"Almost stable" matchings in the Roommates problem

Abstract

An instance of the classical Stable Roommates problem (SR) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an SR instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within n^{\frac{1}{2}-\varepsilon}, for any \varepsilon>0, unless P=NP. If the preference lists contain ties, we improve this result to n^{1-\varepsilon}. Also, we show that, given an integer K and an SR instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an SR instance I.

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Abraham, D.J. and Biro, P. and Manlove, D.F. (2006) "Almost stable"
matchings in the Roommates problem. In, Fleischer, R. and Trippen, G.,
Eds. Proceedings of WAOA 2005: the 3rd International Workshop on
Approximation and Online Algorithms, 6-7 October, 2005 Lecture Notes
in Computer Science Vol 3879, pages pp. 1-14, Palma, Mallorca.




http://eprints.gla.ac.uk/archive/00002816/




Glasgow ePrints Service
http://eprints.gla.ac.uk

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“Almost Stable” Matchings in
the Roommates Problem
David J. Abraham1,?, P´ eter Bir´ o2,??and David F. Manlove3,? ? ?
1Computer Science Department, Carnegie-Mellon University, USA.
2Department of Algebra, and Department of Computer Science and Information
Theory, Budapest University of Technology and Economics, Hungary.
3Department of Computing Science, University of Glasgow, UK.
Abstract. An instance of the classical Stable Roommates problem (sr)
need not admit a stable matching. This motivates the problem of finding
a matching that is “as stable as possible”, i.e. admits the fewest number
of blocking pairs. In this paper we prove that, given an sr instance with n
agents, in which all preference lists are complete, the problem of finding
a matching with the fewest number of blocking pairs is NP-hard and not
approximable within n
lists contain ties, we improve this result to n1−ε. Also, we show that,
given an integer K and an sr instance I in which all preference lists are
complete, the problem of deciding whether I admits a matching with
exactly K blocking pairs is NP-complete. By contrast, if K is constant,
we give a polynomial-time algorithm that finds a matching with at most
(or exactly) K blocking pairs, or reports that no such matching exists.
Finally, we give upper and lower bounds for the minimum number of
blocking pairs over all matchings in terms of some properties of a stable
partition, given an sr instance I.
1
2−ε, for any ε > 0, unless P=NP. If the preference
1Introduction
The Stable Roommates problem (sr) is a classical combinatorial problem that
has been studied extensively in the literature [3,9,7,4,15,8]. An instance I of sr
contains an undirected graph G = (A,E) where A = {a1,...,an} and m = |E|.
We assume that G contains no isolated vertices. We interchangeably refer to
the vertices of G as the agents, and we refer to G as the underlying graph of I.
The vertices adjacent to a given agent ai∈ A are the acceptable agents for ai,
denoted by Ai. If aj ∈ Ai, we say that ai finds aj acceptable. (Note that the
?Part of this work was done whilst at Department of Computing Science, University
of Glasgow, and Max-Planck-Institut f¨ ur Informatik. Email: dabraham@cs.cmu.edu.
??Partially supported by the Center for Applied Mathematics and Computational
Physics, and by the Hungarian National Science Fund (grant OTKA F 037301).
Email: pbiro@cs.bme.hu.
? ? ?Supported by the Engineering and Physical Sciences Research Council (grant
GR/R84597/01), and by Royal Society of Edinburgh/Scottish Executive Personal
Research Fellowship. Email: davidm@dcs.gla.ac.uk.

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acceptability relation is symmetric, i.e. aj∈ Aiif and only if ai∈ Aj.) Moreover
we assume that in I, ai has a linear order ≺aiover Ai, which we refer to as
ai’s preference list. If aj≺aiak, we say that aiprefers ajto ak. Given aj∈ Ai,
define rankai(aj) = 1 + |{ak∈ Ai: ak≺aiaj}|.
Let M be a matching in I. If {ai,aj} ∈ M, we say that aiis matched in M
and M(ai) denotes aj, otherwise ai is unmatched in M. A blocking pair with
respect to M is an edge {ai,aj} ∈ E\M such that (i) either aiis unmatched in
M, or aiis matched in M and prefers ajto M(ai), and (ii) either ajis unmatched
in M, or ajis matched in M and prefers aito M(aj). Let bpI(M) denote the set
of blocking pairs with respect to M in I (we omit the subscript if the instance
is clear from the context). Matching M is stable in I if bpI(M) = ∅.
Gale and Shapley [3] showed that an instance of sr need not admit a stable
matching (see for example the sr instance Irin Figure 1 where r = 1). Irving [7]
gave an O(m) algorithm that finds a stable matching or reports that none exists,
given an instance I of sr. The algorithm in [7] assumes that in I, all preference
lists are complete (i.e. Ai= A\{ai} for each ai∈ A) and n is even, though it is
straightforward to generalise the algorithm to the problem model defined here
(i.e. the case of incomplete lists) [4]. Henceforth we denote by src the special
case of sr in which all preference lists are complete.
As the problem name suggests, an application of sr arises in the context of
campus accommodation allocation, where we seek to assign students to share
two-person rooms, based on their preferences over one another. Another appli-
cation occurs in the context of forming pairings of players for chess tournaments
[10]. Very recently, a more serious application of sr has been studied, involving
pairwise kidney exchange between incompatible patient-donor pairs [14]. Here,
preference lists can be constructed on the basis of compatibility profiles between
patients and potential donors.
Empirical results [12] suggest that, as n increases, the probability that a
random sr instance with n agents admits a stable matching decreases steeply.
Equivalently, as n grows large, these results suggest that an arbitrary matching in
a random sr instance with n agents is likely to admit at least one blocking pair.
In practical situations, a blocking pair {ai,aj} of a given matching M need not
always lead to M being undermined by aiand aj, since these agents might not
realise that together they block M. For example, in situations where preference
lists are not public knowledge, there may be limited channels of communication
that would lead to the awareness of blocking pairs in practice. Nevertheless, it
is reasonable to assert that the greater the number of blocking pairs of a given
matching M, the greater the likelihood that M would be undermined by a pair
of agents in practice. Hence, given an sr instance that does not admit a stable
matching, one may regard a matching that admits 1 blocking pair as being
“more stable” than a matching that admits 10 blocking pairs, for example. This
motivates the problem of finding, given an sr instance I with no stable matching,
a matching in I that admits the fewest number of blocking pairs [11,2]. Such a
matching is, in the sense described here, “as stable as possible”.
Given an sr instance I, define bp(I) = min{|bpI(M)| : M is a matching in I}.
Define min-bp-sr to be the problem of finding, given an sr instance I, a match-

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ing M in I such that bp(M) = bp(I). (Note that, if I is an src instance where n
is even, clearly M must be a perfect matching in I.) In Section 2, we show that
min-bp-sr is NP-hard and very difficult to approximate. In particular we show
that min-bp-sr is not approximable within n
The result holds even for complete preference lists.
We also consider the variant srt of sr in which preference lists may include
ties. Ties arise naturally in practical applications: for example in the kidney
exchange context, two donors may be equally compatible for a given patient.
We also denote by srtc the special case of srt in which all preference lists
are complete. The definition of a blocking pair in the srt and srtc cases is
identical to that given for sr (however the term “prefers” in the sr definition
is interpreted as “strictly prefers” in the presence of ties). (Note that in [8],
stable matchings in srt and srtc are referred to as weakly stable matchings,
where three stability definitions are given; however weak stability is the more
commonly-studied notion in the literature.) Clearly an instance of srtc need
not admit a stable matching. Moreover it is known [13,8] that the problem of
deciding whether a stable matching exists, given an instance of srtc, is NP-
complete. Let min-bp-srt denote the variant of min-bp-sr in which preference
lists may include ties. In Section 2, we show that min-bp-srt is not approximable
within n1−ε, for any ε > 0, unless P=NP. The result holds even if all preference
lists are complete, there is at most one tie per list, and each tie has length 2.
We now remark on the format of the inapproximability results that we present
for min-bp-sr and min-bp-srt. We implicitly assume that a given instance I of
the former problem is unsolvable, so that bp(I) ≥ 1. Recall that the solvability or
otherwise of I can be determined in O(m) time [7,4]. Hence bp(I) can be regarded
as the objective function for measuring performance guarantee. On the other
hand, given an instance I of min-bp-srt, we do not assume that I is unsolvable,
since the problem of deciding whether this is the case is NP-complete [13,8].
Hence possibly bp(I) = 0, and therefore we use opt(I) to measure performance
guarantee, where opt(I) = 1 + bp(I). In fact our inapproximability result for
min-bp-srt shows that, given any ε > 0, it is NP-hard to distinguish between
the cases that I admits a stable matching, and bp(I) ≥ n1−ε.
We also consider the case that we require a matching to admit exactly K
blocking pairs. Define exact-bp-sr to be the problem of deciding, given an
sr instance I and an integer K, whether I admits a matching M such that
bp(M) = K. In Section 2 we show that exact-bp-sr is NP-complete (even
for complete preference lists). However by contrast, in Section 3, we prove that
exact-bp-sr is solvable in polynomial time if K is a constant. In particular we
give an O(mK+1) algorithm that takes as input an sr instance I and a constant
integer K, and finds a matching M in I such that bp(M) = K, or reports that no
such matching exists. We show how to adapt this algorithm to find a matching
M in I such that bp(M) ≤ K, or report that no such matching exists.
We next give a remark regarding related work. An alternative method has
been considered in the literature for coping with instances of sr that do not
admit a stable matching. Tan [16] defined a stable partition in a given instance I
of sr, which is a generalisation of the concept of a stable matching in I. Following
1
2−ε, for any ε > 0, unless P=NP.

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a4i+1 : a4i+2 a4i+3 a4i+4
a4i+2 : a4i+3 a4i+1 a4i+4
a4i+3 : a4i+1 a4i+2 a4i+4
a4i+4 : a4i+1 a4i+2 a4i+3
(0 ≤ i ≤ r − 1)
M1
M2
r= {{a4i+1,a4i+2} : 0 ≤ i ≤ r − 1}
r= M1
r∪ {{a4i+3,a4i+4} : 0 ≤ i ≤ r − 1}
Fig.1. Instance Ir of sr and two matchings M1
r,M2
rin Ir.
[12], a stable partition is a permutation Π of A satisfying the following two
properties (which implicitly assume that if ai is a fixed point of Π then ai is
appended to his own preference list):
(i) for each ai∈ A, aidoes not prefer Π−1(ai) to Π(ai);
(ii) if aiprefers ajto Π−1(ai) then ajdoes not prefer aito Π−1(aj).
Tan [16] showed that every instance I of sr admits a stable partition, and he also
gave an O(n2) algorithm for finding such a structure in I. Moreover, starting
from a stable partition, Tan [17] showed how to construct, also in O(n2) time,
a largest matching M in I with the property that the matched pairs in M are
stable within themselves. However such a matching may only be half the size of
a maximum (cardinality) matching in I. Yet in many applications we seek to
match as many agents as possible, and as discussed above, in order to satisfy this
property, in many cases a certain number of blocking pairs may be tolerated.
For example, suppose that r ≥ 1 and consider the sr instance Irand example
matchings M1
insoluble src instances with 4 agents, Tan’s algorithm is bound to construct
a matching M in Ir of size r (such as M1
|bpIr(M)| ≥ 2r. However M2
and |bpIr(M2
many blocking pairs.
In Section 4, for a given sr instance I, we give upper and lower bounds for
bp(I) in terms of some properties of a stable partition in I.
r,M2
ras shown in Figure 1. Since Ir is built up from r copies of
r). Any such matching M satisfies
ris a solution to min-bp-sr in Ir, where |M2
r)| = r. In particular M1
r| = 2r
ris half the size of M2
rand admits twice as
2 Inapproximability of min-bp-sr and min-bp-srt
In this section we present reductions showing the NP-hardness and inapprox-
imability of each of min-bp-sr and min-bp-srt. Define min-mm (respectively
exact-mm) to be the problem of deciding, given a graph G and integer K,
whether G admits a maximal matching of size at most (respectively exactly)
K. Our reductions utilise the NP-completeness of exact-mm in cubic graphs,
which we now establish.
Lemma 1. exact-mm is NP-complete, even for cubic graphs.
Proof. Clearly exact-mm belongs to NP. To show NP-hardness, we reduce from
min-mm, which is NP-complete even for cubic graphs [6]. Let G (a cubic graph)