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On Matching Robustness and Geometric Stable MarriagePowerPoint Presentation

On Matching Robustness and Geometric Stable Marriage

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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

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

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

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

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

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

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

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

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

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

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

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]

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]

3D Stable Roommates

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

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

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

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

- solely based on dist to travel

IQ

The 2-Body Problem

FindingStable Matching:Successive Closest Pair

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)

Geometric Stable Roommateswith Ties

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

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

Super-Stable Matching

- CP graph is a perfect matching
- Stable Matching exists
- same argument as without ties

- Stable Matching exists
- o.w.
- no Stable Matching
- CP graph has vertex of degree 2
- willing to switch

- no Stable Matching

a

b

Strongly-Stable Matching

- CP graph has a perfect matching
- Stable Matching exists
- same argument as without ties

- Stable Matching exists
- o.w.
- no Stable Matching
- vertex a not matched in CP graph
- a wants to switch
- a’s neighbor in CP doesn’t mind

- no Stable Matching

a

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

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

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

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

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

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])

- find maximum matchings in CP graphs

- weakly-SM always exists
- 3D Stable Roommates: 2-Stable Matching always exists, in P
general 3D 2-Stable Roommates is NP-complete

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

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

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