MAC and Combined Heuristics: Two Reasons to Forsake FC and CBJ on Hard Problems

1
MAC and Combined HeuristicsTwo Reasons to
Forsake FC (and CBJ?) on Hard Problems

• Christian Bessière and Jean-Charles Régin
• Presented by
• Suddhindra Shukla (Fall 2002)

2
Forward Checking

• Looks ahead each time an assignment is made.
• Revises each uninstantiated variable with the
  value of the current variable.
• Reduces thrashing.

3
MAC

• A more sophisticated look-ahead schema.
• Looks ahead with full arc consistency.
• When a variable is instantiated, the remaining
  CSP (i.e., variables not yet instantiated) is
  made AC.

4
Context

• Many search algorithms for solving CSPs.
• Empirical evaluations so far have shown FC or
  FC-CBJ to be the best techniques.
• Sabin Freuder 94? MAC is better for
• large practical problems (commercial tools).
• large hard (_at_phase transition) random problems

5
Review Choices while searching

• What level of look-ahead to do?
• Which variable to instantiate next?
• What value to use for instantiation?
• What kind of backtrack scheme to adopt?

6
Review level of look-ahead

• Level of filtering to be performed before
  instantiating a variable
• FC is a good compromise between
• pruning effect and overhead involved.

7
Review variable ordering

• Variable ordering affects size of search tree
• Static variable orderings (SVO) based on FFP.
• Minimum width ordering (minw)
• Maximum degree heuristic (deg)
• Maximum cardinality ordering (card)
• Minimum domain (dom)
• Haralick Elliot 80
• DVO is better than SVO.
• dom is the best variable ordering heuristic.

8
Review value ordering

• Not as much researched as variable ordering
• No simple generic principle,
• perhaps get quickly to solution
• promise selection criterion of Geelen 94 did
  not attract (many) FC users.
• In fact, others exist
• Criticality Keng Ying
• min-conflict Minton ,
• but not very popular

9
Review Backtrack scheme

• Different approaches to intelligent backtracking
• Backjumping
• Conflict-directed backjumping
• Graph-directed Frost Dechter 91
• dynamic backtracking Ginsberg 96, etc.
• Prosser showed that FC-CBJ is the champion,
  experiments on Zebra problem
• FC-CBJ-dom so far considered most efficient

10
The authors argue..

• Dom (DVO) is not perfect as it seems.
• Value ordering is important.
• The definition of hard problems
• Problems _at_ phase transition,
• tightness is order parameter Tco

11
Experimental conditions

• Parameters (N, D, p1, p2)
• Old p1, p2 are probabilities
• New p1, p2 are proportions
• Authors use (N, D, C, T)
• C number of constraints
• T number of forbidden tuples

12
Measures of performance

• 1. Number of constraint checks.
• For MAC, using AC-7
• constraint checks (classical) list
  checks
• CPU time.
• Number of backtracks performed
• directly related to number of nodes visited

13
AC-7
14
Experiment Details

• Tables 100 instances per parameter
• Figures 50 instances per parameter
• Report mean values,
• Median values questionable near Tco

15
MAC is better than FC-CBJ

• FC demonstrated champion on very easy or very
  small problems
• Dechter and Meiri it is conceivable that on
  larger, more difficult instances , intensive
  preprocessing algorithms may actually pay off.
• Sabin and Freuder showed that MAC can outperform
  FC on hard instances of CSPs.

16
Evidence favoring MAC

• European commercial tools use MAC
• Radio link frequency assignment problems 95
• Smith Grant 95 on exceptionally hard
  problems, MAC vs FC-dom ,detects insolvability
  quickly.
• Authors show experimentally
• FC too weak to be effective on hard problems.
• MAC is more effective on hard and large problems.
• MACs overhead is outweighed by pruning power.

17
FC-CBJ-domdeg-mc VsMac-domdeg-mc
18
FC-CBJ-domdeg-mc Vs Mac-domdeg-mc
19
cpu Time Ratio With (50,d,95,Tco)
20
Observations

• As D grows, MAC-domdeg-mc outperforms
  FC-CBJ-dom-mc more.
• 3 times faster when D is smaller than 10 to 26.
• 26 times faster when D reaches 40.

21
cpu time ratio with number of constraints(30,10,C,
Tco)
22
Observations

• MACs performance increases till the constraint
  graph contains approximately a third of the
  possible number of constraints.
• After this FC-CBJ becomes less and less worse
  as constraints grows till the complete graph.

23
Combined DVOs

• dom/deg.
• When the constraint is sparse a lot of useful
  information is lost by DVO.
• Caught by SVO based on the structure of the
  constraint graph.

24
(No Transcript)
25
Observations

• When random problems with increasing density are
  solved by different versions of MAC
• dom is a poor heuristic at low densities while
  deg is very efficient on the same problem.
• When graph becomes dense, dom is better.

26
Observations (contd)

• With domdeg ,size of domains has main influence
  on ordering .The degree of variables is used only
  in cases where ties are found.
• Drawback domdeg is not as good deg in sparse
  constraint networks.
• New dvo is dom/deg ie size of remaining domain
• to degree of variable
• Has behavior of domdeg in networks where dom
  was good and of deg in networks where deg was
  better.

27
(No Transcript)
28
(No Transcript)
29
dom/deg vs domdeg

• To show that dom/deg is more advantageous when
  domains are larger
• two heuristics
• 1. MAC-domdeg-mc
• 2. MAC-dom/deg-mc.
• On problem instances with increasing domain
  size.

30
MAC-domdeg-mc vs MAC-dom/deg-mc (50,D,95,Tco)
31
CBJ Becomes Useless

• According to evolution pattern we have
• FC
• FC-CBJ
• MAC-CBJ

32
CBJ Becomes Useless(contd)

• R.M. Haralick and G.L. Elliot. Increasing tree
  search efficiency for constraint satisfaction
  problems. Artificial Intelligence, 14263313,
  1980.
• Look ahead to the future in order not to worry
  about the past.
• Research community agreed that if a good variable
  ordering heuristic is used then CBJ is unlikely
  to generate large backjumps and its savings are
  likely to be minimal because variables that have
  conflicts with past assignments are likely to be
  instantiated sooner.

33

• B. Smith and S.A. Grant. Sparse constraint graphs
  and exceptionally hard problems. In Pro-ceedings
  IJCAI95, pages 646651, Montreal, Canada, 1995.
• for most problems, the ordering given by dom
  ensures that chronological backtracking usually
  results in backtracking to the real culprit for a
  failure, so that informed backtracking does not
  add very much.

34

• Haralick and Elliot
• the more we will perform look-ahead, the
  less we will have to worry about looking back.
• MAC-CBJ cannot simply be claimed to be an
  improvement on MAC.
• Lot of problems were found on which FC-CBJ-dom
  outperformed FC-dom by at least one order of
  magnitude, only one instance was found on which
  MAC-CBJ-dom significantly outperformed MAC-dom
  (Smith and Grant).

35
(No Transcript)
36
Conclusions

• MAC outperforms FC and FC-CBJ on relatively hard
  and large random instances of CSPs
• New variable ordering heuristic dom/deg,
• combines information on domain sizes and
  constraint graph structure.
• Its efficiency was proved when compared with
  domdeg, the most efficient previous heuristic.
• CBJ is almost always use-ess when combined with a
  procedure achieving as much look-ahead as
  MAC-dom/deg-mc. The time overhead is too heavy to
  be outweighed by the small number of constraint
  checks and backtracks saved.

37
Comments on Paper

• Authors focus on hard problems and with low
  constraint density.
• Do not explore high densities
• Do not explore problems problems with low
  tightness or outside hard area

38
Tightness
density