Geometric Stable Roommates

1. Geometric Stable Roommates Valentin Polishchuk Helsinki Institute for Information Technology, University of Helsinki Joint work with Esther Arkin Applied Math and Statistics, Stony Brook University Boris AronovComputer and Information Science, Polytechnic University Kobus BarnardComputer Science, the University of Arizona Kevin CooganComputer Science, the University of Arizona Alon Efrat Computer Science, the University of Arizona Joseph Mitchell Applied Math and Statistics, Stony Brook University

2. Stable Roommates y’ v x’ • V – people, |V| = n (even) • Pair up to be roommates 2 people to a room • u,v in V incur cost w(u,v) if roommates • blocking pair x,y in V x,y not roommates w(x,y) < w(x,x’) w(x,y) < w(y,y’) • will switch to become roommates • Assign so that no blocking pair w(u,v) u y x

3. Matching M(v) • G = (V, E, w), w:E→R • Mµ E Every v in V incident to 1 edge in M • (u,v) in M u,v arematched, partners u = M(v), v = M(u) v

4. Weights → Preferences M(v) • G = (V, E, w), w:E→R • w(u,v) – “partnership cost” (u,v) in M → u,v incur cost w(u,v) • w(u,v) < w(u,x) → u prefers v to x v

5. Blocking Pair M(u) M(v) • (u,v) not in M • w(u,v) < w(v,M(v)) • w(u,v) < w(u,M(u)) ↓ rather partner with each other No blocking pair – M is stable u v

6. Stable Roommates • Given G = (V, E, w) • Find Stable Matching • Interpretation: V – roommates-to-be partners = roommates 2 persons to a room no one will switch roommate

7. Related: Stable Marriage • Given G = (W U M, E, w) • Find Stable Matching • Interpretation: W– women, M – men partners = wife and husband no one will switch spouse

8. Related: 3D Stable Roommates • V – roommates-to-be 3 persons to a room • Each v in V • ranks pairs {u,x} • “Matching” – decomposition of V into triples

9. Related: 3D Stable Roommates • Blocking triple v,u,x not in one room v prefers {u,x} to the pair of v’s current roommates u prefers {v,x} to the pair of u’s current roommates x prefers {v,u} to the pair of x’s current roommates ↓ rather match up with each other • No blocking triple – matching is stable • 3D Stable Roommates Problem Given ranking of the pairs Find stable matching u v x

10. Minimum-Regret Stable Matching • w(u,v) – “regret” of u,v • Regret of M Σv in V w(v,M(v)) • Find Stable Matching of minimum-regret

11. Assumption: No Ties • w(u,v) < w(u,x) → u prefers v to x • blocking pair (u,v) w(u,v) < w(v,M(v)) w(u,v) < w(u,M(u)) M(u) M(v) u v

12. With Ties( w(u,v) = w(u,x) for v ≠ x )3 definitions of blocking pair

13. Super-Stable Matching M(u) M(v) • blocking pair (u,v) w(u,v) ≤ w(v,M(v)) w(u,v) ≤ w(u,M(u)) No blocking pair – M is super-stable u v

14. Strongly-Stable Matching M(v) M(u) • blocking pair (u,v) w(u,v) < w(v,M(v)) w(u,v) ≤ w(u,M(u)) No blocking pair – M is strongly-stable u v

15. Weakly-Stable Matching M(v) M(u) • blocking pair (u,v) w(u,v) < w(v,M(v)) w(u,v) < w(u,M(u)) No blocking pair – M is weakly-stable u v

16. Computational Complexity of Finding Stable Matching

17. Stable Marriage • SM exists in any instance • Ɵ(n2) to find SM [Gale, Shapley’62] • also with ties • min-regret: O(n3) [Feder’92, Irving,Leather,Gusfield’87]

18. Stable Roommates • Stable Matching notalways exist • no ties: Ɵ(n2) [Gusfield,Irving,‘85,’89] • with ties: NP-complete[Ronn’90, Irving,Manlove’02] • min-regret: NP-complete[Feder’92]

19. 3D Stable Roommates • NP-complete[Ng,Hirschberg’91, Subramanian’94]

20. NP-Hardness Proofs Specific, “uncorrelated” w Introduce “consistency” into lists [Ng,Hirshberg’92] Master lists [Irving,Manlove,Scott’06] preferences, w, come from a centralized source

21. This talk: Geometric Stable Roommates(a way to introduce consistency)

22. Geometric Stable Roommates v • Participants = points in Rd • w(u,v) = |uv| |uv| u

23. Applications $ • Classification (bi-partite) • matching objects to observations • Distance in some space of character features • how much alike 2 persons are • Finding chess opponents • solely based on dist to travel • Finding life-time partners • solely based on dist to travel finally solving 2BP – most open problem in Academia IQ The 2-Body Problem

24. FindingStable Matching:Successive Closest Pair

25. No Ties b a • a,b – Closest Pair in V {a,b} = arg minx,y in V|xy| • a = M(b) in any SM • o.w., both want to switch • V ← V\{a,b} and recurse Successive Closest Pairs ↓ Stable Matching O(n log n) time[Bespamyatnikh’95] Optimal (from Element Uniqueness)

26. Geometric Stable Roommateswith Ties

27. 3 notions of stability Blocking pair (willing to switch) if after switching each is at least as good as before • super-Stable Matching (at least) one is strictly better, one is at least as good as before • strongly-Stable Matching both are strictly better than before • weakly-Stable Matching

28. Closest-Pair (CP) Graph • a,b – closest pairs in V |ab| = minx,y in Vd(x,y) • no ties: one edge • with ties: arbitrary graph

29. Super-Stable Matching • CP graph is a perfect matching • Stable Matching exists • same argument as without ties • o.w. • no Stable Matching • CP graph has vertex of degree 2 • willing to switch a b

30. Strongly-Stable Matching • CP graph has a perfect matching • Stable Matching exists • same argument as without ties • o.w. • no Stable Matching • vertex a not matched in CP graph • a wants to switch • a’s neighbor in CP doesn’t mind a

31. Weakly-Stable Matching Always exists • Algorithm: Find maximal matching in CP V ← V \ the maximal matching • Proof: none of the matched could be blocking • same argument as without ties • remove them • eventually everybody gets matched a

32. 3D Stable Roommates 3 people to a room Open… Can find 2-Stable Matching…

33. α-Stable Matching {x,y} – blocking pair: after switching each improves by at least factor of α won’t bother switching if improvement is small Stable Matching = 1-Stable Matching α α 1 x y

34. Finding α-Stable Matching: Computational Complexity LB on complexity of Stable Matching ↓for any α>1↓ LB on complexity of α-Stable Matching 3D Stable Roommates is NP-complete ↓ ↓ 3D 2-Stable Roommates is NP-complete

35. Geometric3D 2-Stable Roommates

36. Definition v’ x • V – points in the plane • x in V ranks pairs {y,z} by |xy|+|xz| • “Matching”: break V into triples (rooms) v in V cost = |vv’|+|vv’’| y v v’’ z

37. Blocking Triple z’’ y’ x’ x x,y,z not in one room |xy|+|xz| < ½ (|xx’| + |xx’’|) |yx|+|yz| < ½ (|yy’| + |yy’’|) |zx|+|zy| < ½ (|zz’| + |zz’’|) • will switch Geometric 3D 2-Stable Roommates Problem: Given points = people Assign people to rooms 3 people to a room no blocking triple (2-Stable 3D-Matching) y z’ y’’ x’’ z

38. Solution Successive minimum-perimeter triangles Find min-perimeter triangle, match, remove, recurse

39. Gives 2-Stable Matching • abc – min-perimeter triangle • a,x,y – blocking , (x,y) ≠ (b,c) |ax|+|ay| < ½ (|ab|+|ac|) perimeter(axy) = |ax|+|ay|+|xy| < < |ab|+|ac| < perimeter(abc) b c a x y

40. Geometric Stable Roommates: Results • No ties: always exists unique Stable Matching • true love is crucial • With ties: in P • weakly-SM always exists (general SR: weakly-SM is NP-complete[Ronn’90, Irving, Manlove’02]) • min-regret • find maximum matchings in CP graphs (general SR: min-regret-SM is NP-complete[Feder’92]) • 3D Stable Roommates: 2-Stable Matching always exists, in P general 3D 2-Stable Roommates is NP-complete

41. Open • α-Stable Matching general and geometric • Geometric 3D Stable Roommates 1-Stable Macthing • A hard problem? • Another geometric interpretation: • men: pts in 2D • women: directions • rank: projection • similarly for men rankings “canonical” instance (all women have same preference)O(n log n) [Ɵ(n2) in general] $ IQ

44. preference list of p in 1-SM: a b c d … ↓ w(p,a) = 1, w(p,b) = α, w(p,c) = α2, w(p,d) = α3… switch = improve by at least α

45. Parametric MWM, SP(t)