1. “Almost stable” matchings in the Roommates problem
David Abraham
Computer Science Department
Carnegie-Mellon University, USA
Péter Biró
Department of Computer Science and Information Theory
Budapest University of Technology and Economics, Hungary
David Manlove Department of Computing Science University of Glasgow, UK

2. Stable Roommates Problem (SR)
D Gale and L Shapley, “College Admissions and the Stability of Marriage”, American Mathematical Monthly, 1962
Input: 2n agents; each agent ranks all 2n-1 other agents in
strict order
Output: a stable matching
A matching is a set of n disjoint pairs of agents
A blocking pair of a matching M is a pair of agents {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. Example SR Instance (1) Example SR instance I1: 1: 3 2 4 2: 4 3 12: 3: 4:

4. Example SR Instance (1) Example SR instance I1: 1: 3 2 4 2: 4 3 1
The matching is not stable as {1,3} blocks.
Example SR instance I1: 1: 2: 3: 4:
Stable matching in I1: 1: 2: 3: 4:

5. Example SR Instance (2) Example SR instance I2: 1: 2 3 4 2: 3 1 42: 3: 4:

6. Example SR Instance (2) Example SR instance I2: 1: 2 3 4 2: 3 1 42: 3: 4: 1: 2: 3: 4: 1: 2: 3: 4:
The three matchings containing the pairs {1,2}, {1,3}, {1,4} are blocked by the pairs {2,3}, {1,2}, {1,3} respectively.
 instance I2 has no stable matching.

7. Application: kidney exchange

8. Application: kidney exchange

9. Application: kidney exchange
A. Roth, T. Sönmez, U. Ünver,
Pairwise Kidney Exchange,
Journal of Economic Theory, to appear p2 p1

10. Application: kidney exchange
A. Roth, T. Sönmez, U. Ünver,
Pairwise Kidney Exchange,
Journal of Economic Theory, to appear p2 p1
Create a vertex for each donor- patient pair
Edges represent compatibility
Preference lists can take into
account degrees of compatibility
(d1 , p1)
(d2 , p2)

11. 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:
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 the Stable Marriage problem
Phase 2: elimination of “rotations”

12. Empirical results
Instance
size 4 20 50
100
200
500
1000
2000
4000
6000
8000 %
soluble
96.3
82.9
73.1
64.3
55.1
45.1
38.8
32.2
27.8
25.0
23.6
Experiments based on taking average of s1 , s2 , s3 where sj is number of soluble instances among 10,000 randomly generated instances, each of given size 2n
Results due to Colin Sng
% soluble
Instance size

13. Coping with insoluble SR instances: 1. Coalition Formation Games
Generalisation of SR
Partition agents into sets of size 1
Notions of B-preferences / W-preferences
Cechlárová and Hajduková, 1999
Cechlárová and Romero-Medina, 2001
Cechlárová and Hajduková, 2002 1: 2: 3: 4: 5: 6:
Example:
insoluble SR instance
{1,3,6}, {2,4,5} is a
partition of the agents

14. Coalition Formation Games: B-preferences
Given a partition P of the agents, define P(i) to be the set
containing agent i
Given a set of agents S, define best(i,S) to be the best
agent in S according to i ’s preferences
A set of agents C is said to be a blocking coalition of P if i
prefers best(i,C) to best(i,P(i)) for all iC
{1,3,6}, {2,4,5} is a partition
of the agents
{1,2,3} is a blocking coalition 1: 2: 3: 4: 5: 6:

15. Coalition Formation Games: B-stability
A partition P is B-stable if it admits no blocking coalition
{1,5,6}, {2,3,4} is a B-stable partition of the agents 1: 2: 3: 4: 5: 6:
Every CFG instance admits a B-stable partition
Such a partition may be found in linear time
Also results for W-stability

16. Coping with insoluble SR instances: 2. Stable partitions
Tan 1991 (Journal of Algorithms)
Generalisation of a stable matching
Permutation  of the agents such that:
each agent i does not prefer  -1(i) to (i)
if i prefers j to -1(i) then j does not prefer i to  -1(j)
Example:
=1,4,6, 2,3,5
(i) shown in green
 -1(i) shown in red 1: 2: 3: 4: 5: 6:
E.g. 4 prefers 5 to  -1(4)=1
5 prefers  -1(5)=3 to 4

17. Properties of stable partitions
Every SR instance admits a stable partition
A stable partition can be found in linear time
There may be more than one stable partition, but:
any two stable partitions contain exactly the same odd cycles
An SR instance I is insoluble if and only if a stable partition in I contains an odd cycle
Any stable partition containing only even cycles can be decomposed into a stable matching
Tan 1991 (Journal of Algorithms)
A stable partition can be used to find a maximum matching such that the matched pairs are stable within themselves
Tan 1991 (International Journal of Computer Mathematics)

18. Maximum matching where the matched pairs are stable within themselves
Given a stable partition 
Delete a single agent from each odd cycle
Decompose each even cycles into pairs 1: 2: 3: 4: 5: 6:
Example
=1,4,6, 2,3,5

19. Maximum matching where the matched pairs are stable within themselves
Given a stable partition 
Delete a single agent from each odd cycle
Decompose each even cycles into pairs
Example 1: 2: 3: 4: 5: 6:
=1,4,6, 2,5

20. Maximum matching where the matched pairs are stable within themselves
Given a stable partition 
Delete a single agent from each odd cycle
Decompose each even cycles into pairs
Example 1: 2: 3: 4: 5: 6:
=1,4, 2,5

21. Maximum matching where the matched pairs are stable within themselves
Given a stable partition 
Delete a single agent from each odd cycle
Decompose each even cycles into pairs
Example 1: 2: 4: 5:
Matching M={{1,4},{2,5}} is stable in the smaller instance
A maximum matching such that the matched pairs are stable within themselves can be constructed in linear time from a stable partition

22. Coping with insoluble SR instances: 3. “Almost stable” matchings
The following instance I3 of SR is insoluble 1: 2: 3: 4: 5: 6:
Stable partition
1,2,3, 4,5,6
Let bp(M) denote the set of blocking pairs of matching M 1: 2: 3: 4: 5: 6:
|bp(M2)|=12 1: 2: 3: 4: 5: 6:
|bp(M1)|=2

23. Coping with insoluble SR instances: Method 2 vs method 31: 2: 3: 4:
Recall the insoluble instance I2 of SR:
Tan’s algorithm for method 2 constructs a matching of size 1, e.g. M1={{1,2}}
Any such matching M has |bp(M)|2 1: 2: 3: 4:
Method 3: consider e.g. M2={{1,2},{3,4}}
|bp(M2)|=1 1: 2: 3: 4:
So |M2|=2|M1| and |bp(M2)|=½|bp(M1)|

24. Hardness results for SR
Let I be an SR instance
Define bp(I)=min{|bp(M)|: M is a matching in I}
Define MIN-BP-SR to be problem of computing bp(I), given an SR instance I
Theorem 1: MIN-BP-SR is not approximable within n½-, for any  > 0, unless P=NP
Define EXACT-BP-SR to be problem of deciding whether I admits a matching M such that |bp(M)|=K, given an integer K
Theorem 2: EXACT-BP-SR is NP-complete

25. Outline of the proof Using a “gap introducing” reduction from EXACT-MM
Given a cubic graph G=(V,E) and an integer K, decide whether G admits a maximal matching of size K
EXACT-MM is NP-complete, by transformation from MIN-MM (Minimization version), which is NP-complete for cubic graphs
Horton and Kilakos, 1993
Create an instance I of SR with n agents
If G admits a maximal matching of size K then I admits a matching with p blocking pairs, where p=|V|
If G admits no maximal matching of size K, then bp(I) > p n½-

26. Preference lists with ties
Let I be an instance of SR with ties
Problem of deciding whether I admits a stable matching is NP-complete
Ronn, Journal of Algorithms, 1990
Irving and Manlove, Journal of Algorithms, 2002
Can define MIN-BP-SRT analogously to MIN-BP-SR
Theorem 3: MIN-BP-SRT is not approximable within n1-, for any  > 0, unless P=NP, even if each tie has length 2 and there is at most one tie per list
Define EXACT-BP-SRT analogously to EXACT-BP-SR
Theorem 4: EXACT-BP-SRT is NP-complete for each fixed K  0

27. Polynomial-time algorithm
Theorem 5: EXACT-BP-SR is solvable in polynomial time if K is fixed
Algorithm also works for possibly incomplete preference lists
Given an SR instance I where m is the total length of the preference lists, O(mK+1) algorithm finds a matching M where |bp(M)|=K or reports that none exists
Idea:
For each subset B of agent pairs {ai, aj} where |B|=K
Try to construct a matching M in I such that bp(M)=B
Step 1: O(mK) subsets to consider
Step 2: O(m) time

28. Outline of the algorithm
Let {ai, aj}B where |B|=K
Preference list of ai : … … ak … … aj … …
If {ai, aj}bp(M) then {ai, ak}M
Delete {ai, ak} but must not introduce new blocking pairs
Preference list of ak : … … ai … … aj … …
If {ai, ak} B then {aj, ak}M
Delete {ak, aj} and mark ak
Check whether there is a stable matching M in reduced SR instance such that all marked agents are matched in M

29. Interpolation of |bp(M)|
Clearly |bp(M)|½(2n)(2n-2)=2n(n-1) for any matching M
Is |bp(M)| an interpolating invariant? That is, given an SR instance I, if I admits matchings M1, M2 such that |bp(M1) |=k and |bp(M2)|=k+2, is there a matching M3 in I such that |bp(M3)|=k+1 ?
Not in general! 1: 2: 3: 4: 5: 6:
Instance I3 admits 15 matchings:
9 admit 2 blocking pairs
2 admit 6 blocking pairs
3 admit 8 blocking pairs
1 admits 12 blocking pairs

30. Upper and lower bounds for bp(I)
As already observed, bp(I)2n(n-1)
Let  be a stable partition in an SR instance I
Let C denote the number of odd cycles of length 3 in 
Upper bound: bp(I)  2C(n-1)
Lower bound: bp(I)  C/2

31. kn = max{bp(I) : I is an SR instance with 2n agents}
Open problems
Is there an approximation algorithm for MIN-BP-SR that has performance guarantee o(n2)?
An upper bound is 2C(n-1)
Are the upper/lower bounds for bp(I) tight?
Determine values of kn and obtain a characterisation of In such that In is an SR instance with 2n agents in which bp(In)=kn and
kn = max{bp(I) : I is an SR instance with 2n agents}

32. Further details Acknowledgements - individuals Katarína Cechlárová
Rob Irving
Acknowledgements - funding
Centre for Applied Mathematics and Computational Physics, Budapest University of Technology and Economics
Hungarian National Science Fund (grant OTKA F )
EPSRC (grant GR/R84597/01)
RSE / Scottish Executive Personal Research Fellowship
RSE International Exchange Programme
More information
D.J. Abraham, P. Biró, D.F.M., “Almost stable” matchings in the
Roommates problem, to appear in Proceedings of WAOA ’05: the 3rd
Workshop on Approximation and Online Algorithms, Lecture Notes in
Computer Science, Springer-Verlag, 2005