# #1 the stable roommates problem with ties david manlove joint work with rob irving university of...

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#1 The Stable Roommates Problem with Ties David Manlove Joint work with Rob Irving University of Glasgow Department of Computing Science Supported by: EPSRC grants GR/M13329 and GR/R84597/01 Nuffield Foundation award NUF-NAL-02 RSE/Scottish Executive Personal Research Fellowship

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#1

The Stable RoommatesProblem with Ties

David Manlove

Joint work with Rob Irving

University of GlasgowDepartment of Computing Science

Supported by:

EPSRC grants GR/M13329 and GR/R84597/01Nuffield Foundation award NUF-NAL-02RSE/Scottish Executive Personal Research Fellowship

#2

Stable Roommates (SR): definition

Input: 2n persons; each person ranks all 2n-1other persons in strict order

Output: a stable matching

Definitions

A matching is a set of n disjoint pairs of persons

A blocking pair of a matching M is a pair of persons {p,q}M such that:

• p prefers q to his partner in M, and• q prefers p to his partner in M A matching is stable if it admits no blocking pair

#3

Stable Roommates (SR): definition

Input: 2n persons; each person ranks all 2n-1other persons in strict order

Output: a stable matching

Definitions

A matching is a set of n disjoint pairs of persons

A blocking pair of a matching M is a pair of persons {p,q}M such that:

• p prefers q to his partner in M, and• q prefers p to his partner in M A matching is stable if it admits no blocking pair

Example SR instance I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

#4

Stable Roommates (SR): definition

Input: 2n persons; each person ranks all 2n-1other persons in strict order

Output: a stable matching

Definitions

A matching is a set of n disjoint pairs of persons

A blocking pair of a matching M is a pair of persons {p,q}M such that:

• p prefers q to his partner in M, and• q prefers p to his partner in M A matching is stable if it admits no blocking pair

Example SR instance I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

#5

Stable Roommates (SR): definition

Input: 2n persons; each person ranks all 2n-1other persons in strict order

Output: a stable matching

Definitions

A matching is a set of n disjoint pairs of persons

A blocking pair of a matching M is a pair of persons {p,q}M such that:

• p prefers q to his partner in M, and• q prefers p to his partner in M A matching is stable if it admits no blocking pair

Example SR instance I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

The matching is not stable as {1,3} blocks.

#6

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

#7

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

#8

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

#9

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

#10

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

1: 3 2 4 2: 1 3 4 3: 2 1 4 4: 1 2 3

#11

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

1: 3 2 4 2: 1 3 4 3: 2 1 4 4: 1 2 3

#12

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

1: 3 2 4 1: 3 2 4 2: 1 3 4 2: 1 3 4 3: 2 1 4 3: 2 1 4 4: 1 2 3 4: 1 2 3

#13

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

1: 3 2 4 1: 3 2 4 2: 1 3 4 2: 1 3 4 3: 2 1 4 3: 2 1 4 4: 1 2 3 4: 1 2 3

#14

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Example SR instance I2: 1: 3 2 42: 1 3 43: 2 1 44: 1 2 3

1: 3 2 4 1: 3 2 4 2: 1 3 4 2: 1 3 4 3: 2 1 4 3: 2 1 4 4: 1 2 3 4: 1 2 3

The three matchings containing the pairs {1,4}, {2,4}, {3,4} are blocked by the pairs {1,2}, {2,3}, {3,1} respectively.

instance I2 has no stable matching.

SR was first studied in 1962 by Gale and Shapley

#15

Stable marriage (SM) problem

SM is a special case of SR

Input: n men and n women; each person ranks all n members of the opposite sex in orderOutput: a stable matching

Definitions:

A matching is a set of n disjoint (man,woman) pairs

A blocking pair of a matching M is an unmatched (man,woman) pair (m,w) such that:

• m prefers w to his partner in M, and• w prefers m to her partner in M A matching is stable if it admits no blocking pair

• Every instance of SM admits a stable matching• Such a matching can be found by an efficient algorithm known as the Gale/Shapley algorithm

#16

Reduction from SM to SR

An instance I of SM can be reduced to an instance J of SR such that set of stable matchings in I = set of stable matchings in J

SM instance I:U={m1, m2,..., mn} is the set of men, and W={w1, w2,..., wn} is the set of women

SR instance J:People comprise men in U and women in W

mi’s list in J is his list in I together with the men in U\{mi} appended in arbitrary order.

wi’s list in J is her list in I together with the women in W\{wi} appended in arbitrary order.

Proposition: Set of stable matchings in I = set of stable matchings in J

Open question: can we reduce SR to SM?

#17

Efficient algorithm for SR

• Knuth (1976): is there an efficient algorithm for deciding whether there exists a stable matching, given an instance of SR?

• Irving (1985) “An efficient algorithm for the ‘Stable Roommates’ Problem”, Journal of Algorithms, 6:577-595

given an instance of SR, decides whether a stable matching exists; if so, finds one

• Algorithm is in two phases

Phase 1: similar to GS algorithm for SM

Phase 2: elimination of “rotations”

• if some list becomes empty during Phase 1 no stable matching exists

• if all lists are of length 1 after Phase 1 we have a stable matching, else:

• if some list becomes empty during Phase 2 no stable matching exists

• else Phase 2 finds a stable matching

#18

Stable Roommates with Ties (SRT)

• Ties are permitted in any preference list

Example SRT instance: 1: (2 3) 4 2: 1 3 4 3: (1 4) 2 4: 3 1 2

• More than one stability definition is possible

Super-stability

A matching M in instance J of SRT is super-stable if there is no pair {p,q}M such that

• p strictly prefers q to his partner in M or is indifferent between them• q strictly prefers p to his partner in M or is indifferent between them

In any instance of SRT with no ties, super-stabilityis the same as classical stabilityRecall: an SR instance can have no stable matching

#19

Stable Roommates with Ties (SRT)

• Ties are permitted in any preference list

Example SRT instance: 1: (2 3) 4 2: 1 3 4 3: (1 4) 2 4: 3 1 2

• More than one stability definition is possible

Super-stability

A matching M in instance J of SRT is super-stable if there is no pair {p,q}M such that

• p strictly prefers q to his partner in M or is indifferent between them• q strictly prefers p to his partner in M or is indifferent between them

In any instance of SRT with no ties, super-stabilityis the same as classical stabilityRecall: an SR instance can have no stable matching

#20

Stable Roommates with Ties (SRT)

• Ties are permitted in any preference list

Example SRT instance: 1: (2 3) 4 2: 1 3 4 3: (1 4) 2 4: 3 1 2

• More than one stability definition is possible

Super-stability

A matching M in instance J of SRT is super-stable if there is no pair {p,q}M such that

• p strictly prefers q to his partner in M or is indifferent between them• q strictly prefers p to his partner in M or is indifferent between them

In any instance of SRT with no ties, super-stabilityis the same as classical stabilityRecall: an SR instance can have no stable matching

#21

SRT: super-stability (cont)

• Algorithm for finding a super-stable matching if one exists, given an SRT instance

Irving and Manlove (2002) “The Stable Roommates Problem with Ties”, Journal of Algorithms, 43:85-105

#22

SRT: super-stability (cont)

• Algorithm for finding a super-stable matching if one exists, given an SRT instance

Irving and Manlove (2002) “The Stable Roommates Problem with Ties”, Journal of Algorithms, 43:85-105

• Algorithm is in two phases

Phase 1: similar to SR case– sequence of proposals and deletions– persons become provisionally assigned to / assigned from other persons

Example SRT instance:

1: (6 4) 2 5 32: 6 3 5 1 43: (5 4) 1 6 24: 2 6 5 (1 3)5: 4 2 3 6 16: 5 1 4 2 3

#23

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 2: 6 3 5 1 4 3: (5 4) 1 6 2 4: 2 6 5 (1 3) 5: 4 2 3 6 1 6: 5 1 4 2 3

#24

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 2: 6 3 5 1 4 3: (5 4) 1 6 2 4: 2 6 5 (1 3) 5: 4 2 3 6 1 6: 5 1 4 2 3 1

#25

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 2: 6 3 5 1 4 3: (5 4) 1 6 2 4: 2 6 5 (1 3) 5: 4 2 3 6 1 6: 5 1 4 2 3 1

#26

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 2: 6 3 5 1 4 3: (5 4) 1 6 2 4: 2 6 5 (1 3) 5: 4 2 3 6 1 6: 5 1 4 2 3 1

#27

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6,4 2: 6 3 5 1 4 3: (5 4) 1 6 2 4: 2 6 5 (1 3) 1 5: 4 2 3 6 1 6: 5 1 4 2 3 1

#28

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6,4 2: 6 3 5 1 4 3 3: (5 4) 1 6 2 2 4: 2 6 5 (1 3) 1 5: 4 2 3 6 1 6: 5 1 4 2 3 1

#29

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6,4 2: 6 3 5 1 4 3 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 1 5: 4 2 3 6 1 3 6: 5 1 4 2 3 1

#30

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6,4 2: 6 3 5 1 4 3 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 1 5: 4 2 3 6 1 3 6: 5 1 4 2 3 1

#31

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6,4 2: 6 3 5 1 4 3 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 1 5: 4 2 3 6 1 3 6: 5 1 4 2 3 1

#32

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6,4 2: 6 3 5 1 4 3 3: (5 4) 1 6 2 5,4 2 4: 2 6 5 (1 3) 1,3 5: 4 2 3 6 1 3 6: 5 1 4 2 3 1

#33

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6,4 2: 6 3 5 1 4 3 3: (5 4) 1 6 2 5,4 2 4: 2 6 5 (1 3) 5: 4 2 3 6 1 3 6: 5 1 4 2 3 1

#34

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 2: 6 3 5 1 4 3 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 5: 4 2 3 6 1 3 6: 5 1 4 2 3 1

#35

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 2: 6 3 5 1 4 3 4 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 2 5: 4 2 3 6 1 3 6: 5 1 4 2 3 1

#36

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 2: 6 3 5 1 4 3 4 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 2 5 5: 4 2 3 6 1 4 3 6: 5 1 4 2 3 1

#37

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 6 2: 6 3 5 1 4 3 4 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 2 5 5: 4 2 3 6 1 4 3 6: 5 1 4 2 3 1 1

#38

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 6 2: 6 3 5 1 4 3 4 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 2 5 5: 4 2 3 6 1 4 3 6: 5 1 4 2 3 1 1

#39

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: (6 4) 2 5 3 6 6 2: 6 3 5 1 4 3 4 3: (5 4) 1 6 2 5 2 4: 2 6 5 (1 3) 2 5 5: 4 2 3 6 1 4 3 6: 5 1 4 2 3 1 1

#40

Phase 1 of Algorithm SRT-Super

while (some person p has a non-empty list and p is not assigned to anyone) {

for (each q at the head of p's list) {assign p to q;/* q is assigned from p */for (each strict successor r of p in q's list) {

if (r is assigned to q)break the assignment;

delete the pair {q,r}; }if ( 2 persons are assigned to q) {

break all assignments to q;for (each r tied with p in q's list)

delete the pair {q,r}; }

}}

Example SRT instance: assigned to/from 1: 6 6 6 2: 3 5 4 3 4 3: 5 2 5 2 4: 2 5 2 5 5: 4 2 3 4 3 6: 1 1 1

#41

After Phase 1:

• Lemma: if some person’s list is empty then no super-stable matching exists

• Lemma: if every person’s list has length 1 then the lists specify a super-stable matching

• If nobody’s list is empty and some person’s list has length >1 then we move on to Phase 2

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#42

After Phase 1:

• Lemma: if some person’s list is empty then no super-stable matching exists

• Lemma: if every person’s list has length 1 then the lists specify a super-stable matching

• If nobody’s list is empty and some person’s list has length >1 then we move on to Phase 2

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

Phase 2: “reject and reactivate Phase 1”

– select a person p whose list is of length >1– let p be rejected by first person on his list– reactivate Phase 1

#43

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#44

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#45

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#46

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty

#47

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty– Reinstate the preference lists to their status after original Phase 1

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#48

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty– Reinstate the preference lists to their status after original Phase 1– Let p reject the last person on his list– Reactivate Phase 1

• Let person 2 reject person 4

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#49

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty– Reinstate the preference lists to their status after original Phase 1– Let p reject the last person on his list– Reactivate Phase 1

• Let person 2 reject person 4

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#50

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty– Reinstate the preference lists to their status after original Phase 1– Let p reject the last person on his list– Reactivate Phase 1

• Let person 2 reject person 4

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#51

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty– Reinstate the preference lists to their status after original Phase 1– Let p reject the last person on his list– Reactivate Phase 1

• Let person 2 reject person 4

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#52

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty– Reinstate the preference lists to their status after original Phase 1– Let p reject the last person on his list– Reactivate Phase 1

• Let person 2 reject person 4

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

#53

• Let person 2 be rejected by person 3

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Person 3’s list is empty– Reinstate the preference lists to their status after original Phase 1– Let p reject the last person on his list– Reactivate Phase 1

• Let person 2 reject person 4

1: 62: 3 5 43: 5 24: 2 55: 4 2 36: 1

– Reactivation of Phase 1 terminates

#54

1: 62: 33: 24: 55: 46: 1

• All lists now have length 1

• This corresponds to the matching {1,6}, {2,3}, {4,5}

1: (6 4) 2 5 32: 6 3 5 1 43: (5 4) 1 6 24: 2 6 5 (1 3)5: 4 2 3 6 16: 5 1 4 2 3

• The matching is super-stable with respect to the original lists

#55

Phase 2 of Algorithm SRT-Super

T := preference lists generated by Phase 1;while (some person p’s list in T is of length > 1) {

let q be the person assigned from p ; let r be the person assigned to p;

form T1 from T by letting p be rejected by q;form T2 from T by letting p reject r;

reactivate Phase 1 on T1;reactivate Phase 1 on T2;if (no list in T1 is empty)

T := T1;else if (no list in T2 is empty)

T := T2;else

output “no super-stable matching exists”;}output T; /* which is a super-stable matching */

#56

Correctness of Algorithm SRT-Super

Lemma 1: if T contains an embedded super-stable matching then either T1 or T2 contains an embedded super-stable matching

Corollary 1: if T contains an embedded super-stable matching and T1 has an empty list then T2 contains an embedded super-stable matching

Corollary 2: if each of T1 and T2 has an empty list then T contains no embedded super-stable matching

Lemma 2: if T contains an embedded super-stable matching and T1 has no empty list then T1 contains an embedded super-stable matching

T

T1 T2

#57

Analysis of Algorithm SRT-Super

• Assume standard data structures:

preference lists represented as doubly- linked lists embedded in an array ranking lists: for each person p, rankp(q) gives the position of q in p’s list

• These may be initialised in O(n2) time at outset

• With these data structures each deletion may be carried out in constant time

• Maintain a pointer to the last element in each person’s preference list

• Maintain a stack of persons waiting to propose • Phase 1 is O(n+d) where d is the number of pairs deleted

#58

Analysis of Algorithm SRT-Super (cont)

• Computation of T1 and T2 during iteration i of main loop in Phase 2 is O(di

1+di2) where di

j

is number of pairs deleted during each reactivation of Phase 1

Maintain two copies of the reduced lists at each iteration Initially make two copies, T1 and T2, of the preference lists - O(n2) time Thereafter, simulate parallel applications of Phase 1 on each of T1 and T2

Ensure that number of deletions from T1

never exceeds number of deletions from T2 by >1 (and vice versa)

Maintain a stack of deletions from each E.g. if a list in T1 becomes empty, undo deletions of T1 and apply deletions of T2

• Hence Phase 2 is O(n+d) where d is the total number of (non-redundant) pairs deleted

• Overall Algorithm SRT-Super has O(n2) complexity

#59

SRT: strong stability

A matching M in instance J of SRT is strongly stable if there is no pair {p,q}M such that

• p strictly prefers q to his partner in M • q strictly prefers p to his partner in M or is indifferent between them

Example SRT instance:

1: 4 2 3 2: (1 3) 4 3: 1 (4 2)4: 2 3 1

The matching is strongly stable.

• Polynomial-time algorithm for finding a strongly stable matching, or reporting that none exists, given an instance of SRT

Scott (2004), PhD thesis (in preparation)

#60

SRT: weak stability

A matching M in instance J of SRT is weakly stable if there is no pair {p,q}M such that

• p strictly prefers q to his partner in M • q strictly prefers p to his partner in M

Example SRT instance: 1: 3 2 4 2: (1 3) 4 3: 2 1 4 4: 1 2 3

The matching is weakly stable.

• The problem of deciding whether an instance of SRT admits a weakly stable matching is NP-complete

Ronn (1990) “NP-complete Stable Matching Problems”, Journal of Algorithms, 11:285-304

Irving and Manlove (2002) “The Stable Roommates Problem with Ties”, Journal of Algorithms, 43:85-105

#61

Other extensions of SR

• Stable Crews Problem (SC)

– Each person p can have one of two roles, e.g. (A) Captain (pA) or (B) Navigator (pB)

1: 4B 3A 2A 2B 3B 4A 2: 1B 4A 3B 4B 1A 3A

3: 1A 2A 2B 4A 4B 1B

4: 2A 1B 2B 3B 1A 3A

E.g. person 2 prefers person 1 as a navigator to person 4 as a captain

#62

Other extensions of SR

• Stable Crews Problem (SC)

– Each person p can have one of two roles, e.g. (A) Captain (pA) or (B) Navigator (pB)

role 1: B 4B 3A 2A 2B 3B 4A 2: A 1B 4A 3B 4B 1A 3A

3: A 1A 2A 2B 4A 4B 1B

4: B 2A 1B 2B 3B 1A 3A

E.g. person 2 prefers person 1 as a navigator to person 4 as a captain

– A matching M is a disjoint set of pairs of persons with roles

– each person is in exactly one pair

– each pair has exactly one Captain (A) and exactly one Navigator (B)

– e.g. {1B, 3A}, {2A, 4B}

#63

– {XA,YB} is a blocking pair of M if (1) X prefers YB to his partner in M (2) Y prefers XA to his partner in M

– A matching is stable if it admits no blocking pair

E.g. {2A, 3B} blocks the previous matching

role1: B 4B 3A 2A 2B 3B 4A 2: A 1B 4A 3B 4B 1A 3A

3: A 1A 2A 2B 4A 4B 1B

4: B 2A 1B 2B 3B 1A 3A

role1: B 4B 3A 2A 2B 3B 4A 2: A 1B 4A 3B 4B 1A 3A

3: B 1A 2A 2B 4A 4B 1B

4: A 2A 1B 2B 3B 1A 3A

– The above matching is stable

#64

– An instance I of SR can be reduced to an instance J of SC such that every stable matching in I corresponds to a stable matching in J and vice versa

– Some instances of SC do not admit a stable matching

– There is an efficient algorithm to find a stable matching, if one exists, given an instance of SC

– If crews are of size 3, the problem of deciding whether a stable matching exists is NP-complete

Cechlárová and Ferková (2004) “The Stable Crews Problem”, Discrete Applied Mathematics, 140 (1-3): 1-17

#65

• Stable Fixtures Problem (SF)

– Choose a schedule of fixtures based on teams having preferences over each other

– Two given teams can play each other at most once

– Each team i has a capacity ci , indicating the maximum number of fixtures it can play

– A fixture allocation A is a set of unordered pairs of teams {i, j} such that, for each i, team i is scheduled to play at most ci fixtures

– Example SF instance:

team capacity preferences1: 2 2 3 4

2: 2 1 3 4 3: 2 1 2 44: 2 1 2 3

#66

• Stable Fixtures Problem (SF)

– Choose a schedule of fixtures based on teams having preferences over each other

– Two given teams can play each other at most once

– Each team i has a capacity ci , indicating the maximum number of fixtures it can play

– A fixture allocation A is a set of unordered pairs of teams {i, j} such that, for each i, team i is scheduled to play at most ci fixtures

– Example SF instance:

team capacity preferences1: 2 2 3 4

2: 2 1 3 4 3: 2 1 2 44: 2 1 2 3

– The above fixture allocation is: {1, 2}, {1, 4}, {2, 3}, {3, 4}

#67

– A blocking pair of a fixture allocation A is a pair of teams {i , j}A such that:

i is assigned fewer than ci teams in A or i prefers j to at least one of its assignees and j is assigned fewer than cj teams in A or j prefers i to at least one of its assignees

– A fixture allocation is stable if it admits no blocking pair

– Example SF instance:

team capacity preferences1: 2 2 3 4

2: 2 1 3 4 3: 2 1 2 44: 2 1 2 3

– The above fixture allocation is not stable as {1,3} is a blocking pair

#68

– Example SF instance:

team capacity preferences1: 2 2 3 4

2: 2 1 3 4 3: 2 1 2 44: 2 1 2 3

– The above fixture allocation is stable

– SR is a special case of SF (in which each team has capacity 1)

– There is an efficient algorithm to find a stable fixtures allocation, if one exists, given an instance of SF

Irving and Scott (2004), “The Stable Fixtures Problem”, to appear in Discrete Applied Mathematics

#69

• Stable Activities Problem (SA)

– Generalization of SC in which number of roles (“activities”) is arbitrary– Problem is formulated in terms of a multigraph

vertex for each person edge eA={u,v} indicates that u finds acceptable activity A with v u has a linear order over N(u)

• Stable Multiple Activities Problem (SMA)

– Generalization of SA in which each vertex v has a bound b(v)– In any matching v is incident to at most b(v) edges

– In both cases we seek a stable matching– Each of SA and SMA is reducible to SR

Cechlárová and Fleiner (2003) “On a generalization of the stable roommates problem”, EGRES Technical Report No. 2003-03

#70

Summary

• Stable Roommates problem (SR)

• Stable Roommates problem with Ties (SRT)

– super-stability– strong stability– weak stability

• Other extensions:– Stable Crews problem (SC)– Stable Fixtures problem (SF)– Stable Activities problem (SA)– Stable Multiple Activities problem (SMA)

Open questions

• Reduction from SR to SM?

• Characterise Algorithm SRT-Super in terms of rotations, as in the SR case

• SA / SMA with ties