Solution Counting Methods for Combinatorial Problems

1
Solution Counting Methodsfor Combinatorial
Problems

• Ashish Sabharwal Cornell University
• Based on joint work with
• Carla Gomes, Willem-Jan van Hoeve,Lukas Kroc,
  Bart Selman
• INFORMS, Oct 2008, Washington, D.C.

2
Context

• Constraint Satisfaction Problems (CSPs)
• In particular, Boolean Satisfiability or SAT
• Given a Boolean formula F in conjunctive normal
  forme.g. F (a or b) and (?a or ?c or d) and
  (b or c)determine whether F is satisfiable
• NP-complete
• widely used in practice, e.g. in hardware
  software verification, design automation, AI
  planning,
• How many satisfying assignments does F have?
• F, the model count of F, the solution count of
  F
• SAT is P-complete

3
Model Counting for SAT

• Inspired by the success of SAT solvers, a lot of
  activity in the last few years in attacking the
  solution counting problem
• Aside success of SAT scalability, industrial
  applications, black-box nature and standardized
  input making it easy for users
• Many different approaches, many different
  counting goals
• A zoo of techniques!
• This talk to give a brief overview of these
  techniques, many of which are contributed by our
  group at Cornell
• Further reading and refs Model Counting chapter
  in the upcoming Handbook of Satisfiability (draft
  available on my webpage)
• with Carla Gomes and Bart Selman

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What shall we count?
E.g., F has N1000 variables and 10150 2500
solutions
0
F
2N
0
F
Exact count
Strict (?,?) guarantee
Estimate, no guarantees
Lower bound
Upper bound (appears hard!)
5
Problem Space why are upper bounds hard?
E.g., F has N1000 variables and 10150 2500
solutions
0
F
2N

• Number of solutions is often a miniscule fraction
  of the search space size
• Limits our ability to reason about upper bounds
• E.g., after having searched half the space, could
  still have 2999 potential solutions remaining in
  the worst case! (off by a factor of 2499)
• Probabilistic methods work better for lower
  bounds
• E.g., if expected value true count, Markovs
  ineq. says,cant get high numbers too often
  because 0s cant compensate enough
• reverse Markovs ineq. doesnt help can get low
  numbers too oftenbecause a single 2N can
  compensate for a lot of low numbers!

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The Zoo of Counting Methods
Solutioncounting
Exact methods
Approximate methods
FPRASMCMC sampling
Practical boundswith a guarantee
Estimation withoutany guarantee
Only thecount
Count manyby-products
U
DPLL-stylebacktracksearch
Knowledgecompilation
Backtr. search randomization statistics
L
L
Using backtr.-free space
Sampling randomization
L
L
U
FPTbranch-width,tree-width,
Sampling multipliers
XOR streamlining(randomized)
Belief prop. randomization
Note not an exhaustive listing
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I. Exact Methods
Exact methods
Only thecount
Count manyby-products
CDP, Birnbaul-Lozinskii-99 relsat,
Bayardo-Pehoushek-00 cachet, Sang et
al-04 sharpSAT, Thurley-06
DPLL-stylebacktracksearch
Knowledgecompilation
FPTbranch-width,tree-width,
tree-width Gottlob-Scarcello-Sideri-02 branch-
width Bacchus-Dalmao-Pitassi-03 cluster-width
Fischer-Makowsky-Ravve-08
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Knowledge Compilation for Counting

• Main idea convert F into a different form from
  which one can easily read off the solution count
  (and many other quantities of interest)
• d-DNNF Deterministic, Decomposable Negation
  Normal Form
• Think of the formula as a directed acyclic graph
  (DAG)
• Negations allowed only at the leaves (NNF)
• Children of AND node dont share any variables
  (different components)
• Children of OR node dont share any solutions
• Once converted to d-DNNF, can answer many queries
  in linear time
• Satisfiability, tautology, logical equivalence,
  solution counts,
• Any query that a BDD could answer
• Our recent result can count number of clusters
  of solutions how many different kinds/families
  of solutions are there?

DNNF, c2d, Darwiche et al. 2001-05
can multiplythe counts
can addthe counts
To appearin NIPS-08
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II. ApproximateMethods
Karp-Luby-85Karp-Luby-Madras89
Approximate methods
FPRASMCMC sampling
Practical boundswith a guarantee
Estimation withoutany guarantee
SampleMinisat, Gogate-Dechter-07
U
Backtr. search randomization statistics
L
L
Using backtr.-free space
Sampling randomization
L
L
U
Sampling multipliers
XOR streamlining(randomized)
Belief prop. randomization
MiniCount, CPAIOR-08
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XOR Streamlining for Bounds on F

• Main idea rather than modifying the algorithm
  for solving, modify the problem, run the solver,
  deduce the count
• Randomized algorithm, expected value true count
• Can be converted into bounds with correctness
  guarantees
• Lower bounds easier in practice (XORs of any
  length work)
• Upper bounds possible but not so easy
• Empirical evidence can get by with very short
  XORs
• Can be extended to general CSPs

Mbound, AAAI-06
Off-the-shelfSAT Solver
CNF formula
Streamlinedformula
Model count
Random XORconstraints
ideal when systematic search works well!
SAT-07
AAAI-07 see Willems talk
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Sampling for Estimates Lower Bound

• Main idea find a balanced variable one
  thatappears roughly equally as True and as
  Falsein solutions fix to one value, count
  thatsub-problem, re-scale with appropriate
  multiplier
• Finding balanced variables not so easy
• Use solution sampling ideal when local search
  works well!
• Use Belief Propagation for marginal prob.
  estimates ideal
  when message passing works well!
• Randomize the process expected value true
  count, as before!
• Great lower bounds, but variance too high for
  good upper bounds

x?
T
F
E.g., count FxT, scale up by factor
100/60
40 ofsolutions
60 ofsolutions
ApproxCount, Wei-Selman-05
BPCount, CPAIOR-08
SampleCount, IJCAI-07
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The Zoo of Counting Methods
Solutioncounting
Exact methods
Approximate methods
FPRASMCMC sampling
Practical boundswith a guarantee
Estimation withoutany guarantee
Only thecount
Count manyby-products
U
DPLL-stylebacktracksearch
Knowledgecompilation
Backtr. search randomization statistics
L
L
Using backtr.-free space
Sampling randomization
L
L
U
FPTbranch-width,tree-width,
Sampling multipliers
XOR streamlining(randomized)
Belief prop. randomization
Note not an exhaustive listing