MAC and Combined Heuristics:Two Reasons to Forsake FC (and

CBJ?) on Hard Problems

Christian Bessière and Jean-Charles Régin

Presented by

Suddhindra Shukla (Fall 2002)

Forward Checking

Looks ahead each time an assignment is made.

Revises each uninstantiated variable with the value of the current variable.

Reduces thrashing.

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.

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 (@phase transition) random problems

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?

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.

Review: variable ordering

Variable ordering affects size of search treeStatic 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.

Review: value orderingNot 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

Review: Backtrack schemeDifferent approaches to intelligent backtracking

BackjumpingConflict-directed backjumpingGraph-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

The authors argue..

Dom (DVO) is not perfect as it seems.

Value ordering is important.

The definition of hard problemsProblems @ phase transition,

tightness is order parameter Tco

Experimental conditions

1. Parameters (N, D, p1, p2)

• Old: p1, p2 are probabilities

• New: p1, p2 are proportions

2. Authors use (N, D, C, T)

• C: number of constraints• T: number of forbidden tuples

Measures of performance1. Number of constraint checks. For MAC, using AC-7: #constraint checks (classical) + #list checks

2. CPU time.

3. Number of backtracks performeddirectly related to number of nodes visited

AC-7

Experiment Details

Tables: 100 instances per parameter

Figures: 50 instances per parameter

Report mean values, Median values questionable near Tco

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 CSP’s.

Evidence favoring MAC

European commercial tools use MACRadio 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.MAC’s overhead is outweighed by pruning power.

FC-CBJ-dom+deg-mc VsMac-dom+deg-mc

FC-CBJ-dom+deg-mc Vs Mac-dom+deg-mc

cpu Time Ratio With (50,d,95,Tco)

Observations

As D grows, MAC-dom+deg-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.

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

Observations

MAC’s 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.

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.

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.

Observations (contd)With dom+deg ,size of domains has main influence on ordering .The degree of variables is used only in cases where ties are found.

Drawback: dom+deg is not as good deg in sparse constraint networks.New dvo is dom/deg ie size of remaining domain

to degree of variableHas behavior of dom+deg in networks where dom was good and of deg in networks where deg was better.

dom/deg vs dom+deg

To show that dom/deg is more advantageous when domains are larger :

two heuristics 1. MAC-dom+deg-mc 2. MAC-dom/deg-mc.

On problem instances with increasing domain size.

MAC-dom+deg-mc vs MAC-dom/deg-mc (50,D,95,Tco)

CBJ Becomes Useless

According to evolution pattern we have

FC

FC-CBJ

MAC-CBJ

CBJ Becomes Useless(contd)

R.M. Haralick and G.L. Elliot. Increasing tree search efficiency for constraint satisfaction problems. Artificial Intelligence, 14:263–313, 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.

B. Smith and S.A. Grant. Sparse constraint graphs and exceptionally hard problems. In Pro-ceedings IJCAI’95, pages 646–651, 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”.

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).

ConclusionsMAC outperforms FC and FC-CBJ on relatively hard and large random instances of CSPsNew variable ordering heuristic: dom/deg,

combines information on domain sizes and constraint graph structure. Its efficiency was proved when compared with dom+deg, 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.

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

Constraint

Low High

low MAC

high

Tightness

density