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More on BT Search. Foundations of Constraint Processing CSCE421/821, Fall 2004: www.cse.unl.edu/~choueiry/F04-421-821/ Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 123B [email protected] Tel: +1(402)472-5444. Outline. Lookahead Variations of backtrack search

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More on BT Search

Foundations of Constraint Processing

CSCE421/821, Fall 2004:

www.cse.unl.edu/~choueiry/F04-421-821/

Berthe Y. Choueiry (Shu-we-ri)

Avery Hall, Room 123B

[email protected]

Tel: +1(402)472-5444

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outline
Outline
  • Lookahead
  • Variations of backtrack search
  • Backtrack search for optimization

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lookahead
Lookahead
  • Rationale
    • As decisions are made (conditioning) eliminate inconsistent choices in future sub-problem
    • Domain annihilation of a future variable avoids expansion of useless portions of the tree
  • Techniques
    • Partial: forward-checking (FC), directional arc-consistency (DAC)
    • Full: Maintaining arc-consistency (MAC)
  • MAC: more pruning at the cost of more consistency checks
  • Empirical results:

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outline1
Outline
  • Lookahead
  • Variations of backtrack search
  • Backtrack search for optimization

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variations on bt search
Variations on BT search
  • Bounded number of backtracks search
  • Bounded backtrack-depth search
  • Limited discrepancy search
    • 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
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
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
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
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
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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outline2
Outline
  • Lookahead
  • Variations of backtrack search
  • Backtrack search for optimization

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bt search for optimization
BT search for optimization

Courtesy of Markus Fromherz

  • Branch & bound
    • Application to over-constrained CSPs
  • Binary search
  • Iterative deepening
  • Etc.

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branch bound
Branch & Bound
  • Branch & bound
    • Find a first solution, compute its quality, call it the incumbent
    • Search for other solutions, comparing them with the incumbent
    • As soon a better solution is found, make it the incumbent
    • Continue until you run out of time, patience, or solutions

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b b over constrained csps
B&B: 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
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/2]=[u,l’]
    • If there is, set the bounds [u, u+l’/2] and search for a solution
    • If there is not, set the bounds to [l’, l+l’/2] and search for a solution…
  • Restart search with progressively narrower lower and upper bounds on the solution

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iterative deepening
Iterative deepening
  • Restart search with an increasing upper limit on the solution quality until a solution is found

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rest of the course
Rest of the course?
  • Local search
  • Binary vs. non-binary
  • Phase Transition
  • All-diff constraint (Shasha?)
  • Backtrack-free, backtrack-bounded search
  • Temporal CSPs
  • Interchangeability
  • Dynamic CSPs

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