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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

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On matching robustness and geometric stable marriage

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
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
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
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
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 roommates1
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
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
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 roommates1
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
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
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
With ties( w(u,v) = w(u,x) for v ≠ x )3 definitions of blocking pair


Super stable matching
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
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
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
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 roommates2
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
3D Stable Roommates

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


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


Geometric stable roommates
Geometric Stable Roommates

v

  • Participants = points in Rd

  • w(u,v) = |uv|

|uv|

u


Applications
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


Finding stable matching successive closest pair
FindingStable Matching:Successive Closest Pair


No ties
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)



3 notions of stability
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
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 matching1
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


Strongly stable matching1
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


Weakly stable matching1
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


3d stable roommates1
3D Stable Roommates

3 people to a room

Open…

Can find 2-Stable Matching…


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 stable matching computational complexity
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


Geometric 3d 2 stable roommates
Geometric3D 2-Stable Roommates


Definition
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
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


Solution
Solution

Successive minimum-perimeter triangles

Find min-perimeter triangle, match, remove, recurse


Gives 2 stable matching
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
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


On matching robustness and geometric stable marriage
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


On matching robustness and geometric stable marriage

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