Approximating Maximum Satisfaction in Group Formation Sean Munson, Grant Hutchins Discrete Math, Olin College 14 December 2004

goal Maximize happiness, or satisfaction Input –Preferences: -1, 0, or 1 for each person –Worked with before –Work style preferences Satisfaction: how many preferences you meet

satisfaction –For each team, the sum of how each person feels about the each other person in the group. –Maximize this for each set of teams = -2

possible combinations in our class Teams of 4: Teams of 2:

approximations

form teams of three based on preferences

chain approximation

chain approximation

chain approximation

Fast, simple Variation: choose on most popular or pickiness Problem: only looks at one person’s preferences at a time

group approximation, teams of 3

group approximation Includes everyone’s preferences Still explores very little space

hill climbing

C B DE G H I F A 1043

hill climbing C B DE G H I F A 1043

hill climbing C B DE G H I F A 1043

hill climbing C B DE G H I F A 1065

results: prediction accurate?

results

alternative: nearest neighbors Represent preferences of each student as a 32-number string. Match students based on the distance between their preference strings (d = 12)

Requires calculating distance between each pair of strings. Total comparisons alternative: nearest neighbors

nearest neighbors by work style Three questions. describe students’ working styles –Timeliness (early to late) –Group style (individual to always in team) –Focus (hardcore to relaxed) Answers assigned values from 0 to 2 Each person falls on a coordinate in 3-space

(0,0,0) (2,2,2) (0,2,0) (2,0,0) (2,0,2) (0,0,2) (0,2,2) (2,2,0) nearest neighbors approximation

results: work style nearest neighbors Outperforms random Beaten by chain, group heuristics

future work? Assess more seed and convergent processes Weight edges based on prior experience Break ties with work style Conduct long-term study to evaluate performance of formed teams Evaluate effects of number of preferences expressed

questions