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).

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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).

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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

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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

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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)

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Where we are

• Introduction

• Look-forward

• Look-backward

• Packed truth tables

• SPOT: how to use the look-ahead polynomials together with superresolution.

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Look Forward

• Why?– To make informed decisions

• How?– Abstract representation based on look-ahead

polynomials

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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.

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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

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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

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3p(1-p)2 for MAX-CSP({22})

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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.

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The general case MAX-CSP(G)

G = {R1, … }, tR(F) = fraction of constraints in F that use R.

x = p

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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?

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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

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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)

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The Magic Number

• u = 4/9

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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

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tisf

ied

3p(1-p)2 for MAX-CSP({22})

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Produce the Magic Number

• Use an optimally biased coin– 1/3 in this case

• In general: min max problem

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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.

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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.

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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

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Where we are

• Introduction

• Look-forward

• Look-back

• Packed truth tables

• SPOT: how to use the look-ahead polynomials

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Look Backward

• Why?– to avoid past mistakes

• How?– Transition system based on superresolution

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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).

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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. }

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Properties of TS

• TS finds the maximum in a finite number of steps.

• It creates a proof that we indeed found the maximum.

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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.

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Packed Truth Tables

22 254

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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

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There is hope that the look-ahead polynomials are useful

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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)

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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?

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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.

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end for now

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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

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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

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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

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tisf

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maximize 3x(1-x)2

PhD March 2007 60

Organization of Solver

look back look forward

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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).

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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.

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Transition Rules

• Update:

M || F || SR || N → M || F || SR || M

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

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Transition Rules

• Restart:

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

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Transition Rules

• Finale:

M || F || SR || N → M || F || SR || N

• if Φ SR or unsat(N,F) = 0.

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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).

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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