The Goldilocks Problem Tudor Hulubei Eugene C. Freuder Department of Computer Science University of...

The Goldilocks Problem

Tudor Hulubei

Eugene C. FreuderDepartment of Computer Science

University of New Hampshire

Sponsor: Oracle

Introduction

“So away upstairs she went to the bedroom, and there she saw three beds. There was a very big bed for Father bear, but it was far too high. The middle-sized bed for Mother bear was better, but too soft. She went to the teeny, weeny bed of Baby bear and it was just right.”

-- Goldilocks and the Three Bears

Introduction(continued)

• A lot of work in Constraint Satisfaction has been focused on finding solutions to hard problems.

• Many problems of practical interest are not difficult, but have a huge number of solutions, few of which are acceptable from a practical standpoint.

• We are proposing a value ordering heuristic that will guide the search towards acceptable solutions.

Motivation and Applications

• Provide vendors with tools that can be used to suggest upgrades (upselling) and alternative solutions to customers. This can be achieved by combining a matchmaker or deep-interview strategy with our heuristic that guides the search towards acceptable solutions.

• Provide a way for vendors to promote particular configurations, implement company policies, etc.

Example

• Solution: v1=1, v2=1, v3=-1• SW=(0.8+0.1+0.8)+(0.7+0.5)=2.9• MinSW=(0.2+0.1+0.8)+(0.1+0.2)=1.4• MaxSW=(0.8+0.7+0.9)+(0.9+0.5)=3.8

v1

v3v2

c12 c13

v1={0=0.2, 1=0.8}v2={1=0.1, 2=0.7}v3={-1=0.8, 4=0.9}c12={(0,1)=0.1, (1,1)=0.7, (1,2)=0.9}c13={(0,-1)=0.2, (0,4)=0.3, (1,-1)=0.5}

Definitions

• Weights:– Represent “goodness”.

– Are associated with:• every value in the domain of a variable.

• every pair of allowed values in a constraint.

• Solution Weight:– Defined as the sum, over all the variables and constraints, of the

weights associated with the values and pairs of values involved.

– Lower and upper limits :• MinSW (computed as the sum of the minimum weights of the values

and pair of values in all the variables and constraints).

• MaxSW (computed as the sum of the maximum weights of the values and pair of values in all the variables and constraints).

Definitions(continued)

• Acceptable Solutions:– Given two positive real numbers MinASW and MaxASW (the

minimum/maximum acceptable solution weight) s.t.:MinSW MinASW MaxASW MaxSW,

a solution is considered acceptable if:

MinASW SW MaxASW.

– The ideal solution weight (IdealSW) is defined as the center of the [MinASW,MaxASW] range.

• Active Constraints:– During the search, a constraint is considered “active” if it involves

at least one variable that has not yet been instantiated.

The acceptable-weight Heuristic

• One way of looking for acceptable solutions is by parsing the entire search space and stop at the first solution whose weight is acceptable.

• We can do better than that. We can guide the search towards solutions with acceptable weights, using a special value ordering heuristic called “acceptable-weight”.

• The heuristic will dynamically compute the average of the individual weight (AIW) that the remaining variables and constraints would contribute to the global solution weight, should it equal IdealSW:

( IdealSW - SW(SolvedSubproblem) )

AIW = ---------------------------------------------------------------------------------

(number of uninstantiated variables + number of active constraints)

The acceptable-weight Heuristic (continued)

• The acceptable-weight heuristic will select the value that will minimize the absolute difference between SW” and IdealSW”.

• After the assignment of v4, AIW is recomputed, to compensate for the amount we were off compared to the IdealSW”.

v1[1]

v2[6]

v3[4]

v4[1,5]

v6[2,3]

v5[1,2]

v1={1=0.1}, v2={6=0.2},v3={4=0.4}, v4={1=0.1, 5=0.8}c14={(1,1)=0.1, (1,5)=0.2}c24={(6,1)=0.8, (6,5)=0.9}c34={(4,1)=0.6, (4,5)=0.7}

c14

c24

c34

c45

c46

Pastvariables

Futurevariables

Currentvariable

AIW=0.4 IdealSW”=(0.1+0.2+0.4)+AIW+3*AIW=2.3Options for v4: v4=1: SW”=(0.1+0.2+0.4+0.1)+(0.1+0.8+0.6)=2.6 v4=5: SW”=(0.1+0.2+0.4+0.8)+(0.2+0.9+0.7)=3.3 The heuristic selects v4=1.

P”={{v1,v2,v3}, {c14,c24,c34}}.Note: v5 & v6 are not considered.

The acceptable-weight Heuristic(continued)

• The strategy behind the acceptable-weight heuristic is two-fold:– Locally, we try to make sure that each

subproblem centered around the current variable has a weight that is in line with the global IdealSW.

– Globally, by constantly adjusting the AIW we try to control the overall deviation of the solution weight.

Experimental Results

Variables: 100Domain size: 5Density: 0.00Tightness: 0.00Range size: 0.05MAC range: 0.455-0.54MAC+acceptable-weight range: 0.226-0.79


Experimental Results(continued)




Variables: 100Domain size: 5Density: 0.0651Tightness: 0.25

Range size: 0.1MAC range: 0.43-0.55MAC+acceptable-weight range: 0.44-0.54


Variables: 100

Domain size: 5

Difficulty peak for tightness=0.25 is at density=0.0651.

Future Work

• Potential Improvements:– Look ahead, skip subtrees of the search space where an

acceptable weight is not achievable.

– Consider future variables when estimating the solution weight of the local subproblem centered around the current variable.

– Avoid getting stuck in an interval when the heuristic doesn’t seem capable of escaping it.

Conclusions

• The acceptable-weight heuristic presented here is designed to guide the search towards solutions with acceptable weights, when solutions can be ranked on a given dimension.

• Experiments show that MAC+acceptable-weight guides the search towards acceptable solutions very quickly, much faster than MAC alone.

• Even around the difficulty peak, MAC+acceptable-weight works well, slightly faster than MAC, with a very similar coverage.

The Goldilocks Problem Tudor Hulubei Eugene C. Freuder Department of Computer Science University of...

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