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

2. 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

3. 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 +1 +1 = -2

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

5. approximations

6. form teams of three based on preferences

7. 0 0 1 1 0 3 -2 -5 chain approximation

8. 0 0 1 1 0 3 -2 -5 chain approximation

9. 0 0 1 1 0 3 -2 -5 chain approximation

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

11. group approximation, teams of 3

12. 0 0 1 1 0 3 -2 -5

13. 3 0 0 1 1 0 -2 -5

14. 3 0 0 1 1 0 -2 -5

15. 3 0 0 1 1 0 -2 -5

16. 3 0 0 1 1 0 -2 -5

17. 3 0 0 1 1 0 -2 -5

18. 3 0 0 1 1 0 -2 -5

19. 3 0 0 1 1 0 -2 -5

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

21. hill climbing

22. C B DE G H I F A 1043

23. hill climbing C B DE G H I F A 1043

24. hill climbing C B DE G H I F A 1043

25. hill climbing C B DE G H I F A 1065

26. results: prediction accurate?

27. results

28. alternative: nearest neighbors Represent preferences of each student as a 32-number string. Match students based on the distance between their preference strings. 0110000 0 011 0 10100 000 01110 1121010011110011 (d = 12)

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

30. 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

31. (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

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

40. 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

41. questions