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On Matching Robustness and Geometric Stable Marriage - PowerPoint PPT Presentation

On Matching Robustness and Geometric Stable Marriage. Valentin Polishchuk Helsinki Institute for Information Technology, University of Helsinki Joint work with Esther Arkin Applied Math and Statistics, Stony Brook University

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On Matching Robustness and Geometric Stable Marriage

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

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

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

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

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

• 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

• Given G = (W U M, E, w)

• Find Stable Matching

• Interpretation:

W– women, M – men

partners = wife and husband

no one will switch spouse

• V – roommates-to-be

3 persons to a room

• Each v in V

• ranks pairs {u,x}

• “Matching” – decomposition of V into triples

• 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

• w(u,v) – “regret” of u,v

• Regret of M

Σv in V w(v,M(v))

• Find Stable Matching

of minimum-regret

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

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

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

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

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

• 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]

• 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]

• NP-complete[Ng,Hirschberg’91, Subramanian’94]

Specific, “uncorrelated” w

Introduce “consistency” into lists [Ng,Hirshberg’92]

Master lists [Irving,Manlove,Scott’06]

preferences, w, come from a centralized source

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

v

• Participants = points in Rd

• w(u,v) = |uv|

|uv|

u

\$

• 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

FindingStable Matching:Successive Closest Pair

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)

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

• a,b – closest pairs in V

|ab| = minx,y in Vd(x,y)

• no ties: one edge

• with ties: arbitrary graph

• 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

• 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

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

3 people to a room

Open…

Can find 2-Stable Matching…

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

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

Geometric3D 2-Stable Roommates

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

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

Successive minimum-perimeter triangles

Find min-perimeter triangle, match, remove, recurse

• 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

• 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

• 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

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