Integrating Systematic and Local Search: A New Strategy for MaxSAT

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Integrating Systematic and Local SearchA New
Strategy for MaxSAT

• Lukas Kroc, Ashish Sabharwal,Carla P. Gomes,
  Bart Selman
• Cornell University
• IJCAI 2009, Pasadena, CA
• July 17, 2009

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Combinatorial Search Paradigms
Context Constraint Satisfaction Problems (CSPs),
Boolean Satisfiability (SAT)

• Local search
• Typically focused random walk
• A bunch of bells and whistles
• Great on under-constrainedinstances, and more

• Systematic search
• Typically branch-and-bound
• A bunch of bells and whistles
• Great on highly structuredinstances

Other, less mainstream, paradigms toomessage
passing, knowledge compilation, streamlining,

• Generally believed to have complementary
  strengths
• Would like to harness the power of bothparadigms
  through a hybrid method!
• (especially with dual-core / multi-core hardware)

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Integrating Systematic Local Search

• Several attempts
• E.g., for SAT Mazure-Sais-Gregoire 98,
  Gregoire-Mazure-Piette 07, Hamadi-Jabbour-Sais
  08, etc.
• More work in various contexts arguably limited
  success so far
• Challenges
• What kind of information to pass between the two
  solvers?
• When and how to exchange information
  efficiently? Should one process wait for the
  other? Dovetail? Run in parallel and
  synchronize?
• How to effectively use exchanged information?
• Finding the right balance holds the key!
• This work continuously exchange light-weight
  information (one-way)

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SAT and MaxSAT

• SAT Given a Boolean formula such as
• find, if possible, a truth assignment such that
  F is satisfied.
• MaxSAT Given a Boolean formula,find a truth
  assignment that satisfies as many constraints as
  possible
• Many applications in combinatorial and
  probabilistic reasoning(e.g., critical failure
  paths as a tool within Markov Logic Networks)
• Traditional approaches for MaxSAT
• Systematic search complete but limited
  scalability (e.g., MiniMaxSat Heras et al. 07,
  Msuf Marques-Silva et al.08, MaxSatz Li et
  al.07)
• Local search scalable but sometimes far from
  optimum (e.g., Walksat Selman-Kautz-Cohen 96,
  GLS Mills et al. 00, SAPS Hutter-Tompkins-Hoos
  02-03, Adaptg2wsatp Li-Wei-Zhang 07)

x y SatCls0 0 3 0 1 4 1 0 4 1 1 5
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Local Search Natural for MaxSAT?

• Local search methods walk on a landscape composed
  of truth assignments
• Height number of unsatisfied constraints
• At every point in time, have a (sub-optimal)
  MaxSAT solution!
• anytime solution to MaxSAT
• Bottleneck local minima, just as in local search
  for SAT

credit reactive-search.org
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Systematic Search 2 Key Features

• The DPLL process, and extensions, for SAT
• Search guided by
• Well-informed heuristics
• Clauses falsified or made critical during
  search
• Efficiency relies heavily on the ability to avoid
  local inconsistencies and prune the search space
• Unit propagation if a0, b0, and have
  clause (a or b or ?c), better force c1 right
  away
• Clause learning if a search branch reaches a
  contradiction, learn why this happened and
  never let this happen again

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Systematic Search 2 Key Features

• Unfortunately, unit propagation and clause
  learning arenot suitable for DPLL-based MaxSAT
  solvers
• For MaxSAT, it may be best to ignore the current
  local inconsistency and try to satisfy the rest
  of the formula as much as you can
• Systematic DPLL-style MaxSAT solvers necessarily
  do not employ traditional unit propagation or
  clause learningnote may use more sophisticated
  resolution-based inference
• Cost? Systematic MaxSAT solvers are
  significantly less scalable than
  systematic SAT solverse.g., verification
  instance cmu-bmc-barrel6.cnf best
  MaxSAT solvers need 20-30 minutes
  MiniSat needs 2 seconds

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The Question(s)
Can state-of-the-art systematic SAT techniques be
used as (incomplete) effective heuristics for
MaxSAT?

• Perhaps as a guidance mechanism for local search?
• To zoom down to near-satisfying assignments?

Yes! this talk, also at IJCAI-09 Effort driven
by local search, guided by DPLL running in
parallel
Yes! recently presented at SAT-09 Effort
driven by relaxed DPLL search, followed by
local search if needed
Aside What do best possible near-solutions of
unsatisfiable industrial SAT
instances (100K variables, 100K-1M clauses) look
like? 1000s of unsatisfied clauses
or just a few?
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Contribution Hybrid Solver MiniWalk

• Context MaxSAT instances non-trivial to prove
  unsatisfiable
• Otherwise method falls back to pure local search
• Main idea
• Use local search as the main MaxSAT search method
• Loosely couple it with systematic search for
  guidance
• Key features
• Both solvers run in parallel (no one waits for
  the other to finish)
• Shared memory to continuously communicate very
  little information a single partial truth
  assignment
• Ideal for multi-core processors
• Very easy to code just a few lines on top of
  Walksat and Minisat

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The Hybrid Solver MiniWalk

• Systematic (Minisat) and local search (Walksat)
  run in parallel, using a shared memory to
  continuously exchange variables fixed by Minisat
• Key hybridization constraint Local search not
  allowed to flip variables against Minisats
  current variable setting

Gory details of the implementation // declare
shared memory include ltsys/shm.hgtint sharedMem
NULLlong shmKey 0x11112222 // any unique
IDint shmId shmget (shmKey, \ sizeof(int)
(nVars1), IPC_CREAT0600)sharedMem (int)
shmat(shmId, NULL, 0) // within Walksat check
against sharedMem// before any flip // within
Minisat write to sharedMem// when any variable
is (un)assigned a value
Note loose coupling Walksat could continue
exploring the c1 sub-space
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The Hybrid Solver MiniWalk

• Results
• Surprisingly good performance in practice (on
  instances non-trivial to prove unsatisfiable)
• Tested on all 52 unsat. industrial instances from
  SAT Race-08 (more on this later)
• Substantially better quality solutions, often
  within secondsMiniWalk often finds a solution
  with very few unsatisfied clauses best
  systematic methods time out pure local search
  suggests 100s or 1000s of unsatisfied clauses
• Insights
• New kind of search behavior than pure local
  search
• Some correlation with long backtracks and
  restarts of the DPLL part

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Experimental Evaluation
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Experimental Setup

• Comparison against
• Systematic MaxSAT solvers with best performance
  inMaxSAT Evaluation 2007
  (maxsatz, msuf)
• Local search MaxSAT solvers from UBCSATwith best
  performance on our suite (saps, adaptg2wsatp)
• Time limit 1 hour

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Experimental Results
Most unsat instances from SAT Race 2008
competition have assignments with only ONE
unsatisfied clause!(not detected by any other
solver) a modified version, recently
presented at SAT-09, finds such assignments
for 51 out of all 52 instances
Time limit 1 hour
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Experimental Results, contd.
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A Note on Benchmarks

• Industrial SAT Race instances clearly have
  interest, but
• would have liked to compare on standard MaxSAT
  benchmarks from MaxSAT competitions, etc.
• Unfortunately, they are too easy for current
  SAT solvers(proved unsatisfiable in a couple of
  seconds)
• MiniWalk falls back to pure Walksat
• Is this really a true reflection of MaxSAT
  problem domains?
• Or a chicken-and-egg problem?
• Perhaps there arent hard-to-prove-unsatisfiable
  MaxSAT benchmarks because no current MaxSAT
  solvers can handle them?

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A Note on Design Choices

• Which DPLL and local search solvers to use?
• Relative speed and freedom of the two solvers
  play a crucial role
• Minisat Walksat combination worked the best
• Artificially slowing down Minisat can sometimes
  help
• Rsat perhaps too fast? branch memory adds
  too much focus?
• Saps / Adaptg2wsatp too focused to let DPLL
  guide them?
• Why not an exact MaxSAT solver rather than
  Minisat?
• Too slow does not provide enough variation over
  time
• Restart rate, Walksat parameters, etc. used
  defaults
• Perhaps a role for automatic parameter tuning?

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MiniWalk Further Insights
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A Different Search Behavior

• Local search gradual descent to good solution
  relatively low noise
• MiniWalk stays high but makes very steep dives

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Correlation with Backtracks / Restarts
Loose correlationwould like to understand
better!
Note plots overlaid to visually highlight x-axis
correlation curves have different
units on y axis
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A Note on Alternatives

• Would more time have helpedother solvers?
• In retrospect, how about SAT re-encoding allowing
  one violated clause?
• Create F satisfiable iff F has a near-solution
  violating 1 clause
• Need one new variable per clause of F ? 100K
  additional vars
• Need a way to count violated clauses naïve
  way (100K)2 clauses cascading count
  additional variables, limited propagation
• ? doesnt scale

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Summary and Future Directions

• DPLL-based systematic SAT solvers can provide
  effective guidance for MaxSAT search
• Good at recognizing deep near-solutions,helping
  put local search on the right track for MaxSAT
• Creation of MaxSAT benchmarks that admit
  near-solutionsand/or are hard to prove
  unsatisfiable?
• Do these single bottleneck constraints give any
  useful info.?
• Note different random seeds often yield
  different bottleneck constraints
• Effective ways to exploit information flow in the
  other direction? (Walksat to Minisat)
• Reproducibility of hybrid runs
• Automatic parameter tuning