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5年弱前 (2013/07/22)にアップロードinテクノロジー

国際シンポジウム「International Symposium on Scheduling 2013」で発表したスライドです

- 教室割り当て問題とその解法6年弱前 by Yutaro Ikeda
- 数理工学モデル化実習発表資料6年弱前 by Yutaro Ikeda
- problemsolved.key約6年前 by Yutaro Ikeda

- APPLICATIONS OF POPULAR MATCHINGS

ON CAMPUS

Yutaro Ikeda, Ayumi Igarashi, Maiko Shigeno

Graduate School of Systems and Information Engineering,

University of Tsukuba

13年7月19日金曜日 - APPLICATIONS OF POPULAR MATCHINGS ON CAMPUS

INTRODUCTION

ALGORITHM

Outline

RESULT

CONCLUSION - APPLICATIONS OF POPULAR MATCHINGS ON CAMPUS

INTRODUCTION

ALGORITHM

Outline

RESULT

CONCLUSION - Assignment problem in real world

An assignment problem has many applications in several

areas of scheduling.

Task assignment

Bed assignment in hospitals

Facility assignment

Resident assignment

etc... - Our Problem

G = (A∪P;E) is a bipartite graph.

Every applicant has an ambition to be assigned to some posts

No post has any wish.

one-sided preference list

p

a

1

1

1

2

2

p

applicants

a

2

posts

2

１

2

p

a

１

3

3

p4 - Our Problem

G = (A∪P;E) is a bipartite graph.

Every applicant has an ambition to be assigned to some posts

No post has any wish.

1st preference of a1 :P1

one-sided preference list

2nd preference of a1:P2

p

a

1

1

1

2

2

p

applicants

a

2

posts

2

１

2

p

a

１

3

3

p4 - Our Problem

G = (A∪P;E) is a bipartite graph.

Every applicant has an ambition to be assigned to some posts

No post has any wish.

preference ...

one-sided preference list

p

a

1

1

1

2

2

p

applicants

a

2

posts

2

１

2

p

a

１

3

3

p4 - Minimum-cost matching

An eﬀective and widely used method is

a minimum-cost matching algorithm.

a

2

p

1

1

1

4

a

2

p

2

１

2

applicants

posts

3

a

1

p

3

3

2

a4

１

p4 - Minimum-cost matching

An eﬀective and widely used method is

a minimum-cost matching algorithm.

Min-cost matching M:6

a

2

p

1

1

1

a

2

p

2

１

2

applicants

posts

4

3

a

1

p

3

3

2

a4

１

p4 - Is a minimum-cost matching “BEST”?

Let’s see another matching.

Min-cost matching M:6

a

2

p

1

1

1

a

2

p

2

１

2

applicants

posts

4

3

a

1

p

3

3

2

a4

１

p4 - Is minimum-cost matching “BEST”?

Let’s see another matching.

Min-cost matching M:6

Matching M’:7

a

2

p

a

2

p

1

1

1

1

1

1

a

2

p

a

2

p

2

１

2

2

１

2

3

3

4

4

a

1

p

a

1

p

3

3

3

3

2

2

a

a4

１

p

4

１

p4

4 - Is minimum-cost matching “BEST”?

the applicant who prefer M to M’ : a3

the applicant who prefer M’ to M: a1,a2

Min-cost matching M:6

Matching M’:7

a

2

p

a

2

p

1

1

1

1

1

1

a

2

p

a

2

p

2

１

2

2

１

2

4

3

4

3

a

1

p

a

1

p

3

3

3

3

2

2

a

a4

１

p

4

１

p4

4 - Is minimum-cost matching “BEST”?

the applicant who prefer M to M’ : a3

the applicant who prefer M’ to M: a1,a2

Min-cost matching M:6

Matching M’:7

:7

a

2

p

a

2

p

1

1

1

1

1

1

M’ is more popular than M

a

2

p

a

2

p

2

１

2

2

１

2

3

3

4

4

a

1

p

a

1

p

3

3

3

3

2

2

a

a4

１

p

4

１

p4

4 - Is minimum-cost matching “BEST”?

the applicant who prefer M to M’ : a3

the applicant who prefer M’ to M: a1,a2

min-cost matching

“ :6

popula

Matchi

r matching” ng M’:7

a

2

p

a

2

p

1

1

1

1

1

1

a

2

p

a

2

p

2

１

2

2

１

2

3

3

4

4

a

1

p

a

1

p

3

3

3

3

2

2

a

a4

１

p

4

１

p4

4 - Popular matching

A “popular matching” is an alternative way to assign.

(Abraham et al. 2007)

p

a

1

１

1

2

2

p2

applicants

a

posts

2

１

2

p3

a

１

3

p4 - Popular matching

A “popular matching” is an alternative way to assign.

(Abraham et al. 2007)

I explain detail of “p

p

opular matching”

a

1

１

1

2

2

p2

applicants

a

posts

2

１

2

p3

a

１

3

p4 - a prefers matching M to M’

An applicant a prefers matching M to M’

(denoted by M M’) , if

I. a is matched in M and unmatched in M’

or

II. a is matched in both M and M’, and

a prefers the post matched by M to the post matched by M’

p

p

a

1

１

1

１

1

a

2

1

2

2

p2

2

p2

a2

a2

１

2

１

2

p3

p3

a

１

１

3

a3

p4

p4

M

M’ - a prefers matching M to M’

An applicant a prefers matching M to M’

(denoted by M M’) , if

I. a is matched in M and unmatched in M’

or

II. a is matched in both M and M’, and

a prefers the post matched by M to the post matched by M’

p

p

a

1

１

1

１

1

a

2

1

2

2

p2

2

p2

M M’

a2

a2

１

2

１

2

p3

p3

a

１

１

3

a3

p4

p4

M

M’ - a prefers matching M to M’

An applicant a prefers matching M to M’

(denoted by M M’) , if

I. a is matched in M and unmatched in M’

or

II. a is matched in both M and M’, and

a prefers the post matched by M to the post matched by M’

p

p

a

1

１

1

１

1

a

2

1

2

2

p2

2

p2

a2

a2

１

2

１

2

p3

p3

a

１

１

3

a3

p4

p4

M

M’’ - a prefers matching M to M’

An applicant a prefers matching M to M’

(denoted by M M’) , if

I. a is matched in M and unmatched in M’

or

II. a is matched in both M and M’, and

a prefers the post matched by M to the post matched by M’

p

p

a

1

１

1

１

1

a

2

1

2

2

p2

2

p2

a

M M’

2

a2

１

2

１

2

p3

p3

a

１

１

3

a3

p4

p4

M

M’’ - “More popular” relation

M is more popular than M'

If |{a 2 A|M

a M 0}| > |{a 2 A|M

a M 0}|

*if the number of applicants that prefer M′ to M exceeds

the number of applicants that prefer M to M′.

p

p

１

a

１

1

a

1

1

1

2

2

2

2

2

2

2

p

2

p

2

a

a

2

2

2

2

2

１

１

１

p

p

3

3

a

１

a

１

3

3

１

p

１

p

4

4

a

a

4

4

M

M

M’ - “More popular” relation

M is more popular than M’

|{a 2 A|M a M0}| > |{a 2 A|M a M0}|

p

p

１

a

1

１

a

1

1

1

2

2

2

2

2

2

2

p

2

p

2

2

a

2

a

2

2

2

１

１

１

p

p

3

3

a

１

a

１

3

3

１

p

１

p

4

4

a

a

4

4

M

M

M’

preference of each applicant

# of applicant

a

a

a

a

1

2

3

4

M M’

M‘ M - “More popular” relation

M is more popular than M’

|{a 2 A|M a M0}| > |{a 2 A|M a M0}|

p

p

１

a

1

１

a

1

1

1

2

2

2

2

2

2

2

p

2

p

2

2

a

2

a

2

2

2

１

１

１

p

p

3

3

a

１

a

１

3

3

１

p

１

p

4

4

a

a

4

M

4

M

M’

preference each applicant

# of applicant

a

a

a

a

1

a2

a3

4

M M’

1

2

3

4

M

M

same

M’

M

M’

M‘ M - “More popular” relation

M is more popular than M’

|{a 2 A|M a M0}| > |{a 2 A|M a M0}|

p

p

１

a

1

１

a

1

1

1

2

2

2

2

2

2

2

p

2

p

2

2

a

2

a

2

2

2

１

１

１

p

p

3

3

a

１

a

１

3

3

１

p

１

p

4

4

a

a

4

4

M

M

M’

preference each applicant

# of applicant

a

a

a

a

1

a2

a3

4

M M’

2

1

2

3

4

M

same

M’

M

M‘ M

1 - “More popular” relation

M is more popular than M’

|{a 2 A|M a M0}| > |{a 2 A|M a M0}|

p

p

１

a

1

１

a

1

1

1

2

2

2

2

2

2

2

p

2

p

2

2

a

2

a

2

2

2

１

１

１

M is mo p

re popular than M’

p

3

3

a

１

a

１

3

3

１

p

１

p

4

4

a

a

4

4

M

M

M’

preference each applicant

# of applicant

a

a

a

a

1

a2

a3

4

M M’

2

1

2

3

4

M

same

M’

M

M‘ M

1 - Definition of “popular matching”

A matching M is popular

if there is no matching that is more popular than M.

The popular matching problem[Abraham et al., 2007]

determine whether a given instance admits a popular matching

find such a matching if one exists

characterization of popular matchings

Theorem (Abraham et al. 2007)

A matching M is popular if and only if

(i) every f-post is matched in M

and (ii) M is an applicant-complete matching

on the reduced graph G′ - “f-post”, ”s-post”

f(a) : a’s 1st preference post.

set of f-posts : { f(a) | a ∈ A }

s(a) : a’s 1st non-f-post.

set of s-posts : { s(a) | a ∈ A }

p

１

a

1

1

2

3

4

p2

2

3

a

１

p3

2

4

p4 - “f-post”, ”s-post”

f(a) : a’s 1st preference post.

set of f-posts : { f(a) | a ∈ A }

s(a) : a’s 1st non-f-post.

set of s-posts : { s(a) | a ∈ A }

p

１

f(a

a

1

1)

1

2

3

4

p2

2

3

a

１

p

f(a2)

3

2

4

p4 - “f-post”, ”s-post”

f(a) : a’s 1st preference post.

set of f-posts : { f(a) | a ∈ A }

s(a) : a’s 1st non-f-post.

set of s-posts : { s(a) | a ∈ A }

p

１

f-post

a

1

1

2

3

4

p2

2

3

a

１

p

f-post

3

2

4

p4 - “f-post”, ”s-post”

f(a) : a’s 1st preference post.

set of f-posts : { f(a) | a ∈ A }

s(a) : a’s 1st non-f-post.

set of s-posts : { s(a) | a ∈ A }

p

１

f-post

a

1

1

2

3

4

p2

s(a1)

2

3

a

１

p

f-post

3

2

4

p4 - “f-post”, ”s-post”

f(a) : a’s 1st preference post.

set of f-posts : { f(a) | a ∈ A }

s(a) : a’s 1st non-f-post.

set of s-posts : { s(a) | a ∈ A }

s(a2)

p

１

f-post

a

1

1

2

3

4

p2

s(a1)

2

3

a

１

p

f-post

3

2

4

p4 - “f-post”, ”s-post”

f(a) : a’s 1st preference post.

set of f-posts : { f(a) | a ∈ A }

s(a) : a’s 1st non-f-post.

set of s-posts : { s(a) | a ∈ A }

s(a2)

p

１

f-post

a

1

s(a2)

1

2

3

4

p2

s(a1)

2

3

a

１

p

f-post

3

2

4

p4 - Reduced graph

A reduced graph G′ is the induced subgraph of G

by {(a,f(a)) | a ∈ A} ∪{(a,s(a)) | a ∈ A}

f(a1)

p

１

a

1

s(a2)

1

2

3

4

p2

s(a1)

2

3

f(a

a

１

p

2)

3

2

4

p4 - Reduced graph

A reduced graph G′ is the induced subgraph of G

by {(a,f(a)) | a ∈ A} ∪{(a,s(a)) | a ∈ A}

reduced graph

G′

f(a1)

p

１

a

1

s(a2)

1

2

3

4

p2

s(a1)

2

3

f(a

a

１

p

2)

3

2

4

p4 - “applicant-complete”

A matching is applicant-complete

if every applicant is matched to a post.

a

p

1

1

a

p2

2

a

p3

3

a4

p4 - “applicant-complete”

A matching is applicant-complete

if every applicant is matched to a post.

not applicant complete

a

p

1

1

a

p2

2

a

p3

3

a4

p4 - “applicant-complete”

A matching is applicant-complete

if every applicant is matched to a post.

applicant complete

a

p

1

1

a

p2

2

a

p3

3

a4

p4 - Definition of “popular matching”

Theorem (Abraham et al. 2007)

A matching M is popular if and only if

(i) every f-post is matched in M

(ii) M is an applicant-complete matching on the reduced graph G′

We can find a popular matching in polynomial time. - Our purpose

Awareness of the issues

There are few applications of popular matchings

to real-world assignment problems

Purpose

We verify the availability of popular matchings

in real-world assignment problems - APPLICATION OF POPULAR MATCHINGS ON CAMPUS

INTRODUCTION

ALGORITHM

Outline

RESULT

CONCLUSION - Requirement in real-world problems

The trouble when we apply popular matchings

to real-world problems

There are instances which do not have any popular matchings.

Our problems require any applicant-complete matching

which is an appropriate nearly popular.

Solutions to this problem

In order to find an applicant-complete nearly popular matching,

we employ a multi-phase algorithm. - A multi-phase algorithm

-Find a maximum matching M

on the reduced graph such that every f-post is matched.

If M is applicant-complete then

initial phase

-Stop

Otherwise

-Go to the recurrence phase

-Delete applicants and posts matched by M from G

-Reconstruct the reduced graph G′

-Find a maximum f -post-complete matching M′ in

G′

recurrence

-Replace M by M ∪ M′

phase

If M is applicant-complete then

-Stop

Otherwise

-Go to the recurrence phase - A multi-phase algorithm

initial phase

-Find a maximum matching M

on the reduced graph such that every f-post is matched.

If M is applicant-complete then

-Stop

Otherwise

-Go to the recurrence phase - A multi-phase algorithm

initial phase

If M is applicant-complete then

-Stop

Otherwise

matching M

-Go to the recurrence phase

a

p

1

1

a

p

2

2

a

p

3

3

a4

p4 - A multi-phase algorithm

recurrence phase

-Delete applicants and posts matched by M from G

-Reconstruct the reduced graph G′

-Find a maximum f -post-complete matching M′ in G′,

-Replace M by M ∪ M′

If M is applicant-complete then

-Stop

Otherwise

-Repeat to the recurrence phase - A multi-phase algorithm

recurrence phase

-Delete applicants and posts matched by M from G

-Reconstruct the reduced graph G′

matching M

p

a1

1

a2

p2

a3

p3

a4

p4 - A multi-phase algorithm

recurrence phase

-Delete applicants and posts matched by M from G

-Reconstruct the reduced graph G′

matching M

a

p

1

1

a

p

2

2

a

p

3

3

a

p

4

4 - A multi-phase algorithm

recurrence phase

-Delete applicants and posts matched by M from G

-Reconstruct the reduced graph G′

matching M

f(a2)

a

p

1

1 s(a1)

a

p

2

2

a

p f(a1)

3

3

reduced graph

s(a2)

G′

a

p

4

4 - A multi-phase algorithm

recurrence phase

-Delete applicants and posts matched by M from G

-Reconstruct the reduced graph G′

repeat the same

f(a2)

a

process as inis(a

ti 1)

al phase

p

1

1 s(a1)

a

p

2

2

a

p f(a1)

3

3

reduced graph

s(a2)

G′

a

p

4

4 - A multi-phase algorithm

recurrence phase

-Delete applican

Rep ts and p

ea

osts ma

t the r

tched b

ecurrenc y M from G

e phase

-Rec

un

onstr

til we uc

fi t the reduced gr

nd an applicaph G

ant-′complete matching

✔

The algorithm always terminates

f(a2)

repeat the same

because a

a

process as inis(a

ti 1)

al phase

p

1

1

t least 2 applicants are matched

s(a1)

f-post

in each recurrence phase

a

p

2

2

f-post

f(a1)

a

s(a2)

p

3

3

reduced graph

s(a2)

G′

a

p

4

4 - APPLICATIONS OF POPULAR MATCHINGS ON CAMPUS

INTRODUCTION

ALGORITHM

Outline

RESULT

CONCLUSION - Evaluation of popular matching

We verify the availability of popular matchings in real-world

three assignment problems which arise in our campus

by following 2 approach.

We investigate the existence of popular matchings

We compare popularity between a matching obtained

by our multi-phase algorithm and a min-cost matching

The popularity is evaluated by

✔ the number of applicants who prefer the Mp to Mmc

Mp : A min-cost matching among at most 10 nearly popular matchings

obtained by our algorithm

Mmc : A most popular matching among at most 10 min-cost matchings - APPLICATIONS

Assignment of classrooms

Assignment of advisers

Laboratory Assignment

Additional experiments - APPLICATIONS

✔ Assignment of classrooms

Assignment of advisers

Laboratory Assignment

Additional experiments - ASSIGNMENT OF CLASSROOMS

BACKGROUND

In our campus, classrooms are assigned to some clubs for

activities out side of class.

We made inquiry about preference lists for clubs and

make classrooms assignment.

club(applicant)

classroom(post)

the number

20

60 - ASSIGNMENT OF CLASSROOMS

RESULT＆DISCUSSION

In this case, the obtained matching Mp is almost popular,

since there are only two clubs which is not assigned any

classroom at the initial phase.

The popularity between Mp and Mmc does not have any

diﬀerence.

In both matchings Mp and Mmc,

18 clubs are assigned the same classrooms.

One club prefers Mp to Mmc and one club prefers Mmc to Mp. - APPLICATIONS

Assignment of classrooms

✔ Assignment of advisers

Laboratory Assignment

Additional experiments - ASSIGNMENT OF ADVISERS

BACKGROUND

In our department, every student is assigned to a professor to

take advices on his/her graduation thesis.

According to student’s preference lists,

a coordinator assigns each student to a professor.

A capacity stands for the number of students who can be

assigned the same professor.

student(applicant) professor (post)

the number

56

22 - ASSIGNMENT OF ADVISERS

Bias of preference lists - Comparing popularity for the ASSIGNMENT OF ADVISERS

RESULT

# o

# f

o s

f tud

s

ents

tud

ents w

ith

capacity

Mp Mmc

Mp Mmc

4

3

11

5

1

3

6

1

2

7

0

1

8

0

2

It is surprising that Mmc is more popular than Mp for every

capacity. Indeed, preference lists of students tend to be hard

to admit popular matchings, because the similar choices arise

from several students in the same research fields.

*Using the average of 10 result obtained - APPLICATIONS

Assignment of classrooms

Assignment of advisers

✔ Laboratory Assignment

Additional experiments - LABORATORY ASSIGNMENT

BACKGROUND

In our department, students are assigned to each laboratory.

We made inquiry about preference lists for students and

make a laboratory assignment.

A capacity stands for the number of students who can be

assigned the same laboratory.

student(applicant) laboratory(post)

the number

54

23 - LABORATORY ASSIGNMENT

Bias of preference lists - Comparing popularity for the ASSIGNMENT OF LABORATORY

RESULT

# o

# f

o s

f tud

s

ents

tud

ents w

ith

capacity

Mp Mmc

Mp Mmc

4

9

8

5

6

10

6

4

8

7

2

2

8

2

1

9

2

1

It depends on the capacity whether Mp is more popular than Mmc.

As in the case of the adviser assignment, a few laboratories are ranked in many

preference lists, which causes to admit no popular matching.

However, the number of choices ranked in a preference list is larger than

one for the adviser assignment.

This fact may derive the tendency that Mp is more popular than Mmc.

*Using the average of 10 result obtained - APPLICATIONS

Assignment of classrooms

Assignment of advisers

Laboratory Assignment

✔ Additional experiments - ADDITIONAL EXPERIMENTS

We generate artificial two types preference lists as follows.

For each applicant, generate number pi for post i (i = 1,...,50)

by uniform random numbers.

(1) uniform preference lists:

The preference list of the applicant is given by the

decreasing order of the value of pi.

(2) biased preference lists:

The preference list of the applicant is given by the

decreasing order of the value of pi × i. - ADDITIONAL EXPERIMENTS

Two types preference lists are generated as follows. For each

applicant, generate number pi for post i (i = 1,...,50) by uniform

random numbers.

(1) uniform preference lists:

The preference list of the applicant is given by the

decreasing order of the value of pi.

(2) biased preference lists:

The preference list of the applicant is given by the

decreasing order of the value of pi × i. - ADDITIONAL EXPERIMENTS

Two types preference lists are generated as follows. For each

applicant, generate number pi for post i (i = 1,...,50) by uniform

random numbers.

(1) uniform preference lists:

The preference list of the applicant is given by the

decreasing order of the value of pi.

(2) biased preference lists:

The preference list of the applicant is given by the

decreasing order of the value of pi × i. - ADDITIONAL EXPERIMENTS

Bias of uniformed preference lists - ADDITIONAL EXPERIMENTS

Bias of biased preference lists - Comparing popularity for the uniformed & biased preference lists

UNIFORMED

BIASED

RESULT

# o

# f

o s

f tud

s

ents

tud

ents w

ith

# o

# f

o s

f tud

s

ents

tud

ents w

ith

capacity

Mmc Mp

Mp Mmc

Mmc Mp

Mp Mmc

1

0.0

0.0

0.0

0.0

2

4.1

27.9

10.0

7.9

3

4.2

12.7

15.2

14.4

4

1.0

12.0

20.5

15.6

5

0.1

12.4

20.7

16.5

6

0.0

12.6

20.5

14.9

7

0.0

12.5

18.2

13.8

8

0.0

12.3

17.0

12.3

9

0.0

12.3

15.5

11.3

10

0.0

12.2

14.5

9.8

*Using the average of 10 result obtained - APPLICATION OF POPULAR MATCHINGS ON CAMPUS

INTRODUCTION

POPULAR MATCHING

Outline

APPLICATIONS

CONCLUSION - Summary

In this research, we applied popular matchings to three real

world assignment problems with one-side preference lists.

We showed a multi-phase algorithm which gives a suitable

matching even if any popular matching does not exist.

Because real-world problems require some matching nearly

popular when the instance has no popular matching.

We compared the results obtained by popular matchings and

by min-cost matchings. - Conclusion & Future work

In some real-world problems, an assignment obtained by our

multi-phase algorithm is not more popular than one by min-

cost matching algorithm. This is because a few posts are

ranked in high priority of many applicants.

As we observed at the experiments for artificial data, we

conjecture that our nearly popular matching is better than

min-cost matchings when applicants have uniform

preference lists. Proving this hypotheses is our future work. - Reference

Abraham, D. J. and K. Chechla rova, and D. F. Manlove, and K. Mehlhoen. (2004).

Pareto-optimality in house allocation problems. The 15th Annual International Symposium

on Algorithms and Computation, Lecture Notes in Computer Science, Vol. 3341, pp. 3–15.

Abraham, D. J. and R. W. Irving, and T. Kavitha, and K. Mehlhorn. (2007). Popular

matchings. SIAM Journal on Computing, Vol. 37, pp. 1030–1045.

Irving, R. W. and T. Kavitha, and K. Mehlhone, and D. Michail, and K. Paluch. (2006).

Rank-maximal matchings. ACM Transactions on Algorithms, Vol. 2, pp. 602–610.

Kavitha, T. and M. Nasre. (2009). Optimal popular matchings. Discrete Applied

Mathematics, Vol. 157, pp. 3181–3186.

Gardenfors, P. (1975). Match making: Assignments based on bilateral preferences.

Behavioural Sciences, Vol. 20, pp. 166–173.

Lawler, E. L. (1976). Combinatorial Optimization: Net- works and Matroids, Holt, Rinehart

and Winston. Mahdian, M. (2006). Random popular matchings. Proceedings of the 7th

ACM Conference on ElectronicCommerce, pp. 238–242.

McCutchen, R. M. (2008). The Least-Unpopularity-Factor and Least-Unpopularity-Margin

Criteria for Matching Problems with One-Sided Preferences The 8th Latin American

Symposium, Lecture Notes in Computer Science, Vol. 4957, pp. 593–604 - Thank You

For

Listening - Reference Materials
- ADDITIONAL EXPERIMENTS

Although Mahdian (Mahdian 2006) and Abraham et al.

(Abraham et al. 2007) said that popular matchings do exist

with good probability when preference lists are randomly

constructed, our three applications do not admit popular

matchings.

Moreover, our multi-phase algorithm based on popular

matchings sometimes gives a result not more popu- lar than

one by a min-cost matching algorithm. - ADDITIONAL EXPERIMENTS

One of the reasons seems to be that posts ranked in

preference lists are biased.

To examine the influence of such biased preference lists, we

create artificially data with 50 applicants and 50 posts.

Each applicant ranks all the posts by strict order. - IMPLEMENTATION

Although a matching represents a one-to-one assignment,

our applications require one-to-some assignments.

We transform such one-to-some assignment problems to

matching problems by copying multiple nodes.

参考資料

結果のSectionで触れる - The criteria by which a matching is chosen

Please See this example.

a

１

a1

1

p

1

p1

1

a

2

p

2

p

2

2

a2

2

a3

3

p3

a3

2

p3

4

a4

2

p4

a4

p4

M

M

1

2 - The criteria by which a matching is chosen

Although there is no diﬀerence in popularity between

M1 and M2, it is obvious that M1 is more desirable.

Corresponding to this example, we choose a min-cost

matching among near popular matchings obtained by the

multi-phase algorithm.

We denote this obtained matching by MP.

MP

a

１

a1

1

p

1

p1

1

a

2

p

2

p

2

2

a2

2

a3

3

p3

a3

2

p3

4

a4

2

p4

a4

p4

M

M

1

2 - The criteria by which a matching is chosen

Although there is no diﬀerence in popularity between

M1 and M2, it is obvious that M1 is more desirable.

Corresponding to this example, we choose a min-cost

matching among near popular matchings obtained by the

multi-phase algorithm.

元の論文を見て差が

We denote this obtained matching by MP.

顕著な例

MP

a

１

p

a

1

p

1

1

1

1

a

2

p

a

2

p

2

2

p

2

2

a

2

8

9

a3

3

p3

p

a3

2

p3

p

4

a

2

p4

4

a4

p4

MP

M

M

1

2 - The criteria by which a matching is chosen

Please See next example.

a

１

p

a

p

1

1

1

1

3

a

3

p

2

2

a

p

2

2

1

a

1

3

2

p3

p

a3

p3

p

M’

M’

1

2 - The criteria by which a matching is chosen

Min-cost matchings, although M ′

′

2 is more popular than M1 .

Thus, we need to compare popularity among min-cost

matchings in order to find a desired matching.

We call a matching M most popular among a collection of

matchings, if the number of times that M becomes more

popular when comparing it with other matching is largest. By

comparing all pairs of candidate ten min-cost matchings, we

choose the most popular matching.

We denote this obtained matching by MMC.

a

１

p

a

p

1

1

1

1

3

a

2

p

2

2

a

p

2

2

1

a

1

3

2

p3

p

a3

p3

p

M’

M’

1

2 - “M(a)”

The post to which an applicant a is matched in M is

denoted by M(a).

p

a

１

1

1

2

M1(a1)

2

p2

a2

１

2

M1(a3)

p3

a

１

3

M1(a2)

p4

M1 - a prefers matching M to M’

An applicant a prefers matching M to M’

(denoted by M M’) , if

I. a is matched in M and unmatched in M’

or

II. a is matched in both M and M’, and a prefers M(a) to M’(a).

p

p

a

1

１

1

１

1

a

2

1

2

2

p2

2

p2

a2

a2

１

2

１

2

p3

p3

a

１

１

3

a3

p4

p4

M

M’