THE STUDY OF STABLE MARRIAGE PROBLEM WITH TIES AND INCOMPLETE BOUNDED LENGTH PREFERENCE LIST UNDER SOCIAL STABILITY

International Journal of Computer Science & Information Technology (IJCSIT) Vol 8, No 1, February 2016
DOI:10.5121/ijcsit.2016.8106 75
THE STUDY OF STABLE MARRIAGE PROBLEM WITH
TIES AND INCOMPLETE BOUNDED LENGTH
PREFERENCE LIST UNDER SOCIAL STABILITY
Ashish Shrivastava and C. Pandu Rangan
Department of Computer Science and Engineering, Indian Institute of Technology,
Madras, Chennai, India
ABSTRACT
We consider a variant of the Stable Marriage Problem where preference lists of man/woman may be
incomplete, may contain ties and may have bounded length in presence of a notion of social stability. In
real world matching applications like NRMP and Scottish medical matching scheme such restrictions can
arise very frequently where set of agents (man/woman) is very large and providing a complete and strict
order preference list is practically in-feasible. In presence of ties in preference lists, there exist three
different notion of stability, weak stability, strong stability and super stability. The most common solution is
to produce a weakly stable matching. It is a fact that in an instance of Stable Marriage problem with Ties
and Incomplete list (SMTI), weakly stable matching can have different sizes. This motivates the problem of
finding a maximum cardinality weakly socially stable matching.
In this paper, we find maximum size weakly socially stable matching for a special instance of Stable
Marriage problem with Ties and Incomplete bounded length preference list under Social Stability. The
motivation to consider this instance is the known fact, finding maximum size weakly socially stable
matching in any larger instance of this problem is NP-hard.
KEYWORDS
The Stable Marriage Problem, Socially Stable Matching, Bipartite Matching, Stable Marriage Problem
with Ties and Incomplete list, Two Sided Matching, Matching under Preference.
1. INTRODUCTION
The Stable marriage problem was first introduced by Gale and Shapley in 1962 [1]. The classical
instance I of the stable marriage problem has a set of n men U, a set of n women W and
preference lists of men over women and vice versa. Each preference list contains all members of
opposite sex in a strict order. A man mi and a woman wj are called acceptable to each other in I
instance I if mi is in preference list of wj and wj is in preference list of mi. Let α is the set of all
acceptable pairs in the instance I. A matching M is a set of independent pairs (mi, wj) such that mi
∈ U and wj ∈ W. If (mi, wj) ∈ M, we say that mi is matched to wj in M and vice versa and we
denote M (mi) = wj and M (wj) = mi.
A pair (mi,wj) ∉ M is called a blocking pair for matching M if both mi and wj prefer each other to
their partners in M. A matching M is called a stable matching iff there is no blocking pair with
respect to M. Gale and Shapley gave a deferred acceptance algorithm and proved that every

76
instance I of the stable marriage problem admits a stable matching which can be found in
polynomial time [1].
The largest and one of the best known applications of Hospitals Residents problem is National
Resident Matching Program (NRMP) and Scottish medical matching scheme which match
graduated medical students (residents) with their preferred hospitals on the basis of both side
preference lists.
The research work in the field of The Stable Marriage Problem has a long history. As we have
mentioned earlier, the first problem on stable marriage problem was introduced by Gale and
Shapley in 1962. After that lots of variation on first problem came into the picture. Some major
variations are Stable Marriage problem with Ties (SMT), Stable Marriage problem with
Incomplete list (SMI), Stable Marriage problem with Ties and Incomplete list (SMTI) and Stable
Marriage problem with Bounded length preference lists.
1.1 STABLE MARRIAGE PROBLEM WITH TIES (SMT)
In Stable Marriage problem with Ties, each man can give a preference list over a set of women,
where two or more women can hold the same place (ties) in the preference list and vice-versa. In
SMT there are three notion of stability: weak stability, strong stability and super stability [2, 3]. A
blocking pair (mi, wj) ∉ M with respect to a weakly stable matching M can be defined as follows:
(a) mi and wj are acceptable to each other. (b) mi strictly prefers wj to M(mi) (partner of mi in
matching M) (c) wj strictly prefers mi to M(wj). For an instance I of weakly stable matching
problem, a weakly stable matching M always exist and can be found in polynomial time [3].
A blocking pair (mi, wj) ∉ M with respect to a strongly stable matching M can be defined as
follows: (a) mi and wj are acceptable to each other. (b) mi strictly prefer wj to M(mi) and wj is
indifferent between mi and M(wj) and vice-versa.
A blocking pair (mi, wj) ∉ M with respect to a super stable matching M can be defined as follows:
(a) mi and wj are acceptable to each other. (b) both mi and wj either strictly prefer each other to
their partners M or indifferent between them. There could be an instance I that have neither super
nor strongly stable matching but there is an algorithm which can find super and strong stable
matching in I (if exist) in polynomial time [4]. Among these three stability notions, weak stability
has received most attention in the literature [5-12].
1.2 STABLE MARRIAGE PROBLEM WITH INCOMPLETE LISTS (SMI)
Stable Marriage with Incomplete list (SMI) is another variation of stable marriage problem in
which number of men and women in an instance I need not be same. Each man and woman can
give a preference list over a subset of opposite sex. For an instance I a pair (mi, wj) is called
blocking pair with respect to a matching M if: (a) mi and wj are acceptable to each other (b) mi is
either unmatched in M or prefer wj to M(mi) (c) wj is either unmatched in M or prefer mi to M(wj).
A matching M is called stable if there is no blocking pair with respect to M.
In an instance I of SMI we can partition the set of men and women such that, one partition have
those men and women which have partners in all stable matching and other partition have those
men and women which are unmatched in all stable matching [13].

77
1.3 STABLE MARRIAGE PROBLEM WITH TIES AND INCOMPLETE LISTS (SMTI)
Stable Marriage with Ties and Incomplete list (SMTI) is an extension of classical stable marriage
problem in which number of men and women in an instance I need not be same. Each man gives
a preference list over a subset of women and vice-versa. Each preference list may contain ties
(two or more men/women have same rank). A pair (mi, wj) ∉ M forms a blocking pair with
respect to matching M if (a) Both mi and wj are acceptable to each other and (b) mi is either
unmatched or strictly prefers wj to M(mi) and (c) wj is either unmatched or strictly prefers mi to
M(wj). A matching M is called a weakly stable matching if there is no blocking pair with respect
to M.
It is known that a weakly stable matching in an instance I of SMTI can have different sizes and
finding maximum cardinality weakly stable matching is an NP-hard problem [6]. NP-hardness
holds even if only one tie of size 2 occurs on men's preference list at the tail and women's
preference list contain no ties [6].
1.4 STABLE MARRIAGE PROBLEM WITH BOUNDED LENGTH PREFERENCE LISTS
The idea behind bounded length preference list is, in case of large scale matching problems, the
preference list of at-least one side of agent tend to be short. An example of large scale matching is
Scottish medical matching scheme [14] where each student is required to rank only three
hospitals in their preference list. This variation leads to a question, whether problem of finding
maximum size stable matching becomes simpler? (For an instance, with one side or both sided
bounded preference list).
Suppose (p, q)-MAX SMTI denotes such variation on MAX SMTI problem (finding maximum
size matching in an instance of SMTI) where each man can give at-most p women in his
preference list and each woman can give at-most q men in her preference list. Halldorsson et al.
[7] showed that (4, 7)-MAX SMTI is NP-hard and not approximable within some δ >1 unless P =
NP. A reduction from Minimum Vertex Cover to MAX SMTI, shows that later problem cannot
be approximable within 21/19 unless P = NP [9]. Another study in [15] uses NP-hard restriction
of minimum vertex cover of graph of minimum degree 3 in producing NP-hard result for (5, 5)-
MAX SMTI. Irving et al. [16] shows that (3, 4)-MAX SMTI is NP-hard and not approximable
within δ >1 unless P = NP.
1.5 THE HOSPITALS RESIDENTS PROBLEM
In the classical Hospitals Residents problem, agents are partitioned into the set of hospitals and
set of residents. Each resident gives preference to a subset of hospitals in strict order and vice-
versa. Each hospital hj has a non-negative capacity cj. A matching M is a set of resident hospital
pairs such that each resident ri is matched to at most one hospital and each hospital hj is matched
to at most cj residents. In a matching M, M(ri) denotes hospital assigned to ri and M(hj) denotes a
set of residents assigned to hj in M. A hospital hj is under-subscribed if M(hj) < cj and a resident ri
is free if he/she is not matched. A pair (ri, hj) ∉ M forms a blocking pair with respect to matching
M if (a) either ri is free or prefers hj to M(ri) and (b) either hj is under-subscribed or prefers ri to
one of its residents in M. A matching M is stable if there is no blocking pair with respect to
matching M.

78
The research work in the field of hospital resident problem has a long history. After the seminal
paper of Gale and Shapley [1] in 1962 lots of variation of this problem comes into the picture.
Some of them are The Hospital Resident problem with ties [2] and Hospital Resident problem
with couples [2].
2. RELATED WORK
Another variation of stable marriage problem is socially stable marriage problem. An instance I"
of socially stable marriage problem can be defined by (I, G) where I is an instance of classical
stable marriage problem and G = (U ∪ W, A) is a social network graph. Here U and W are set of
men and women respectively and A is set of man woman pair who knows each other in social
network G. Set A is called the set of acquainted pairs which is the subset of all acceptable pairs
(A ⊆ α). A marriage M is called socially stable marriage if there is no socially blocking pair with
respect to M. A socially blocking pair (mi, wj) ∉ M is defined as follows: (a) both mi and wj
prefers each other to their partner in M and (b) mi and wj are connected in social network G.
In large scale matching like NRMP and Scottish medical matching scheme, social stability is a
useful notion in which members of blocking pair block a matching M only if they know the
existence of each other. Thus the notion of social stability allows us to increase the cardinality of
matching without taking care of those pairs which are not socially connected in social network
graph.
The work in this paper is motivated by the work of Irving et al. [16] where they study about stable
marriage problem with ties and bounded length preference list. They show that if each man's list
is of length at most two and women's lists are of unbounded length with ties, we can find a
maximum size weakly stable matching in polynomial time.
Our work in this paper is also motivated by the work of Askaladis et al. [17] where they study
about socially stable matching problem with bounded length preference list. They gave a O(n3/2
log n) time algorithm for (2, ∞) - MAX SMISS problem. Where (2, ∞)-MAX SMISS problem is
to find a maximum size socially stable matching in an instance of stable marriage problem with
incomplete list under social stability, where each man's list is of length at most two (without ties)
and women's lists are of unbounded length (without ties).
3. OUR CONTRIBUTION
In an instance I of (2, ∞)-MAX SMISS problem if we include ties on both side preference lists,
where the length of a tie could be arbitrary, this instance converts into an instance I’ of (2, ∞)-
MAX SMTISS. In this paper we will show that we can find maximum size weakly socially stable
matching in instance I’ in polynomial time. Due to presence of ties in both side preference lists
there are three notion of stability: weak, strong and super. In this paper we are considering
maximum size weakly stable matching in I’ of (2, ∞)-MAX SMTISS.
As we mention earlier, Irving et al. [16] shows that (3, 4)-MAX SMTI is NP-hard and not
approximable within δ >1 unless P = NP. It follows that the complexity status of (3, ∞)-MAX
SMTI is also NP-hard. Similarly socially stable variation of (3, ∞)-MAX SMTI problem, “(3,
∞)-MAX Weakly SMTISS” is also NP-hard.
Given an instance I’ of (2, ∞)-MAX Weakly SMTISS (Stable Marriage problem with Ties and
Incomplete bounded length preference list under Social Stability), we present an algorithm that

79
gives a maximum size weakly socially stable matching with time complexity O(n3/2
log n), where
n is the total number of men and women in the instance I.
4. STABLE MARRIAGE PROBLEM WITH TIES AND INCOMPLETE BOUNDED
LIST UNDER SOCIAL STABILITY (SMTISS)
An instance of Stable Marriage Problem with Ties and Incomplete bounded list under Social
Stability (SMTISS) can be defined by (I, G) where I is the instance of SMTI and G = (U ∪ W, A),
where A (the set of all acceptable pairs). A man mi and a woman wj are called socially connected
to each other in graph G if (mi, wj) ∈ A. Each preference list is a partial order on a subset of
opposite sex. A matching M is called weakly socially stable if there is no socially blocking pair. A
pair (mi, wj) ∉ M is a socially blocking pair if (a) (mi, wj) ∈ A and (b) mi is either unmatched or
strictly prefers wj to his partner in M and (c) wj is either unmatched or strictly prefers mi to her
partner in M. In general, for any instance I of SMTISS problem, one of the aim is to compute a
maximum cardinality socially stable matching (weakly, strong, super etc). In an incomplete tied
preference list, arbitrary breaking of ties need not always lead to a maximum weakly socially
stable matching. The following example shows that if we break ties arbitrarily we can find
weakly socially stable matching of different sizes.
Example: Men’s preference lists Women’s preference lists
m1: (w1, w2) w1: m1, m2
m2: w1 w2: m1
In above example the underline shows a social connection in G. Here man m1 has a social
connection with woman w1. Observe that if we break the tie of m1 as m1 : w1, w2 then maximum
weakly socially stable matching will be {(m1, w1)} of size 1 and if we break tie of m1 as m1 : w2,
w1 then maximum weakly socially stable matching will be {(m1, w2), (m2, w1)} of size 2. The
above example motivates us to find maximum cardinality weakly socially stable matching in an
instance I of SMTISS.
Observe that if we restrict the length of all ties equal to 1 in an instance I of SMTISS then it will
reduce into an instance I"
of SMISS. Since it is known that finding a maximum cardinality
socially stable matching in an instance of SMISS is NP-complete [17], finding a maximum
cardinality weakly socially stable matching in an instance of SMTISS is also NP-complete.
Askalidis et al. showed that the problem (2, ∞)-MAX SMISS ((2, ∞)-MAX SMTISS with ties
length 1) is solvable in polynomial time [17], this result directed us to a more general version
called (2, ∞)-MAX Weakly SMTISS problem where ties length could be two or more. It may
seem that one can consider that if we break the ties arbitrary and apply (2, ∞)-Max SMISS
algorithm then we can find maximum cardinality weakly socially stable matching for (2, ∞)-
SMTISS instance, but this is not always true. We can verify this by above example.
4.1. ALGORITHM FOR (2, ∞)-MAX WEAKLY SMTISS
The objective of this problem is to find a maximum cardinality weakly socially stable matching in
SMTI instance under social stability, where each man can give a preference list of length at most
two and each woman can give unbounded length incomplete list, with or without ties of any
length. We present an O(n3/2
log n) time algorithm for this problem. Similar to (2, ∞)-MAX

80
SMISS given in [17], this algorithm also completes in three phases. In phase 1 we delete all pairs
which can never belong to any weakly socially stable matching. The intuition behind phase 1 is, if
there is a man mi who is socially connected to his first choice woman wj then any man who is less
preferable than mi in wj preference list, cannot match with wj in any socially stable matching. If it
happens, (mi, wj) will be blocking pair for resultant socially stable matching M.
In phase 2, first we build a graph from the reduced instance from phase 1 and weight each edge
(mi, wj) by rank(wj , mi), where rank(wj , mi) is rank of man mi in wj’s reduced preference list. Now
we construct a minimum weight maximum matching MG in graph. Finally, in phase 3 we settle
those pairs which are matched in phase 2 but will be socially blocking pair for output matching
M.
Lemma 4.1.1. (2, ∞)-MAX weakly SMTISS algorithm terminates.
Proof. We start phase 1 by unmarking all men. Now we mark those men who are unmarked and
have a non-empty reduced list. When every man becomes either marked or having an empty
reduced preference list, phase 1 will terminate. Since a man mi can be marked at most twice
during phase 1 and total number of men in instance (I, G) is finite, phase 1 will terminate. In
phase 2 of algorithm we are finding a minimum weight maximum matching of the reduced
instance, therefore phase 2 will also terminate. In phase 3, each iteration improves the choice of a
man from his second choice woman to his first choice woman and no man obtains worse woman
or becomes unmatched. Since total number of possible improvements for men is finite, therefore
the total number of iterations is also finite and hence phase 3 will also terminate.
Figure 1. (2, ∞)-MAX Weakly SMTISS Algorithm

81
Lemma 4.1.2. Phase 1 of (2, ∞)-MAX Weakly SMTISS Algorithm never deletes a weakly
socially stable pair.
Proof. Suppose (mi, wj) is a weakly socially stable pair which has been deleted during execution
of phase 1 of algorithm 1 such that (mi, wj) ∈ M, where M is a weakly socially stable matching in
(I, G). Suppose this is the first weakly stable pair deleted during phase 1. This deletion was done
because of wj being the first choice of some man mr where (mr, wj) ∈ A, mr's reduced list was not a
tie of length 2 and wj prefers mr to mi. But in that case pair (mr, wj) becomes a social blocking pair
with respect to matching M. This is a contradiction to the fact that M is a weakly socially stable
matching.
Lemma 4.1.3. The matching returned by algorithm (2, ∞)-MAX weakly SMTISS is weakly
socially stable in (I, G).
Proof. Suppose our algorithm outputs the matching M and this matching is not weakly socially
stable in (I, G). It means there exist a pair (mi, wj) which is a socially blocking pair with respect
to M. We can consider following four cases corresponding to a socially blocking pair.
Case (i) both mi and wj are unmatched in M:
We know once a man mi is matched in MG, he will never be unmatched in phase 3. Either mi
remains with his partner in MG or he can improve his partner (if possible) during phase 3.
Therefore, if a man mi is unmatched in M, he was unmatched in MG. Woman wj can either be
unmatched in MG or during phase 3. In first case, suppose a woman wj is unmatched in MG, then
we can increase the size of matching MG by adding the edge (mi, wj), which contradicts the fact
that matching MG is a maximum matching. In second case, suppose a woman wj becomes
unmatched during phase 3, then it means that her partner in MG, say mp1 , had a strict preference
list of length 2 and got his first choice woman, say wq1 , where (mp1, wq1 ) ∈ A. Now again we
have two cases for wq1. In the first case, woman wq1 is unmatched in MG. This leads to an
augmenting path {(mi, wj), (mp1, wj), (mp1, wq1)} in MG, where the first and the last edges are not in
MG. So we can increase the size of the matching MG by one. This is a contradiction to the fact that
MG is maximum matching. In second case, suppose wq1 becomes unmatched during phase 3, then
it means her partner in MG, say mp2, had a strict preference list of size 2 and got his first choice
woman wq2, where mp2 and wq2 had a social connection in (I, G). Again we can observe an
augmenting path {(mi, wj), (mp1, wj), (mp1, wq1) (mp2, wq1), (mp2, wq2)} in MG which contradicts the
fact that MG is maximum matching in (I, G). Similarly, if we keep on doing this operation,
number of men is finite and since every man strictly improves his partner in MG, there exist a
finite number of women who can become unmatched after phase 3. Hence at some time there
exists a man mpr, who is matched with wqr-1 in MG, and switched to his first choice wqr and wqr is
unmatched in MG. Here we can form an augmenting path {(mi, wj), (mp1, wj), (mp1, wq1), (mp2, wq1),
(mp2, wq2), …, (mpr, wqr)}, which leads to a contradiction that MG is a maximum matching.
Case (ii) mi is unmatched in M and wj prefers mi to M (wj):
As explained before mi is unmatched in MG. Woman wj is either matched to M(wj) in MG or
matched to some mp1 in MG and after that matched to M(wj) in phase 3. In first case, if wj is
matched to M(wj) in MG then it leads to contradiction that MG is a minimum weight maximum
matching. We can simply discard the edge (M(wj),wj) from MG and add edge (mi,wj), which
reduces the weight of MG without reducing its cardinality. In second case, wj is matched to some
mp1 in MG, where mp1 ≠ mi ≠ M(wj) and after that matched to M(wj) in phase 3. Now, after phase 3,
wj is not matched with mp1, which means that mp1 got his first choice woman say wq1 in phase 3.
Now woman wq1 is either free or matched to some man in MG. In both cases, using similar
arguments as in Case (i) we can construct an augmenting path and contradict that MG is a
maximum matching.

82
Case (iii) mi is matched to M(mi) in M, mi prefers wj to M(mi) and wj is unmatched in M:
We know that man mi has a strict preference list of size 2. It follows that wj is the first woman of
mi's preference list. Since (mi, wj) is an edge in social graph G, this satisfies the loop condition of
phase 3 and mi will be matched to wj during phase 3. Therefore this case will never occur after
execution of this algorithm.
Case (iv) mi is matched with wk = M(mi), and mi prefers wj to wk and wj is assigned to mr = M(wj)
and wj prefers mi to mr:
We know that length of preference list of mi is 2 and mi is in social connection with wj. mi strictly
prefers wj to wk, which means that wj is first choice of mi. Woman wj strictly prefers mi to mr.
Therefore the loop condition of phase 1 will be true and phase 1 will delete the pair (mr, wj).
Hence this case will never occur in our algorithm.
Since by lemma 1 we know that phase 1 of algorithm never deletes a socially stable pair, in phase
2 we constructed a minimum weight maximum matching MG from the reduced preference list by
phase 1 using algorithm in [18]. During phase 3 we never decrease the size of MG, which follows
that resultant matching M after phase 3 is a maximum cardinality matching. Lemma 3 ensures
that the matching produced by (2, ∞)-MAX weakly SMTISS algorithm is weakly socially stable.
It follows that the algorithm produced a maximum weakly socially stable matching in instance (I,
G). The running time complexity of the algorithm is dominated by phase 2 which constructs a
minimum weight maximum matching in G’
(V, E’
) in time O(√|V| |E’
| log|V|) [18]. Suppose |V| = n
= n1 + n2 is total number of men and women, then the cardinality of set of acceptable pairs is at
most 2n1 = O(n). It follows that the time complexity of (2, ∞)-MAX weakly SMTISS algorithm is
O(n3/2
log n).
Theorem 4.1. For a given instance (I, G) of (2, ∞)-MAX Weakly SMTISS, Algorithm (2, ∞)-
MAX weakly SMTISS produces a maximum size weakly socially stable matching in O(n3/2
log n)
time, where n is the total number of men and women in I.
5. CONCLUSION AND FUTURE WORK
In this paper we have presented an algorithm for an instance of stable marriage problem with ties
and incomplete bounded length preference list under social stability, where each man can give at
most 2 women in his preference list (with or without ties) and each woman can give unbounded
length preference list (with or without ties). Length of ties in women preference list could be 2 or
more. We have found that this instance can be solved in polynomial time and gave an algorithm
having complexity O(n3/2
log n) time, where n is the total number of men and women in I. These
instances are very common in real world scenario like NRMP and Scottish medical matching
scheme where medical students can give small size preference list. It would be interesting to
study about maximum size strongly stable matching and super stable matching in the scenario of
social stability. It would also be interesting to find a parameterized algorithm for a larger
instance. We are leaving these problems as open problems.
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AUTHORS
Ashish Shrivastava is a M.S.(By Research) student in Computer Science and Engineering
department in Indian Institute of Technology, Madras Chennai. His area of research is
Theoretical Computer Science.
C.Pandu Rangan is a Professor in Computer Science and Engineering department in Indian
Institute of Technology, Madras Chennai. His area of research is Theoretical Computer
Science, Cryptography.

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