Foundations of Constraint Processing

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Extensions to BT Search

• Foundations of Constraint Processing
• CSCE421/821, Spring 2011
• www.cse.unl.edu/cse421
• Berthe Y. Choueiry (Shu-we-ri)
• Avery Hall, Room 360
• choueiry_at_cse.unl.edu
• Tel [phone]

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Outline

• Variations of backtrack search
• Backtrack search for optimization

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Variations on BT search

• Bounded number of backtracks search
• Bounded backtrack-depth search
• Limited discrepancy search (LDS)
• Heuristic may be blind at shallowest level of
  search-tree
• Disobey heuristic a given number of times
• Credit-based backtrack search
• Randomized backtrack search ( restart)

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Credit-Based Search

• Start with a given credit (usually n3)
• Assign ½ credit to current assignment, ½ to the
  remaining ones
• Keep going, in a depth-first manner until credit
  is used up, (chronoligically) backtrack from
  there
• ECLiPSe uses it in conjunction with
  backtrack-bounded search

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Randomized BT search
In systematic backtrack search

• Ordering of variables/values determines which
  parts of the solution space are explored
• Randomization allows us to explore wider portion
  of search tree
• Thrashing causes stagnation of BT search
• Interrupt search, then restart

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Restart strategies

• Fixed-cutoff universal strategy
  Luby et al., 93
• Randomization Rapid restarts (RRR) Gomes et
  al., 98
• Fixed optimal cutoff value
• Priori knowledge of cost distribution required
• Randomization geometric restarts (RGR)
  Walsh 99
• Randomization dynamic geometric restarts (RDGR)
  
• 
  
  Guddeti 04
• Bayesian approach
  Kautz et al., 02

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RGR Walsh 99

• Static restart strategy
• As the cutoff value increases, RGR degenerates
  into randomized BT
• Ensures completeness (utopian in our setting)
• But restart is obstructed
• and thrashing reappears ? diminishing the
  probability of finding a solution

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RDGR Guddeti 04

• Randomization Dynamic Geometric Restarts
• Cutoff value
• Depends on the progress of search
• Never decreases, may stagnate
• Increases at a much slower rate than RGR
• Feature restart is less obstructed

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Outline

• Variations of backtrack search
• Backtrack search for optimization
• Courtesy of Markus Fromherz
• Branch and bound
• Binary search
• Iterative deepening

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Branch Bound

• Find a first solution, compute its quality, call
  it the incumbent
• Systematically explore alternative solutions
• While building an alternative solution,
  estimate its quality and compare the estimated
  quality with that of the incumbent
• If a partial solution has no chance of becoming
  better than the incumbent
• Then prune it
• Else, keep building it
• As soon as a better solution is found, make it
  the incumbent
• Continue until you run out of time, patience, or
  solutions

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BB over-constrained-CSPs

• Max-CSP
• Goal minimize the number of broken constraints
  (while instantiating all variables)
• Maximize solution length
• Goal maximize number of variables instantiated
  (while satisfying all constraints)
• We compare the incumbent and the partial solution
  along current path according to
• the number of broken constraints or
• the number of instantiated variables

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Binary search

• Given l, u lower and upper bounds of the quality
  of the solution
• Check whether there is a solution in l,
  u-l/2l,u
• If there is, set the bounds l,u l,u-l/2 and
  apply the mechanism again
• If there is not, set the bounds to u, u and
  apply the mechanism again
• Restart search with progressively narrower lower
  and upper bounds on the solution

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Iterative deepening

• Restart search with an increasing lower limit on
  the solution quality until a solution of an
  acceptable quality is found