PhD March 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 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 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 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: –22: –22: in3 has number 22 M = {1 !2 !3 !4} satisfies all

PhD March 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 3 MAX-CSP({22},f): 22: : : :

PhD March 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 Where we are Introduction Look-forward Look-backward Packed truth tables SPOT: how to use the look-ahead polynomials together with superresolution.

PhD March Look Forward Why? –To make informed decisions How? –Abstract representation based on look-ahead polynomials

PhD March 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 Consider an instance: 40 variables, 1000 constraints (1in3) 1, …,40 22: : Abstract representation: reduce the instance to look-ahead poly. 3p(1-p) 2

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

PhD March Look-ahead Polynomial (Definition) F is a MAX-CSP(G) instance. N is an arbitrary assignment. The look-ahead polynomial la F,N (p) computes the expected fraction of satisfied constraints of F when each variable in N is flipped with probability p.

PhD March The general case MAX-CSP(G) G = {R 1, … }, t R (F) = fraction of constraints in F that use R. x = p

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PhD March 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 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 3 MAX-CSP({22},f): 22: : : :

PhD March Simple example MAX-CSP({22},f): For f <= u: problem has always a solution For f = u +  : problem has not always a solution,    u  critical transition point always (fluid) not always (solid)

PhD March The Magic Number u = 4/9

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

PhD March Produce the Magic Number Use an optimally biased coin –1/3 in this case In general: min max problem

PhD March General Dichotomy Theorem MAX-CSP(G,f): For each finite set G of relations there exists an algebraic number t G For f <= t G : MAX-CSP(G,f) has polynomial solution For f = t G +  : MAX-CSP(G,f) is NP-complete,   t G  critical transition point easy (fluid) hard (solid) due to Lieberherr/Specker polynomial solution: Use optimally biased coin. Derandomize. P-Optimal. 

PhD March 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 N 0 ={!v 1,!v 2,!v 3,!v 4 }

PhD March N 0 ‘ ={v 1,!v 2,!v 3,!v 4 }

PhD March SAT Rank 2 example 9 constraints 14 : : : : : : : : : : 1 2 = or(1 2) 7: 1 3 = or(!1 !3) What is the look-ahead polynomial?

PhD March appmean = 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 Where we are Introduction Look-forward Look-back Packed truth tables SPOT: how to use the look-ahead polynomials

PhD March Look Backward Why? –to avoid past mistakes How? –Transition system based on superresolution

PhD March 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 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 Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

PhD March 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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PhD March 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 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 Where we are Introduction Look-forward Look-back Packed Truth Tables SPOT: how to use the look-ahead polynomials

PhD March 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 Packed Truth Tables

PhD March 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 Where we are Introduction Look-forward Look-back Packed truth tables SPOT: how to use the look-ahead polynomials with superresolution

PhD March Using the look-ahead polynomials Value Ordering –Decide: how to set the variable Variable Ordering –Which variable to set next

PhD March There is hope that the look-ahead polynomials are useful

PhD March 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 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 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 end for now

PhD March Rank 2 example 14 : : : : : : : : : 4 6 0

PhD March appmean is an approximation of the true mean

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PhD March Transition Manager

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PhD March MAX-CSP: Superresolution and P-Optimality Karl J. Lieberherr Northeastern University Boston joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart

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PhD March Binomial Distribution

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

PhD March maximize 3x(1-x) 2

PhD March Organization of Solver look backlook forward

PhD March 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 Transition Rules Decide (D): M || F || SR || N → Mk d || F || SR || N if k is undefined in M, and v(k) occurs in some constraint of F.

PhD March Transition Rules Update: M || F || SR || N → M || F || SR || M if M is complete, and unsat(M,F) < unsat(N,F).

PhD March Transition Rules Restart: M || F || SR || N → { } || F || SR || N

PhD March Transition Rules Finale: M || F || SR || N → M || F || SR || N if Φ SR or unsat(N,F) = 0.

PhD March Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

PhD March Transition Rules

PhD March Transition Rules (cont.)

PhD March Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

PhD March Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where k M d M || F || SR || N → M || F || SR, NewSR || N if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).

PhD March Transition Rules Semi-Superresolution (SSR): NewSR = V (¬k), where kєM’ subset M d M || F || SR || N → M || F || SR, NewSR || N if mistake(M) and UP*(reduce(F,A(NewSR)))

PhD March 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 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 Sudoku