The Evergreen Project: How To Learn From Mistakes Caused by Blurry Vision in MAX-CSP Solving

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The Evergreen ProjectHow 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
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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 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
• 221 2 3 0
• 221 2 4 0
• 221 3 4 0

1in3 has number 22 M 1 !2 !3 !4 satisfies all
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Decision MAX-CSP(G,f)

• MAX-CSP(22,f)
• Given a MAX-CSP(22) instance (a bag of
  constraints using relation 22 1in3) 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 use 1in3 22.
• 221 2 3 0
• 221 2 4 0
• 221 3 4 0
• 22 2 3 4 0

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MAX-CSP

• Search approach Look for forced variables before
  making a decision (as in Sudoku)
• Look-forward make informed decisions
• Abstract representation based on look-ahead
  polynomials
• Look-backward avoid past mistakes
• Transition system based on superresolution

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Organization of Solver
look back
look forward
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Look-ahead polynomial

• 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
to look-ahead poly. 3p(1-p)2
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3p(1-p)2 for MAX-CSP(22)
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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-ahead polynomial?
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excellent peripheral vision
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.

appmean lookahead is an approximation of the
true mean
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Forget about computation ...

• Focus on purely mathematical question first
• Algorithmic solution will follow
• Mathematical question Given a MAX-CSP(G,f)
  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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Simple example
MAX-CSP(22,f) For f lt u problem has
always a solution For f u e problem has not
always a solution, egt0.
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not always (solid)
u critical transition point
always (fluid)
0
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3p(1-p)2 for MAX-CSP(22)
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The Magic Number

• u 4/9

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Look-ahead Polynomial

• 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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General Dichotomy Theorem
MAX-CSP(G,f) For each finite set G of
relations there exists an algebraic number
tG For f lt tG MAX-CSP(G,f) has polynomial
solution For f tG e MAX-CSP(G,f) is
NP-complete, egt0.
polynomial solution Use maximally biased
coin. Derandomize.
hard (solid)
tG critical transition point
easy (fluid)
0
due to Lieberherr/Specker
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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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Where we are

• Introduction
• Look-forward
• Look-back
• Packed truth tables

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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 PNP.
• We improve it now by learning new constraints
  that will influence the polynomial.

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Algorithm plan

• start with assignment N all zero.
• 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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N0 !v1,!v2,!v3,!v4
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N0 v1,!v2,!v3,!v4
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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 (Mk,SR) gt 0 or unsat(Mk,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) lt 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 F 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) gt 0 or unsat(M,F) unsat(N,F).

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Transition Rules
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Transition Rules (cont.)
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Transition Manager
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Where we are

• Introduction
• Look-forward
• Look-back
• Packed truth tables

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Requirements

• 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
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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

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appmean is an approximation of the true mean
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Transition Manager
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MAX-CSPSuperresolution and P-Optimality

• Karl J. Lieberherr
• Northeastern University
• Boston

joint work with Ahmed Abdelmeged, Christine Hang
and Daniel Rinehart
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Binomial Distribution
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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
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maximize 3x(1-x)2
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Organization of Solver
look back
look forward