The stable marriage problem an example

LAW February 2007 1

A 5/3 approximation algorithm for a hard case of stable marriage

Rob Irving

Computing Science DepartmentUniversity of Glasgow

(joint work with David Manlove)

LAW February 2007 2

The stable marriage problem an example

m1: w2 w4 w1 w3 w1: m2 m1 m4

m2: w3 w1 w4 w2: m4 m3 m1

m3: w2 w3 w3: m1 m4 m3 m2

m4: w4 w1 w3 w2 w4: m2 m1 m4

men’s preferences women’s preferences

It's an unstable matching - (m1, w4) is a blocking pair

An example matching - {(m1, w1), (m2, w3), (m3, w2), (m4, w4)}

LAW February 2007 3

The stable marriage problem an example

m1: w2 w4 w1 w3 w1: m2 m1 m4

m2: w3 w1 w4 w2: m4 m3 m1

m3: w2 w3 w3: m1 m4 m3 m2

m4: w4 w1 w3 w2 w4: m2 m1 m4


{(m1, w4), (m2, w1), (m3, w2), (m4, w3)} - a stable matching

LAW February 2007 4

Stable marriage facts

There is always at least one stable matching

The Gale-Shapley algorithm finds such a matching [Gale & Shapley 1962]

There may be many stable matchings exponentially many in some cases

[Knuth 1976]

All stable matchings have the same size and involve the same people [Gale & Sotomayor 1985]

The version of stable marriage described here is usually called "Stable Marriage with Incomplete Lists" (SMI)

LAW February 2007 5

The Hospitals / Residents problem an example

r1: h4 h2 h1 h1: 2: r2 r7 r1 r5 r4 r6 r3

r2: h3 h1 h4 h2: 2: r8 r1 r3 r6

r3: h2 h3 h1 h3: 2: r3 r5 r2 r6 r8 r7 r4

r4: h3 h4 h1 h4: 3: r5 r8 r2 r4 r7 r1

r5: h1 h4 h3

r6: h2 h3 h1

r7: h4 h1 h3

r8: h3 h4 h2

residents’ preferences hospitals’ quotas and preferences

It's unstable - (r3,h2) is a blocking pair

so is (r5,h1)

An example matching -

{(r1,h2),(r2,h3),(r3,h3),(r4,h4),(r5,h4),(r6,h2),(r7,h1),(r8,h4)}

LAW February 2007 6

The Hospitals / Residents problem an example

r1: h4 h2 h1 h1: 2: r2 r7 r1 r5 r4 r6 r3

r2: h3 h1 h4 h2: 2: r8 r1 r3 r6

r3: h2 h3 h1 h3: 2: r3 r5 r2 r6 r8 r7 r4

r4: h3 h4 h1 h4: 3: r5 r8 r2 r4 r7 r1

r5: h1 h4 h3

r6: h2 h3 h1

r7: h4 h1 h3

r8: h3 h4 h2

residents’ preferences hospitals’ capacities and preferences

A stable matching{(r1,h2),(r2,h3),(r3,h2),(r4,h4),(r5,h1),(r6,h3),(r7,h4),(r8,h4)}

LAW February 2007 7

Stable matching in practice The National Resident Matching Program (NRMP)

a scheme that each year matches over 30,000 graduating medical students (residents) to hospitals in the US

Since the early 1950s, the NRMP has used essentially the Gale-Shapley algorithm [Roth 1984]

Owes its success to the stability of its matching

Similar matching schemes in other countries and other contexts

including the ‘SFAS’ scheme in Scotland [Irving 1998]

LAW February 2007 8

Ties in the preference lists

• Stable Marriage with Ties and Incomplete Lists (SMTI) • Hospitals / Residents with Ties (HRT)

• Each preference list can contain one or more ties

• arguably a more realistic model in practical applications • large hospitals cannot reasonably differentiate among all of their many applicants

• Now, a pair blocks a matching only if both members of the pair would be better off if matched together

• sometimes called weak stability

• To find a stable matching, break the ties arbitrarily and apply the Gale-Shapley algorithm

• But now – stable matchings can have different sizes

LAW February 2007 9

SMTI - an illustration

m1: w3 (w1 w2) w4 w1: m1 m2 m3 m4

m2: w1 w3 w2: m1 m4

m3: w3 w1 w3: m4 m3 m1 m2

m4: w1 (w3 w4) w2 w4: m4 m1


A stable matching of size 4:

{(m1, w2), (m2, w1), (m3, w3), (m4, w4)}

A stable matching of size 2: {(m1, w1), (m4, w3)}

LAW February 2007 10

A hard stable matching problem

• For Stable Marriage (and therefore Hospitals Residents) with incomplete lists and ties

• It's NP-hard to find a maximum cardinality stable matching

• even if• all the ties are on one side, and• a tie occurs only at the end of a list, and• all ties are of length 2

[Iwama, Manlove et al 1999]

[Manlove, Irving et al 2002]

• Question: what about polynomial-time approximation?

• Can we decide on a 'good' way to break the ties?


Approximating maximum cardinality SMTI

• Trivially 2-approximable• max cardinality ≤ 2 * min cardinality

• But APX-complete [HIIMMMS 2003]

• And not approximable within 21/19 (unless P = NP)

• 13/7-approximable if all ties are of length 2

• 8/5-approximable if all ties are of length 2 and on just one side

• 10/7 approximable (randomised) if, in addition, ≤ 1 tie per list [Halldórsson et al 2003]

• 15/8 approximable in the general case [Iwama et al, SODA 2007]


A case of practical interest

'Special' HRT (SHRT)• residents' preference lists are strict• hospitals' lists can contain one tie, of arbitrary length, at the end

• reflects the situation where hospitals are asked to rank as many candidates as they reasonably can, then place all others in a single tie at the end of their list

'Special' SMTI (SSMTI)• men's preference lists are strict• women's lists can contain one tie, of arbitrary length, at the end

• MAX-SSMTI is NP-hard (and so MAX-SHRT is too)


Approximating Max-SSMTI and MAX-SHRT

• There is a polynomial-time approximation algorithm for maximum cardinality SSMTI ( and for SHRT) with an approximation guarantee of 5/3.

• Algorithm is quite different from the approach used by previous approximation algorithms for variants of MAX-SMTI

• Consists of three phases:

• Phase 1 : apply a tailored version of the Gale-Shapley algorithm

• Phase 2 : find a maximum cardinality matching of selected subsets

• Phase 3 : break the ties according to a particular rule, then apply standard Gale-Shapley


Algorithm SSMTI-APPROX: Phase 1

assign each person to be free; /* free = not engaged */

while (some woman w is free) and (w has a non-empty list) and (w has an untied man m at the head of her list){ w proposes, and becomes engaged to m;

/* m can never have a 'worse' mate than w; in particular, m is matched in every stable matching */

for each successor w' of w on m's list { if w' is engaged to m break the engagement; delete the pair (m,w') from the preference lists; }} Fact: engaged men are matched in every stable matching



m1 : w6 w1 w2 w10 w5 w8 w1 : m2 m3 (m1 m6 m9)

m2 : w7 w10 w3 w1 w6 w2 : m1 (m3 m5 m7)

m3 : w6 w2 w8 w9 w1 w3 : m2 (m4 m6 m8)

m4 : w7 w9 w8 w3 w4 w5 w4 : m4 m9 (m5 m8 m10)

m5 : w2 w6 w4 w5 w10 w5 : m4 m1 m5 m7

m6 : w1 w3 w6 : m2 (m1 m3 m5)

m7 : w2 w5 w7 : (m2 m4 m10)

m8 : w3 w4 w8 : m1 (m3 m4)

m9 : w1 w4 w9 : (m3 m4)

m10: w7 w10 w4 w10: m5 (m1 m2 m10)



Algorithm SSMTI-APPROX: Phase 1 outcome

m1 : w6 w1 w2 w1 : m3 (m1 m6 m9)

m2 : w7 w10 w3 w2 : m1 (m3 m5 m7)

m3 : w6 w2 w8 w9 w1 w3 : m2 (m4 m6 m8)

m4 : w7 w9 w8 w3 w4 w4 : m4 m9 (m5 m8 m10)

m5 : w2 w6 w4 w5 w5 : m5 m7

m6 : w1 w3 w6 : (m1 m3 m5)

m7 : w2 w5 w7 : (m2 m4 m10)

m8 : w3 w4 w8 : (m3 m4)

m9 : w1 w4 w9 : (m3 m4)

m10: w7 w10 w4 w10: (m2 m10)

red people are engaged, black people are free



Find a maximum cardinality matching F of free men and women;

for each pair (m,w) F promote m from the tie to the head of w's list;

reactivate phase 1;

/* essentially extends the sets of engaged men and women */



m1 : w6 w1 w2 w1 : m3 (m1 m6 m9)

m2 : w7 w10 w3 w2 : m1 (m3 m5 m7)

m3 : w6 w2 w8 w9 w1 w3 : m2 (m4 m6 m8)

m4 : w7 w9 w8 w3 w4 w4 : m4 m9 (m5 m8 m10)

m5 : w2 w6 w4 w5 w5 : m5 m7

m6 : w1 w3 w6 : (m1 m3 m5)

m7 : w2 w5 w7 : (m2 m4 m10)

m8 : w3 w4 w8 : (m3 m4)

m9 : w1 w4 w9 : (m3 m4)

m10: w7 w10 w4 w10: (m2 m10)




m1 : w6 w1 w2 w1 : m3 (m1 m6 m9)

m2 : w7 w10 w3 w2 : m1 (m3 m5 m7)

m3 : w6 w2 w8 w9 w1 w3 : m2 (m4 m6 m8)

m4 : w7 w9 w8 w3 w4 w4 : m4 m9 (m5 m8 m10)

m5 : w2 w6 w4 w5 w5 : m5 m7

m6 : w1 w3 w6 : (m1 m3 m5)

m7 : w2 w5 w7 : (m2 m4 m10)

m8 : w3 w4 w8 : (m3 m4)

m9 : w1 w4 w9 : (m3 m4)

m10: w7 w10 w4 w10: m10 m2



Algorithm SSMTI-APPROX: Phase 2 outcome

m1 : w6 w1 w2 w1 : m3 (m1 m6 m9)

m2 : w7 w10 w3 w2 : m1 (m3 m5 m7)

m3 : w6 w2 w8 w9 w1 w3 : m2 (m4 m6 m8)

m4 : w7 w9 w8 w3 w4 w4 : m4 m9 (m5 m8)

m5 : w2 w6 w4 w5 w5 : m5 m7

m6 : w1 w3 w6 : (m1 m3 m5)

m7 : w2 w5 w7 : (m2 m4 m10)

m8 : w3 w4 w8 : (m3 m4)

m9 : w1 w4 w9 : (m3 m4)

m10: w7 w10 w10: m10 m2




for each woman w break the tie (if any) in w's list, placing the free men ahead of the engaged men;

reactivate phase 1; /* now just classical Gale-Shapley */

return the set of engaged pairs;

• This is obviously a stable matching

• All we have done is decide how to break the ties and then apply Gale-Shapley



m1 : w6 w1 w2 w1 : m3 (m1 m6 m9)

m2 : w7 w10 w3 w2 : m1 (m3 m5 m7)

m3 : w6 w2 w8 w9 w1 w3 : m2 (m4 m6 m8)

m4 : w7 w9 w8 w3 w4 w4 : m4 m9 (m5 m8)

m5 : w2 w6 w4 w5 w5 : m5 m7

m6 : w1 w3 w6 : (m1 m3 m5)

m7 : w2 w5 w7 : (m2 m4 m10)

m8 : w3 w4 w8 : (m3 m4)

m9 : w1 w4 w9 : (m3 m4)

m10: w7 w10 w10: m10 m2



m1 : w6 w1 w2 w1 : m3 m6 m9 m1

m2 : w7 w10 w3 w2 : m1 m7 m3 m5

m3 : w6 w2 w8 w9 w1 w3 : m2 m6 m8 m4

m4 : w7 w9 w8 w3 w4 w4 : m4 m9 m8 m5

m5 : w2 w6 w4 w5 w5 : m5 m7

m6 : w1 w3 w6 : m1 m3 m5

m7 : w2 w5 w7 : m2 m4 m10

m8 : w3 w4 w8 : m3 m4

m9 : w1 w4 w9 : m3 m4

m10: w7 w10 w10: m10 m2



m1 : w6 w1 : m6 m9

m2 : w7 w2 : m7 m3 m5

m3 : w6 w2 w8 w3 : m8

m4 : w7 w9 w4 : m9 m5

m5 : w2 w6 w4 w5 w5 : m5

m6 : w1 w6 : m1 m3 m5

m7 : w2 w7 : m2 m4 m10

m8 : w3 w8 : m3

m9 : w1 w4 w9 : m4

m10: w7 w10 w10: m10

A stable matching of size 10


Visualising the performance guarantee

Engaged men

Free men

Extramen

Extrawomen

M

A

A

Men Women

M = maximum cardinality stable matching

A = stable matching found by algorithm SSMTI-APPROX

Extra men are matched in M but not in A

Extra women must alsobe matched in A



Engaged men

Free men

Extramen

Extrawomen

M

A

A

Men Women

M = maximum cardinality stable matching

A = stable matching found by algorithm SSMTI-APPROX

M

s

r-s

s

s s

r-s

r-s

A

These women mustbe matched in A

x



Extramen

Extrawomen

M

A

A

Men Women

Inequalities:

(i) |A| ≥ |M| - r

(ii) |A| ≥ |M| - x (a consequence of Phase 2)

(iii) |A| ≥ x + r – s

(iv) |A| ≥ r + s

Algebra gives |M| ≤ 5|A|/3

M

s

r-s

s

s s

r-s

r-s

A

These women mustbe matched in A

x


Further observations

• The algorithm extends in a (fairly) straightforward way to SHRT

• with the same performance guarantee of 5/3

• We have no example where the 5/3 bound is realised

• The worst case we have is 8/5

• so maybe a sharper analysis could improve the bound

Open question

• Can these or similar techniques give good approximation algorithms for more general cases?

• e.g. when men also can have a tie at the end of their list


A bad case for Algorithm SSMTI-APPROX

m1 : w5 w2 w1 : m3 (m8 m5)

m2 : w6 w3 w2 : m1 m6

m3 : w5 w7 w8 w1 w3 : m2 m7

m4 : w6 w8 w7 w4 w4 : m4 m8

m5 : w1 w5 : (m3 m1)

m6 : w2 w6 : (m4 m2)

m7 : w3 w7 : (m4 m3)

m8 : w1 w4 w8 : (m3 m4)

Algorithm can return the red matching – size 5The blue matching – size 8 – is also stable

The stable marriage problem an example

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