Backdoor Sets in SAT Instances

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Backdoor Sets in SAT Instances

Ryan WilliamsCarnegie Mellon UniversityJoint
work in IJCAI03 withCarla Gomes and Bart
Selman Cornell University
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Significant progress in Complete search methods!

(Complete always returns SAT or unSAT)
Software and hardware verification complete
methods are critical - e.g. for verifying the
correctness of chip design, using SAT encodings
Current methods can verify automatically the
correctness of a large fraction of a Pentium IV.
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A real world example
(Thanks to Oliver Kullmann)
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Bounded Model Checking instance

i.e. ((? x1) or x7) and ((?x1) or
x6) and etc.
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10 pages later


(x177 or x169 or x161 or x153
or x17 or x9 or x1 or (?x185)) clauses /
constraints are getting more interesting
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4000 pages later

?!! a 59-cnf clause

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Finally, 15,000 pages later
Note that
!!!
The MiniSat solver (EenSorensson) solves this
instance in 2 seconds.
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Gap between Theory and Practice

• The good scaling behavior of state-of-the art SAT
  solvers seems to defy our complexity-theoretic
  intuition that SAT is NP-complete!
• How can we explain this gap between theory and
  practice?
• What makes this possible?
• Our answer Hidden tractable substructure in
  real-world problems.
• Can we make this more precise?
• Proposal We consider structures we call
  backdoor sets.
• Idea came out of study of heavy-tailed
  phenomena in runtime
• distributions for some SAT solvers.

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Backdoor Sets Initial Motivation
Heavy-tailed distributions and Randomization. Cert
ain problems, when solved by randomized
backtracking, yield a runtime distribution that
is heavy-tailed

• Explains why restarting a solver often is an
  effective strategy
• Implies a wide range of possible solution times,
  often including short runs

Prsolution found in time t 1/tc, 0 lt c lt 2
How to explain short runs?
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Explaining short runsBackdoors to tractability

• Informally
• A backdoor set to a given problem instance is
  a subset of its variables such that, once
  assigned values, the remaining instance
  simplifies to a tractable class.
• Formally
• We define notion of a sub-solver
• (handles tractable substructure of problem
  instance)
• backdoor set and strong backdoor set

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Defining a sub-solver
Definition is general enough to encompass many
polynomial time propagation methods. (Also
those for which we do not know a clean
characterization of the tractable
subclass.) Valid for other encoding languages
besides SAT e.g., Mixed Integer Programming and
Constraint Satisfaction Problems
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Defining backdoors
Backdoor set (for satisfiable instances)
Strong backdoor set (applies to satisfiable or
inconsistent instances)
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Backdoors can be surprisingly small

• Backdoors help explain how a solver can get
  lucky on certain runs backdoor sets are
  identified early on in backtracking search.

Most recent Other combinatorial domains. E.g.
Graphplan planning, near constant size backdoors
(2 or 3 variables) in certain domains. (Hoffman,
Gomes, Selman 03) Backdoors capture critical
problem resources (bottlenecks).
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Constraint Satisfaction Problem

• The Constraint Satisfaction Problem (CSP)
• A finite set of n variables is given and with
  each variable is associated a non-empty finite
  domain.
• A constraint on k variables X1,,Xk is a relation
  R(X1,,Xk) ? D1 x x Dk.
• A solution to a CSP is an assignment of values to
  all the variables, satisfying all the
  constraints.
• (Satisfaction of a constraint the relation
  holds)

(Dechter 86, Freuder 82, Mackworth 77, Tsang 93,
van Beek and Dechter 97)
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Explicit Algorithms for Finding/Exploiting
Backdoor Sets

• We cover three kinds of strategies for dealing
  with instances with small backdoor sets
• A deterministic algorithm
• A randomized algorithm
• Provably better worst-case performance over the
  deterministic one
• A heuristic randomized algorithm
• Assumes existence of a good heuristic for
  choosing variables to branch on
• We believe this is close to what happens in
  practice

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Deterministic Generalized Iterative Deepening
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Randomized Generalized Iterative Deepening

• Assumption
• There exists a backdoor whose size is bounded by
  a function of n (call it B(n))
• Idea
• Repeatedly choose random subsets of variables
  that are slightly larger than B(n), searching
  these subsets for the backdoor

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Randomized Generalized Iterative Deepening
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Deterministic Versus Randomized
Suppose variables have 2 possible values (e.g.
SAT)
For B(n) n/k, algorithm runtime is cn
c
Deterministic algorithm
Randomized algorithm
k
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Complete Randomized Depth First Search with
Heuristic

• Assume we have the following.
• DFS, a generic depth first search randomized
• backtrack search solver with
• (polytime) sub-solver A
• Heuristic H that (randomly) chooses variables to
  branch on, in polynomial time
• H has probability 1/h of choosing a
• backdoor variable (h is a fixed constant)
• Call this ensemble (DFS, H, A)

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Polytime Restart Strategy for(DFS, H, A)

• Essentially
• If there is a small backdoor,
• then (DFS, H, A) has a restart strategy that
  runs in polytime.

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Runtime Table for Algorithms
DFS,H,A
B(n) upper bound on the size of a backdoor,
given n variables
When the backdoor is a constant fraction of n,
there is an exponential improvement between the
randomized and deterministic algorithm
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Summary

• Introduced notion of a backdoor set of
  variables.
• More closely captures combinatorics of a problem
  instance, as dealt with in practice.
• Provides insight into restart strategies.
• 3) Backdoors can be surprisingly small in
  practice.
• 4) Search heuristics randomization can be used
  to find them, provably efficiently.