The Evergreen Project: The Promise of Polynomials to Boost CSP/SAT Solvers*

1
The Evergreen Project The Promise of
Polynomials to Boost CSP/SAT Solvers

• Karl J. Lieberherr
• Northeastern University
• Boston

joint work with Ahmed Abdelmeged, Christine Hang
and Daniel Rinehart
Title inspired by a paper by Carla Gomes / David
Shmoys
2
Two objectives

• I want you to become
• better writers of MAX-SAT/MAX-CSP solvers
• UBCSAT (Tompkins/Hoos) can it be improved by what
  you learn today?
• better players of the Evergreen game

3
Introduction

• Boolean MAX-CSP(G) for rank d, G set of
  relations of rank d
• Input
• Input Bag of Constraint CSP(G) instance
• Constraint Relation Set of Variable
• Relation int. // Relation number lt 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. H
• 221 2 3 0
• 221 2 4 0
• 221 3 4 0

1in3 has number 22 M 1 !2 !3 !4 satisfies all
4
Variation

• MAX-CSP(G,f)
• Given a CSP(G) instance H 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 in H.
• Example G 22 of rank 3
• MAX-CSP(22,f) H
• 221 2 3 0
• 221 2 4 0 in MAX-CSP(22,?).
  Highest value for ?
• 221 3 4 0
• 22 2 3 4 0

1in3 has number 22
5
Evergreen(3,2) game
P1 you P2 I

• Two players They agree on a protocol P1 to
  choose a set of 2 relations (G) of rank 3.
• Player P1 chooses CSP(G)-instance H (limited).
• Player P2 solves H and gets paid by P1 the
  fraction that P2 satisfies in H.
• This gives nice control to P1. P1 will choose an
  instance that will minimize P2s profit.
• Take turns.

R2, P2
R1, P1
100 R1, 0 R2
100 R2, 0 R1
6
Protocol choice

• Randomly choose symmetric R1 and symmetric R2
  (independently) between 1 and 255 (Throw two dice
  choosing symmetric relations).

7
Tell me

• How would you react as player P1?
• The relations 22 and 22 have been chosen.
• You must create a CSP(22) instance with 1000
  variables in which only the smallest possible
  fraction can be satisfied.
• What kind of instance will this be?
• What kind of algorithm should P2 use to maximize
  its payoff?

8
Game strategy in a nutshell

• Choose GR1,R2 randomly (both symmetric).
• P1 chooses instance so that payoff is minimized.
• P2 finds solution so that payoff is maximized
  (Solve MAX-CSP(G))
• Take turns Choose G
• P2. Choose instance so that payoff is minimized.

9
Our approach by ExampleSAT 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

14 1 2 or(1 2) 7 1 3 or(!1 !3)
H
Evergreen game maximize payoff find maximum
assignment
10
excellent peripheral vision
0 1 2
3 4 5 6
k
8/9
7/9
Blurry vision

• What do we learn from the abstract representation
  absH?
• 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 blurry in the
  middle.

appmean approximation of the mean (k variables
true)
11
Our approach by Example

• Given a CSP(G)-instance H and an assignment N
  which satisfies fraction f in H
• Is there an assignment that satisfies more than
  f?
• YES (we are done), absH(mb) gt f
• MAYBE, The closer absH() comes to f, the better
• Is it worthwhile to set a certain literal k to 1
  so that we can reach an assignment which
  satisfies more than f
• YES (we are done), H1 Hk1, absH1(mb1) gt f
• MAYBE, the closer absH1(mb1) comes to f, the
  better
• NO, UP or clause learning

absH abstract representation of H
12

• 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

8/9
7/9
abstract representation
3/9
H
0 1 2 3 4 5
6
14 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
6/7 8/9
5/77/9
H0
3/75/9
maximum assignment away from max bias blurry
0 1 2 3 4
5
13

• 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

8/9
7/9
3/8
H
0 1 2 3 4 5
6
clearly above 3/4
14 1 2 014        3 4
014            5 6 0  7
    3            0 7         5      0 7
       3   5      0 7    2     4         
0 7    2         6   0 7          4  
6   0
7/88/9
6/87/9
H1
maximum assignment away from max bias blurry
2/73/8
0 1 2 3 4
5
14
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
8/9
7/9
abstract representation guarantees 7/9
H
7/8 8/9
6/78/9
6/8 7/9
5/77/9
abstract representation guarantees 7/9
abstract representation guarantees 8/9
H0
UBCSAT
H1
NEVER GOES DOWN DERANDOMIZATION
15
rank 2 10 1 or(1) 7 1 2 or(!1 !2)
4/6
4/6
10 1 0 10 2 0 10 3 0 7 1 2 0 7 1 3
0 7 2 3 0
3/6
3/6
abstract representation guarantees 0.625 6
3.75 4 satisfied.
Evergreen game G 10,7 How do you choose
a CSP(G)-instance to minimize payoff? 0.618
0 1 2
3
4/6
4/6
4/6
4/6
5 1 0 10 2 0 10 3 0 13 1 2 0 13 1 3
0 7 2 3 0
3/6
3/6
rank 2 5 1 or(!1) 13 1 2 or(1 !2)
The effect of n-map
16
First Impression

• The abstract representation look-ahead
  polynomials seems useful for guiding the search.
• The look-ahead polynomials give us averages the
  guidance can be misleading because of outliers.
• But how can we compute the look-ahead
  polynomials? How do the polynomials help play the
  Evergreen(3,2) game?

17
Where we are

• Introduction
• Look-forward
• Look-backward
• SPOT how to use the look-ahead polynomials
  together with superresolution.

18
Look Forward

• Why?
• To make informed decisions
• To play the Evergreen game
• How?
• Abstract representation based on look-ahead
  polynomials

19
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.

20
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
to look-ahead polynomial 3p(1-p)2 B1,3(p)
(Bernstein)
21
3p(1-p)2 for MAX-CSP(22)
22
Look-ahead Polynomial(Definition)

• H is a CSP(G) instance.
• N is an arbitrary assignment.
• The look-ahead polynomial laH,N(p) computes the
  expected fraction of satisfied constraints of H
  when each variable in N is flipped with
  probability p.

23
The general case MAX-CSP(G)
G R1, , tR(F) fraction of constraints in
F that use R.
appSATR(x) over all R is a super set of the
Bernstein polynomials (computer graphics,
weighted sum of Bernstein polynomials)
x p
24
Rational Bezier Curves
25
Bernstein Polynomials
http//graphics.idav.ucdavis.edu/education/CAGDNot
es/Bernstein-Polynomials.pdf
26
all the appSATR(x) polynomials
27
Look-ahead Polynomial in Action

• Focus on purely mathematical question first
• Algorithmic solution will follow
• Mathematical question Given a 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?

28
Remember?

• MAX-CSP(G,f)
• Given a CSP(G) instance H 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 in H.
• Example G 22 of rank 3
• MAX-CSP(22,f)
• 221 2 3 0
• 221 2 4 0
• 221 3 4 0
• 22 2 3 4 0

29
Mathematical Critical Transition Point
MAX-CSP(22,f) For f u problem has always
a solution For f u e problem has not always
a solution, egt0.
1
not always (solid)
u critical transition point
always (fluid)
0
30
The Magic Number

• u 4/9

31
3p(1-p)2 for MAX-CSP(22)
32
Produce the Magic Number

• Use an optimally biased coin
• 1/3 in this case
• In general min max problem

33
The 22 reductionsNeeded for implementation
1,0
2,0
22
60
240
3,0
2,1
3,1
1,1
3,0
2,0
3
15
255
3,1
2,1
22 is expanded into 6 additional relations.
0
34
The 22 N-MappingsNeeded for implementation
1
41
134
2
2
0
0
1
1
22
73
146
104
2
2
0
0
97
148
1
22 is expanded into 7 additional relations.
35
General Dichotomy Theorem
MAX-CSP(G,f) For each finite set G of relations
closed under renaming there exists an algebraic
number tG For f tG MAX-CSP(G,f) has
polynomial solution For f tG e MAX-CSP(G,f)
is NP-complete, egt0.
1
hard (solid) NP-complete
polynomial solution Use optimally biased
coin. Derandomize. P-Optimal.
tG critical transition point
easy (fluid) Polynomial
0
due to Lieberherr/Specker (1979, 1982)
implications for the Evergreen game? Are you a
better player?
36
Context

• Ladner Lad 75 if P !NP, then there are
  decision problems in NP that are neither
  NP-complete, nor they belong to P.
• Conceivable that MAX-CSP(G,f) contains problems
  of intermediate complexity.

37
General Dichotomy Theorem(Discussion)
MAX-CSP(G,f) For each finite set G of relations
closed under renaming there exists an algebraic
number tG For f tG MAX-CSP(G,f) has
polynomial solution For f tG e MAX-CSP(G,f)
is NP-complete, egt0.
1
hard (solid), NP-complete exponential,
super-polynomial proofs ??? relies on clause
learning
tG critical transition point
easy (fluid), Polynomial (finding an
assignment) constant proofs (done statically
using look-ahead polynomials) no clause learning
0
38
min max problem
sat(H,M) fraction of satisfied constraints in
CSP(G)-instance H by assignment M
tG min max
sat(H,M)
all (0,1) assignments M
all CSP(G) instances H
39
Problem reductions are the key

• Solution to simpler problem implies solution to
  original problem.

40
min max problem
sat(H,M,n) fraction of satisfied constraints
in CSP(G)-instance H by assignment M with n
variables.
tG lim min
max sat(H,M,n)
all (0,1) assignments M to n variables
n to infinity
all SYMMETRIC CSP(G) -instances H with n
variables
41
Reduction achieved

• Instead of minimizing over all constraint systems
  it is sufficient to minimize over the symmetric
  constraint systems.

42
Reduction

• Symmetric case is the worst-case If in a
  symmetric constraint system the fraction f of
  constraints can be satisfied, then in any
  constraint system the fraction f can be satisfied.

43
Symmetric the worst-case
n variables n! permutations
If in the big system the fraction f is satisfied,
then there must be a least one small
system where the fraction f is satisfied
. .
44
min max problem
sat(H,M,n) fraction of satisfied constraints
in system S by assignment I
tG lim min
max sat(H,M,n)
all (0,1) assignments M to n variables where
the first k variables are set to 1
n to infinity
all SYMMETRIC CSP(G) -instances H with n
variables
45
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.

46
(No Transcript)
47
The Game Evergreen(r,m) for Boolean MAX-CSP(G),
rgt1,mgt0

• Two players They agree on a protocol P1 to
  choose a set of m relations of rank r.
• The players use P1 to choose a set G of m
  relations of rank r closed under renamings.
• Player 1 constructs a CSP(G) instance H with 1000
  variables and at most 2(1000 choose r)
  constraints and gives it to player 2 (1 second
  limit).
• Player 2 gets paid the fraction of constraints
  she can satisfy in H (100 seconds limit).
• Take turns (go to 1).

48
For Evergreen(3,2)
100 R1, 0 R2
100 R2, 0 R1
49
Evergreen(3,2) protocol possibilities

• Variant 1
• Player P2 chooses both relations G
• Player P1 chooses CSP(G) instance H.
• Player P2 solves H and gets paid by P1.
• This gives too much control to P2. P2 can choose
  two odd relations which guarantees P2 a pay of 1
  independent of how P1 chooses the instance H.

50
Evergreen(3,2) protocol possibilities

• Variant 2
• Player P1 chooses a relation R1 (e.g. 22).
• Player P2 chooses a relation R2.
• Player P1 chooses CSP(G) instance H.
• Player P2 solves H and gets paid by P1.
• This gives nice control to P2. P2 will choose a
  relation R2 that will maximize P2s profit.

R2, P2
R1, P1
100 R1, 0 R2
100 R2, 0 R1
51
Problem with variant 2

• P1 can just ignore relation R2.
• Gives P1 too much control because the payoff for
  P2 depends only on R1 chosen by P1.

52
Protocol choice variant 3

• Randomly choose symmetric R1 and symmetric R2
  (independently) between 1 and 255 (Throw two
  dice).

53
Tell me

• How would you react as player P1?
• The relations 22 and 22 have been chosen.
• You must create a CSP(22) instance with 1000
  variables in which only the smallest possible
  fraction can be satisfied.
• What kind of instance will this be? 4/9
• What kind of algorithm should P1 use to maximize
  its payoff? compute optimal k best MAX-CSP
  solver.

symmetric instance with (1000 choose 3)
constraints
54
For Evergreen(3,2)
Tells us how to mix the two relations
100 R1, 0 R2
100 R2, 0 R1
55
Role of tG in the Evergreen(3,2) game

• 1 Instance construction Choose a CSP(G)
  instance so that only the fraction tG can be
  satisfied symmetric formula.
• 2 Choose an algorithm so that at least the
  fraction tG is satisfied. (2 gets paid tG from
  1).

56
Game strategy in a nutshell

• 1. Best Choose tG instance
• 2. Best Gets paid tG
• etc.

57
Additional Information

• Rich literature on clause learning in SAT and CSP
  solver domain. Superresolution is the most
  general form of clause learning with restarts.
• Papers on look-ahead polynomials and
  superresolution http//www.ccs.neu.edu/resea
  rch/demeter/papers/publications.html

58
Additional Information

• Useful unpublished paper on look-ahead
  polynomials http//www.ccs.neu.edu/research/demet
  er/biblio/partial-sat-II.html
• Technical report on the topic of this talk
  http//www.ccs.neu.edu/research/demeter/biblio/POp
  tMAXCSP.html

59
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?

60
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.
• SPOT can be implemented very efficiently.
• Preliminary experimental results are encouraging.
  A lot more work is needed to assess the practical
  value of the look-ahead polynomials.

61
Polynomials for rank 3

• x3 x2 x1 x0 relation
• -1 3 -3 1 1
• 1 -2 1 0 2
• 0 1 -2 1 3
• 1 -2 1 0 4
• 0 1 -2 1 5
• For 2 x(1-x)2x3-2x2x
• maximum at x1/3 1/3(2/3)24/27

62
Recall

• (fg) fg fg
• (f2) 2f f
• For relation 2
• x(1-x)2 (1-x)2 x2(1-x)(-1) (1-x)(1-3x)
• x1 is minimum
• x1/3 is maximum
• value at maximum 4/27

63
Check 2 and 4 should be the same
64
Harold

• concern intension, extension query, predicate
• extension intension(software)
• Harold Ossher confirmed pointcuts

65
The Game Evergreen(r,m) for Boolean MAX-CSP(G),
rgt1,mgt0

• Two players They agree on a protocol P1 to
  choose a set of m relations of rank r.
• The players use P1 to choose a set G of m
  relations of rank r.
• Player 1 constructs a CSP(G) instance H with 1000
  variables and at most 2(1000 choose r)
  constraints and gives it to player 2 (1 second
  limit).
• Player 2 gets paid by player 1 the fraction of
  constraints she can satisfy in H (100 seconds
  limit).
• Take turns (go to 1).

66
Evergreen(3,2)

• Rank 3 Represent relations by the integer
  corresponding to the truth table in standard
  sorted order 000 111.
• choose relations between 1 and 254 (exclude 0 and
  255).
• Dont choose two odd numbers All false would
  satisfy all constraints.
• Dont choose both numbers above 128 All true
  would satisfy all constraints.

67
The Game Evergreen(r,m) for Boolean MAX-CSP(G),
rgt1,mgt0

• Two players They agree on a protocol P1 to
  choose a set of m relations of rank r.
• The players use P1 to choose a set G of m
  relations of rank r.
• Player 1 constructs a CSP(G) instance H with 1000
  variables and at most 2(1000 choose r)
  constraints and gives it to player 2 (1 second
  limit).
• Player 2 gets paid by player 1 the fraction of
  constraints she can satisfy in H (100 seconds
  limit).
• Take turns (go to 1).

68
Requirements for algorithms and properties to work

• Relative P-optimality
• Absolute P-optimality
• G needs to be closed under renaming and
  reductions and n-maps
• Look-ahead polynomials

69
What happens during the solution process

• Maximum of polynomial will be at the boundary,
  say 0. Can be achieved in P. Notice folding
  effect.
• Many superresolvents will be learned until better
  assignment is found.
• Most constraints use an odd relation, a few an
  even relation (if many constraints can be
  satisfied).

70
What happens

• Because the polynomial only depends on a few
  numbers, it is not sensitive to the detailed
  properties of the instance.
• But if one variable has a visible bias towards
  either 1 or 0, polynomials might detect it.
• Adjust the weight of the constraints to bring the
  maximum of the polynomial into the middle so that
  abs(mb) increases.

71
Question for Daniel

• p(x) t1p1(x)t2p2(x)
• mb at 0
• p(mb)
• perturb t1,t2 so that p(x) gets a higher maximum.
  The fraction of t1 should go up if R1 is an
  unsatisfied relation.
• How high can we bring the fraction of satisfied
  constraints this way?

72
Question

• Does this solve the original problem?
• If we get all all satisfied, yes.
• Can force that, by deleting all but one
  unsatisfied and adding them later on.
• Are forced to work with many relations.