The Goldilocks Problem Tudor Hulubei Eugene C. Freuder Department 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=( )+( )=2.9 MinSW=( )+( )=1.4 MaxSW=( )+( )=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 Past variables Future variables Current variable AIW=0.4  IdealSW”=( )+AIW+3*AIW=2.3 Options for v4: v4=1: SW”=( )+( )=2.6 v4=5: SW”=( )+( )=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: 100 Domain size: 5 Density:0.00 Tightness: 0.00 Range size: 0.05 MAC range: MAC+acceptable-weight range: Variables: 100 Domain size: 5 Density: 0.00 Tightness: 0.25 Range size: 0.05 MAC range: MAC+acceptable-weight range:

Experimental Results (continued) Variables: 100 Domain size: 5 Density:0.00 Tightness: 0.25 Range size: 0.1 MAC range: MAC+acceptable-weight range: Variables: 100 Domain size: 5 Density: Tightness: 0.25 Range size: 0.1 MAC range: MAC+acceptable-weight range:

Experimental Results (continued) Variables: 100 Domain size: 5 Density: Tightness: 0.25 Range size: 0.1 MAC range: MAC+acceptable-weight range:

Experimental Results (continued) Variables: 100 Domain size: 5 Difficulty peak for tightness=0.25 is at density=

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.