A NOVEL REPRESENTATION AND ALGORITHMS

FOR (QUASI) STABLE MARRIAGES

B. Y. Zavidovique

Institut d’Electronique Fondamentale, Universit

e Paris 11

atiment 220 - 91405 Orsay Cedex

N. Suvonvorn

Institut d’Electronique Fondamentale, Universit

e Paris 11

atiment 220 - 91405 Orsay Cedex

Guna S. Seetharaman

Air Force Institute of Technology

AFIT/ENG 2950 P Street Wright Patterson AFB OH 45433-7765

Keywords:

stable marriages, matching algorithms.

Abstract:

In this paper, we propose ”stable marriages” algorithms based on a novel representation called marriage table.

After explaining how properties as global satisfaction, sex equality and stability show in the representation,

we deﬁne 3 algorithms corresponding to 3 different scans of the marriage table to meet progressively all

constraints. The performance is evaluated in front of the population size for 200 instances in each case.

That supports qualitative statistic analysis. Two matching examples in image processing are displayed for

illustration.

1 INTRODUCTION

The problem of stable marriage was ﬁrst studied by

Gale and Shapley (Gale and Shapley, 1962). In this

problem, two ﬁnite sub-sets M and W of two re-

spective populations, say men and women, have to

match. Assume n is the number of elements, M =

, m

, ..., m

} and W = {w

, w

, ..., w

}. Each

element x creates its preference list l(x) i.e. it sorts

all members of the opposite sex from most to less pre-

ferred. A matching M is a one to one correspondence

between men and women. If (m, w) is a matched pair

in M , we note M(m) = w and M(w) = m and

is the rank of m in the list of w (resp. ρ

the

rank of w in the list of m) . Man m and woman

w form a blocking pair if (m, w) is not in M but

m prefers w to M(m) and w prefers m to M(w).

If there is no blocking pair, then the matching M is

stable (Abeledo and Rothblum, 1995), (Diamantoudi

et al., 2004). Gale and Shapley proved that there is

always at least one stable matching M whichever the

instance {M , W , l(m), l(w)}. They proposed the al-

gorithm of Gale-Shapley (GS ) to ﬁnd M with com-

plexity O(n

Since then, this optimization problem was con-

stantly one among the most popular in combina-

torics from both theoretical (McVitie and Wilson,

1971), (K. Iwama and Morita, 1999),(D.F. Manlove

et al., 2002), (Gent and Prosser, 2002), (McVi-

tie and Wilson, 1971) and practical points of view

(D. Bianco and Larimer, 2001), (C.P. Teo and Tan,

1999), (T. Kavitha and Paluch, 2004). According to

(K. Iwama and Morita, 1999), the stable marriage

problem was studied and generalized in 4 directions:

(i) stable marriage with complete list and total or-

der, the case of Gale and Shapley; (ii) stable marriage

with incomplete list and total order (Manlove, 1999),

(iii) stable marriage with complete list and indiffer-

ence (Irving, 1994), and (iv) stable marriage with in-

complete list and indifference (K. Iwama and Morita,

1999).

GS has usually two different solutions, men-

optimal and women-optimal depending whom is

asked ﬁrst to choose. Men-optimal brings a stable

matching in which men have the best possible part-

ner and women may have the worst and conversely.

In many applications of such optimization on bi-

partite graphs, as resource scheduling, there might

be reasons why to favour one sub-population: for

instance demand constraints are economically more

Y. Zavidovique B., Suvonvorn N. and S. Seetharaman G. (2005).

A NOVEL REPRESENTATION AND ALGORITHMS FOR (QUASI) STABLE MARRIAGES.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics, pages 63-70

DOI: 10.5220/0001159900630070

 SciTePress

important than supply ones, or teachers constraints

might be more strict than class-rooms ones. In many

other such problems like segment-pairing in robot vi-

sion for stereo reconstruction, motion understanding

or object recognition there is an a priori equal im-

portance of both sets of segments respectively ex-

tracted from a couple of images or from the model

(Bouchafa and Zavidovique, ), (J.L.Lisani et al.,

2001), (Monasse and Guichard, 1998), (Ballester

et al., 1998), (Caselles et al., 1999) : then sex equal-

ity is likely worth accounting for, leading to a fair

algorithm. Moreover, some global satisfaction from

the matching may translate a better balanced solution

among the many possible ones. Neither one is guar-

anteed by GS : the obtained stable matching can be

such that everybody is unsatisﬁed. A last difﬁculty

comes from the order in which men or women are

considered inside their own sub-set, it can inﬂuence

the result.

In this article, we propose a stable marriages al-

gorithm based on a novel representation, called mar-

riage table. The new algorithm ﬁts stable marriages

with complete/incomplete list and total order. It aims

at stability, sex equality and global satisfaction.

2 NOVEL REPRESENTATION OF

THE STABLE MARRIAGES

PROBLEM

In order to build an algorithm that had a chance

to meet the three criteria of stability, sex equality

and global satisfaction, we ﬁrst change representa-

tion. The so-called marriage table translates and sup-

plements the preference lists. Stable matchings are

looked for by scanning this latter array and suitable

properties of the solution are associated to the type of

scan. The marriage table is a table with (n + 1) lines

and (n + 1) columns. Lines (resp. columns) frame

the preference orders of men, {1 · · · p · · · N ∞} (resp.

women, {1 · · · q · · · N ∞}). The cell (p, q) contains

pairs (m, w) such that w is the p

choice of m, and

m is the q

choice of w. Cells can thus contain more

than one pair or none. The cell (p, ∞) (resp. (∞, q))

contains the pairs where the woman is the p

choice

of the man (resp the q

choice of the woman) but the

man does not exist in her preference list (resp. the

woman is not in his preference list). A key feature of

this table in the ”complete list” case is that each line

contains all men once and each column contains all

women once. The ﬁgure 1 shows a typical marriage

table.

The table 1 is the example of an instance of three

men and women. Every man or woman made their

preference list. The ﬁgure 2 is the marriage table and

stable matching established from the population 1.

1 2 3 4 q

(x,y)

∞

Figure 1: Marriage table : the pair (x,y), y is the 3

choice

of x and x is the 4

choice of y

Table 1: An instance of 3 men and women and their prefer-

ence lists

Men Women

1 : C, A, B A : 1, 2, 3

2 : A, C, B B : 2, 3, 1

3 : C, B, A C : 1, 2, 3

One advantage of the marriage table is that satis-

faction and equality of sex show concurrently in the

same representation.

For instance let us deﬁne a global satisfaction by:

S =

(m,w)∈M

(ρ

+ ρ

) (1)

Intuitively, the closer S to zero the greater global

satisfaction: in average more people are satisﬁed. A

solution with maximum global satisfaction would dis-

play matched pairs as close around the origin (table

bottom-left) as mutual exclusion allows. More gen-

erally the table representation is indicative of a result

global satisfaction through the lay out of the selected

couples. Satisfaction is constant along antidiagonals

(straight lines of equation p + q = constant) and de-

creasing with the distance to the origin . And that

provides some criteria to design scans of the marriage

table that could favour better global solutions.

Conversely, sex equality tends to ﬁt the diagonal of

the marriage table. Let us deﬁne it as

S =

(m,w)∈M

|ρ

− ρ

| (2)

Intuitively the closer to the diagonal the more bal-

anced treatment. Elements of a pair in a cell close to

the diagonal are equally satisﬁed or unsatisﬁed, de-

pending on the distance to the origin. The smaller the

greater equity. And again that provides some criteria

to design scans of the marriage table that could favour

more equitable solutions.

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

1 2 3 q

(3,C)

(2,A)(1,C)

(1,A)

(2,C)

(3,B)

(3,A)

(1,B)

(2,B)

∞

(3,A)

1 2 3 q

(3,C)

(2,A)(1,C)

(1,A)

(2,C)

(3,B)

(2,B)

(1,B)

∞

(a) (b)

Figure 2: (a) Marriage table established from the table 1 (b)

Matching result : (1,C), (2,A) and (3,B).

Stability gets a graphic translation too in the mar-

riage table. In the case of complete preference lists, a

blocking situation is represented ﬁgure 3. Assuming

(x, t) and (z, y) were respectively paired, then (x, y)

cannot be in the grey rectangle and (z, t) cannot be in

the dashed one. These constraints will help building a

new algorithm. In the case of uncomplete preference

lists an additionnal blocking situation is when m and

w are not matched in M but m ﬁnds w acceptable and

w ﬁnds m acceptable.

(z, y)

(x, y)

(z, t)

(x, t)

Figure 3: Blocking situation in a marriage table.

3 MARRIAGES AND TABLE

SCANS

Algorithms can now be designed to ﬁnd marriages

that would globally guarantee one or the other prop-

erty. Any selection process is a scan of the marriage

table along which couples are stored or not and then

released or not depending on circumstances. After the

analysis above, suitable scans to meet all constraints

should be zig-zag ones that trade off between the ﬁrst

and second diagonal directions. Considering a priori

symmetries of the marriage table, scanning from left

to right or conversely does not matter statistically and

same for scanning from origin to top or conversely.

Indeed, given an instance, changing man’s lists into

woman’s lists makes it for the right/left invariance

and complementing preferences to the population-

size makes it for the top-bottom invariance. We study

experimentally three strategies here (zigzag ZZ with

man-optimal or woman-optimal, optimal (symmetric)

zigzag OZ , blocked zigzag BZ ) and then compare

the results with GS in a systematic way: 200 in-

stances are built at random for populations of 5, 10,

50, 100, 150 and 200 respectively. Each algorithm is

run on the populations and for each one the following

plots are displayed and analyzed: global satisfaction

vs instances index, fairness vs instances index and, in

case there are, number of blocking pairs vs popula-

tions.

3.1 Zigzag with man-optimal or

woman-optimal (ZZ )

It appears from the global satisfaction graph (ﬁgure

4) that its patterns present some similarity ( min/max,

σ) by sample of 50. We pick up 40 consecutive in-

stances at random for display to show results more in

details. From the ﬁgure 5, one can see that the global

satisfaction obtained by ZZ is better in average than

the one by GS . Its variation is also smaller, meaning

that results from ZZ are more consistent (hence more

reliable to global matching) than by GS .

Figure 4: Comparing global satisfaction between meth-

ods: (a) GS man-optimal (b) GS woman-optimal (c) Zigzag

man-optimal and (d) Zigzag woman-optimal, in case of 150

large populations.

Figure 6 shows the same sample of 40 instances

for sex equality. Trends of that type of plot are very

similar to global satisfaction trends. The jittery pat-

tern of GS s are similar for global satisfaction and sex

equality, with average standard deviation in the range

of 40 to 50%. In both cases, strong variations from

one instance to the next one show qualitative same

tendancies, contrasting with comparatively bounded

variations by ZZ (15 to 20%). In average ZZ per-

forms twice as well as GS (3500 vs. 4800 and 1800

A NOVEL REPRESENTATION AND ALGORITHMS FOR (QUASI) STABLE MARRIAGES

Figure 5: Close up on global satisfaction between meth-

ods: (a) GS man-optimal (b) GS woman-optimal (c) Zigzag

man-optimal and (d) Zigzag woman-optimal

vs. 3800). However the improvement is much more

signiﬁcant regarding sex equality that is genuinely de-

nied by GS . Moreover, if one considers the number

b of instances where ZZ is better than GS , deﬁned as

b =

all instances

[min(GS

,GS

)−max(ZZ

,ZZ

)]

(3)

with

= 1 if x > 0

= 0 else

and

β =

100×b

number of instances

β = 96% for global satisfaction and β = 99% for

sex equality. This percentage depends directly on the

population size. The larger population, the greater im-

provement. Experimentally, beyond 200 large popu-

lations ZZ becomes 100 percent better than GS for

both global satisfaction and sex equality.

Figure 6: Comparison of sex equality between the meth-

ods: (a) GS man-optimal (b) GS woman-optimal (c) Zigzag

man-optimal and (d) Zigzag woman-optimal

However, matching stability is not guaranteed by

ZZ . As a gauge of unstability, ﬁgure 7 (a) and (b)

display the average number of blocking pairs with

their standard deviations vs. the population size. Note

that cases (a) and (b), respectively ZZ man ﬁrst and

ZZ women ﬁrst, are unseparable at that representa-

tion scale. It appears that the larger, the less stable.

Figure 7: The average number of blocking pairs using: (a)

Zigzag man-optimal and (b) Zigzag woman-optimal (c) Op-

timal zigzag (d) Blocked zigzag

3.2 Optimal (Symmetric) zigzag

(OZ )

The primary implementation of a diagonal scan of the

marriage table proves signiﬁcantly better than GS as

for global satisfaction and fairness. But the scan start

direction - men or women ﬁrst - still matters in ex-

treme cases, although its impact is less in general. We

now propose an algorithm (algo.1) that targets opti-

mal zigzag from bottom-left to top-right (forward).

Here again, anti-diagonals of the table are scanned

forward from maximum to minimum global satisfac-

tion but each one is read in swinging from center to

sides meaning maximum to minimum sex equality.

With this algorithm global satisfaction is considered

ﬁrst and then sex equality. Figures 8(c) and 9(c) show

the global satisfaction and sex equality compared with

GS 8(a)(b) and 9(a)(b) respectively. Both global sat-

isfaction and sex equality slightly worsen compared

to ZZ ,but the main result is blocking pair numbers de-

crease signiﬁcantly by about 40% (ﬁgure 7 (c)). Still,

the method does not guarantee any matching stability.

The complexity for all these algorithms so far remains

in O(n

), table building included.

3.3 Blocked zigzag (BZ )

Both previous algorithms provide matching solutions

that are globally satisfactory and fair but unstable:

they might even be such that everybody would like

to move!. In this section an algorithm (algo.2) is de-

signed to meet all three criteria concurrently, to the

price of reasonnable increase in complexity due to

systematic test added. We scan anti-diagonals same

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

1: Optimal zigzag algorithm

begin

foreach anti-diagonal, maximum to miminum

global satisfaction do

foreach diagonal, maximum to miminum

sex equality in alternate directions do

foreach pair (m, w) do

if m et w are free then

Marry m with w

end

as before. In each cell, all pairs are accepted for mar-

riage if their components are free. After all cells have

been considered, the table is then revisited up to com-

plete removal of blocking situations as follows: po-

tential blocking pairs are matched upon detection (test

according to ﬁgure 3) while both blocked couples are

broken and complementary elements are freed. To

overcome cycles in the assignment the number of res-

canning is limited to the population size. Scan di-

rections together with questionning all previous mar-

riages on demand guarantees the better at end. Global

satisfaction and sex equality are comparable to for-

mer ones (ﬁgure 8(d) and 9(d)) if not even better. The

main improvement is matching stability now obtained

in a great majority of cases, with the number of un-

stabilities signiﬁcantly lowered otherwise (see ﬁgure

7(d)). More precisely, the number of blocking pairs

is null until 50 man-or-woman large populations. It

is still 0 in an average 96% of the 200 instances for

populations larger than 50, and its maximum ranges

in the 15 blocking pairs for 100. However, the algo-

rithm complexity is in the order O(n

In ﬁgure 10, the algorithm performances are com-

pared wrt. population size. The improvement β from

GS to BZ increases with the population size. Again,

beyond 100 large populations ZZ becomes 100 per-

cent better than GS for both global satisfaction and

sex equality.

4 APPLICATIONS

For sake of illustration we outline here three im-

age processing applications in stereovision, registra-

tion and motion analysis. Matching relies on level-

lines. Features as simple as junctions or sequences

of line segments are extracted from each image and

then selected into primitives. Each primitive is given

its preference list containing primitives of the other

image (see for instance ﬁgure 11). The preference

list is incomplete and sorted by features similarity

(e.g. contrast, length, relative orientation, relative

2: Blocked zigzag algorithm

begin

while there is a bloking pair and rescan

number < population size do

foreach anti-diagonal, maximum to

miminum global satisfaction do

foreach diagonal, maximum to

miminum sex equality back and forth

foreach pair (m, w) do

if m and w are free then

Marry m with w

foreach anti-diagonal, maximum to

miminum global satisfaction do

foreach diagonal, maximum to

miminum sex equality back and forth

foreach pair (m, w) do

if (m, w) is blocking pair

then

Free m and w and their

spouse

Marry m with w

end

position etc.). Blocked zigzag is then run. Figure

12(a)(b)(c)(d) show the original stereo images and the

features in them respectively. The result of feature

matching by BZ shows as an optical ﬂow in ﬁgure

12(e). And the ﬁgure 12(f) supports comparison with

GS . Results are quite comparable to the naked eye

due to the ”incomplete list” nature of the implemen-

tation.

The same process can do for motion. Figures 13

show the image sequence and the matching result re-

spectively.

5 CONCLUSION

In this paper, we proposed and evaluated ”stable mar-

riages” algorithms based on a novel representation,

called marriage table. Its deﬁnition and properties are

presented. Algorithms follow different scan styles of

the latter table, deﬁning result properties accordingly.

We introduce three different ones to progressively

meet three criteria: global satisfaction, sex equality

and stability. The three criteria together (quasi) satis-

ﬁed change complexity from O(n

) to O(n

). Match-

A NOVEL REPRESENTATION AND ALGORITHMS FOR (QUASI) STABLE MARRIAGES

Figure 8: Comparison of global satisfaction between the

methods: (a) GS man-optimal (b) GS woman-optimal (c)

Optimal zigzag and (d) Blocked zigzag, with 200 instances

of 150 large populations.

Figure 9: Comparison of sex equality between the meth-

ods: (a) GS man-optimal (b) GS woman-optimal (c) Opti-

mal zigzag and (d) Blocked zigzag

ing results obtained are systematically and experi-

mentally compared to GS

′

s. Some application re-

sults in motion analysis, registration and stereovision

are also given just for illustration.

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Figure 10: Performance comparison between GS and BZ

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(a) (b)

(e) (f)

(g)

Figure 11: Stable marriages for MEMS images registration

in electron beam microscopy, (a) 1

ﬁeld part scaned, (b)

ﬁeld part scaned to be super imposed into a larger ﬁeld.

Let us underline the VLSI implementation artefact : this

defect will eventually support the perfect match between

(a) and (b), despiste the cumb ambiquity from periodicity,

(c)(d) the primitives extracted from (a) and (b) with primi-

tives from the defect underlined in the frame, (e) primitives

in the defect that supports the perfect piecing, (f) potential

mates of the primitive underlined in bold in (e) with their

rank, (g) ﬁnal matching results by BZ : the are in perfect

agreement with the known reality

(a) (b)

(e) (f)

(g)

Figure 12: Stable marriages matching for stereovision,

(a)(b) stereo images, (c)(d) features extracted from (a)(b)

respectively, (e) matching results by BZ, (f) matching re-

sults by GS, (g) ﬁnal matching results by BZ

A NOVEL REPRESENTATION AND ALGORITHMS FOR (QUASI) STABLE MARRIAGES

(a)

(b)

(c)

(d)

Figure 13: (a)(b)(c) Image sequence for motion detection,

(d) Matching result by using the stable marriages algorithm.

ICINCO 2005 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION