvers useful for reasoning about biological pathways?
PhD March 2007 47
Conclusions
• Presented SPOT, a family of MAX-CSP solvers based on look-ahead polynomials and non-chronological backtracking.
• SPOT has a desirable property: P-optimal.
• Preliminary experimental results are encouraging.
PhD March 2007 48
end for now
PhD March 2007 49
Rank 2 example
• 14 : 1 2 014 : 3 4 014 : 5 6 0 7 : 1 3 0 7 : 1 5 0 7 : 3 5 0 7 : 2 4 0 7 : 2 6 0 7 : 4 6 0
PhD March 2007 50appmean is an approximation of the true mean
PhD March 2007 51
PhD March 2007 52
Transition Manager
PhD March 2007 53
PhD March 2007 54
MAX-CSP:Superresolution and P-Optimality
Karl J. Lieberherr
Northeastern University
Boston
joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart
PhD March 2007 55
PhD March 2007 56
Binomial Distribution
PhD March 2007 57
PhD March 2007 58
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
PhD March 2007 59
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)
Fra
ctio
n o
f co
nst
rain
ts t
hat
ar
e g
uar
ante
ed t
o b
e sa
tisf
ied
maximize 3x(1-x)2
PhD March 2007 60
Organization of Solver
look back look forward
PhD March 2007 61
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).
PhD March 2007 62
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.
PhD March 2007 63
Transition Rules
• Update:
M || F || SR || N → M || F || SR || M
• if M is complete, and• unsat(M,F) < unsat(N,F).
PhD March 2007 64
Transition Rules
• Restart:
M || F || SR || N → { } || F || SR || N
PhD March 2007 65
Transition Rules
• Finale:
M || F || SR || N → M || F || SR || N
• if Φ SR or unsat(N,F) = 0.
PhD March 2007 66
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).
PhD March 2007 67
Transition Rules
PhD March 2007 68
Transition Rules (cont.)
PhD March 2007 69
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).
PhD March 2007 70
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).
PhD March 2007 71
Transition Rules
• Semi-Superresolution (SSR):
NewSR = V (¬k), where kєM’ subset Md
M || F || SR || N → M || F || SR, NewSR || N
• if mistake(M) and UP*(reduce(F,A(NewSR)))
PhD March 2007 72
Our Approach
• Superresolution & P-Optimality Based MAX-CSP Solver
• Highlights– Optimally Biased Coin (in Abstract Representation)– Clause Learning (in Transition System)– Bitwise Relation Reduction (in Intermediate
Representation)
PhD March 2007 73
Clause Learning• Let’s go beyond what an optimally biased
coin guarantees!• Goal: satisfy the maximum number of
constraints. • Approach: Superresolution.
– When to apply: number of constraints guaranteed to be unsatisfied doesn’t decrease
• A mistake is made.
– Who to blame: the decision literals• They are the culprits.
– How to penalize: add the disjunctions of their negations as a superresolvent
• The gang of culprits is watched.
PhD March 2007 74
Sudoku