Download - SAT 2009 Ashish Sabharwal Backdoors in the Context of Learning (short paper) Bistra Dilkina, Carla P. Gomes, Ashish Sabharwal Cornell University SAT-09

SAT 2009 Ashish Sabharwal

Backdoors in the Context of Learning(short paper)

Bistra Dilkina, Carla P. Gomes, Ashish Sabharwal

Cornell University

SAT-09 Conference

Swansea, U.K., June 30, 2009

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• Boolean Satisfiability or SAT :– Given a Boolean formula F in conjunctive normal form

e.g. F = (a or b) and (¬a or ¬c or d) and (b or c)determine whether F is satisfiable

– NP-complete [note: “worst-case” notion]

– widely used in practice, e.g. in hardware & software verification, design automation, AI planning, …

• Large industrial benchmarks (10K+ vars) are solved within seconds by state-of-the-art complete/systematic SAT solvers

• Even 100K or 1M not completely out of question• Good scaling behavior seems to defy “NP-completeness”!

Real-world problems have tractable sub-structure

“Backdoors” help explain how solvers canget “smart” and solve very large instances

SAT: Gap between theory & practice

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not quite Horn-SATor 2-SAT…


(~500 vars)

Informally:

A backdoor to a given problem is a subset of its variables such that, once assigned values, the remaining instance simplifies to a tractable class.

Formally:define a notion of a poly-time “sub-solver” handles tractable substructure of problem instance e.g. unit prop., pure literal elimination, CP filtering, LP solver, …

• Weak backdoors for finding feasible solutions

• Strong backdoors for finding feasible solutions or proving unsatisfiability

Backdoors to TractabilityBackdoors to Tractability

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A notion to capture “hidden structure”


The notion of backdoors has provided powerful insights, leading totechniques like randomization, restarts, and algorithm portfolios for SAT

Domain Instance Vars Clause %Vars in Bgraph coloring gcp 1500 187556 0.43planning map_50_97 38364 438840 0.25game theory pne 5000 98930.79 0.64car configuration C210_FS_RZ 1755 5764.333 0.70car configuration C210_FW_UT 2024 9720 0.74verification ssa0432-003 435 1027 3.91verification bf2670-001 1393 3434 2.80verification bf1355-638 2177 6768 10.66

Are backdoors small in practice?

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Enough to branch on backdoor variables to “solve” the formula heuristics need to be good on only a few vars


• “Traditional” backdoors are defined for a basic tree-search procedure, such as pure DPLL– Oblivious to the now-standard (and essential) feature of

learning during search, i.e, clause learning for DPLL

• Note: state-of-the-art SAT solvers rely heavily on clause learning, especially for industrial and crafted instances– provably leads to shorter proofs for many unsatisfiable formulas

– significant speed-up on satisfiable formulas as well

Does clause learning allow for smaller backdoorswhen capturing hidden structure in SAT instances?

This Talk: Motivation

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Affirmative answer:

1. First, must extend the notion of backdoors to clause learning SAT solvers: take ‘order-sensitivity’ into account

2. Theoretically, learning-sensitive backdoors for SAT solvers with clause learning (“CDCL solvers”) can be exponentially smaller than traditional strong backdoors

3. Initial empirical results suggesting that in practice,– More learning-sensitive backdoors than traditional (of a given size)– SAT solvers often find much smaller learning-sensitive backdoors

than traditional ones

This Talk: Contribution

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Input: CNF formula FAt every search node:

– branch by setting a variable to True or False;current partial variable assignment:

– consider simplified sub-formula F|

– apply a poly-time inference procedure to F|(e.g. unit prop., pure literal test, failed literal test / “probing”) Contradiction learn a conflict clause Solution declare satisfiable and exit

Not solved continue branching

“sub

-sol

ver”

fo

r S

AT

DPLL Search with Clause Learning

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Traditional Backdoor

Bac

kdoo

r

{Sub-solver

infers solution

x

y

z

w

=1

=0

=1

=1

{B

ackd

oor?

Search Tree to Solution

Contradiction:Conflict clause

learnedEarly contradictiondue to previouslylearned clause

Sub-solver infers solutionwith help from

learned clauses

x

y y

=0 =1

=1=0 =0

Search order matters!Search order matters!

Backdoors and Search with Learning

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Definition [Williams, Gomes, Selman ’03]:

A subset B of variables is a strong backdoor(for F w.r.t. a sub-solver S) if for every truth assignment to variables in B,

S “solves” F|.

Issue: oblivious to “previously” learned clauses; sub-solver must infer contradiction on F| for every from scratch.

“Traditional” Backdoors

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either finds a satisfying assignment for For proves that F is unsatisfiable


Definition:

A subset B of variables is a learning-sensitive backdoor(for F w.r.t. a sub-solver S) if there exists a search order s.t. a clause learning solver

– branching only on the variables in B– in this search order– with S as the sub-solver at each leaf

“solves” F.

New: Learning-Sensitive Backdoors

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either finds a satisfying assignment for For proves that F is unsatisfiable

Theoretical Results


Setup:

• Sub-solver: unit propagation

• Clause learning scheme: 1-UIP

• Comparison w.r.t. traditional strong backdoors

Theorem 1: There are unsatisfiable SAT instances for which learning-sensitive backdoors are exponentially smaller than the smallest traditional strong backdoors.

Theorem 2: There are satisfiable SAT instances for which learning-sensitive backdoors are smaller than the smallest traditional strong backdoors.

Learning-Sensitive Backdoors Can Provably be Much Smaller

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used Rsat for experiments


Proof Idea: Simple Example

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{x} is a learning-sensitive backdoor (of size 1) :

x=0

p1

p2

qa b

contradiction

Learn 1-UIP clause:(q)

x=1a b

contradictionq

r

With clause learning, branching on xin the right order suffices to prove unsatisfiability(x appears only

in a “long” clause)


Proof Idea: Simple Example

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In contrast, without clause learning, must branch onat least 2 variables in every proof of unsatisfiability! every “traditional” strong backdoor is of size ≥ 2

Why?•every variable, in at least one polarity, only in “long” clausese.g., p1, q, r, a do not appear in any 2-clauses

•therefore, no unit prop. or empty clause generation by fixing this variable to this value•therefore, this variable by itself cannot be a strong backdoor


Construct an unsatisfiable formula F on n vars. such that

1. certain long clauses must be used in every refutation(i.e., removing a long clause makes F satisfiable)

2. many variables in at least one polarity appear only in such long clauses with (n) variables Controlled unit propagation / empty clause generation Must branch on essentially all variables of the long clauses to

derive a contradiction Such variables must be part of every traditional backdoor set

3. With learning: conflict clauses from previous branches on O(log n) “key variables” enable unit prop. in long clauses

Proof Idea: Exponential Separation

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Corollary (follows from the proof of Theorem 1) :

There are unsatisfiable SAT instances for which learning-sensitive backdoors w.r.t. one value ordering are exponentially smaller than the smallest learning-sensitive backdoors w.r.t. another value ordering.

Order-Sensitivity of Backdoors

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Experimental evaluation


Learning-Sensitive Backdoors in Practice

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Preliminary evaluation of smallest backdoor size Reporting “best found” backdoors over 5000 runs of Rsat (with clause learning) or Satz-rand (no learning) :

•up to 10x smaller than traditional on satisfiable instances•often 2x or less smaller than traditional on unsatisfiable instances


• Considering only the size of the smallest backdoor does not provide much insight into this question

• One way to assess this difficulty:– How many backdoors are there of a given cardinality?

• Experimental setup:– For each possible backdoor size k, sample uniformly at random

subsets of cardinality k from the (discrete) variables of the problem

– For each subset, evaluate whether it is a backdoor or not

How hard is it to find small backdoor sets with learning?

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Recently reported in a paper at CPAIOR-09(backdoors in the context of optimization problems)


Backdoor Size Distribution

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E.g., for a Mixed Integer Programming (MIP)optimization instance:


Added Power of Learning

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E.g., for a Mixed Integer Programming (MIP)optimization instance:


• Defined backdoors in the context of learning during search (in particular, clause learning for SAT solvers)

• Proved that learning-sensitive backdoors can be smaller than traditional strong backdoors– Exponentially smaller on unsatisfiable instances– Somewhat smaller on satisfiable instances (open?)

• Branching order affects backdoor size as well

Future work: stronger separation for satisfiable instances; detailed empirical study

Summary

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Download - SAT 2009 Ashish Sabharwal Backdoors in the Context of Learning (short paper) Bistra Dilkina, Carla P. Gomes, Ashish Sabharwal Cornell University SAT-09

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