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“Almost stable” matchings in the Roommates problem

David AbrahamComputer Science DepartmentCarnegie-Mellon University, USA

Péter Biró Department of Computer Science and Information TheoryBudapest University of Technology and Economics, Hungary

David ManloveDepartment of Computing ScienceUniversity of Glasgow, UK

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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 orderOutput: 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

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Example SR Instance (1)

Example SR instance I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

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Example SR Instance (1)

Example SR instance I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

Stable matching in I1: 1: 3 2 42: 4 3 13: 2 1 44: 1 3 2

The matching is not stable as {1,3} blocks.

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Example SR Instance (2)

Example SR instance I2: 1: 2 3 42: 3 1 43: 1 2 44: 1 2 3

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Example SR Instance (2)

Example SR instance I2: 1: 2 3 42: 3 1 43: 1 2 44: 1 2 3

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.

1: 2 3 42: 3 1 43: 1 2 44: 1 2 3

1: 2 3 42: 3 1 43: 1 2 44: 1 2 3

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Application: kidney exchange

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Application: kidney exchange

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Application: kidney exchange

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A. Roth, T. Sönmez, U. Ünver, Pairwise Kidney Exchange, Journal of Economic Theory, to

appear

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Application: kidney exchange

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d2

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A. Roth, T. Sönmez, U. Ünver, Pairwise Kidney Exchange, Journal of Economic Theory, to

appear

(d1 , p1) (d2 , p2)

• Create a vertex for each donor-patient pair

• Edges represent compatibility• Preference lists can take into

account degrees of compatibility

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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, 6:577-595

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”

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Empirical results

Instance size

% s

olu

ble

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

Instancesize

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

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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 5 4 3 62: 3 6 5 1 43: 4 5 2 6 14: 2 6 5 1 35: 6 2 3 4 16: 3 1 4 5 2

• Example: insoluble SR instance

• {1,3,6}, {2,4,5} is a partition of the agents

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• 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 iC

1: 2 5 4 3 62: 3 6 5 1 43: 4 5 2 6 14: 2 6 5 1 35: 6 2 3 4 16: 3 1 4 5 2

Coalition Formation Games: B-preferences

• {1,3,6}, {2,4,5} is a partition of the agents

• {1,2,3} is a blocking coalition

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• A partition P is B-stable if it admits no blocking coalition

1: 2 5 4 3 62: 3 6 5 1 43: 4 5 2 6 14: 2 6 5 1 35: 6 2 3 4 16: 3 1 4 5 2

Coalition Formation Games: B-stability

• Every CFG instance admits a B-stable partition

• Such a partition may be found in linear time

• Also results for W-stability

• {1,5,6}, {2,3,4} is a B-stable partition of the agents

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Coping with insoluble SR instances: 2. Stable partitions

• Stable partition

Tan 1991 (Journal of Algorithms) Generalisation of a stable matching Permutation of the agents such that:

1. each agent i does not prefer -1(i) to (i)2. if i prefers j to -1(i) then j does not prefer i to -1(j)

1: 2 5 4 3 62: 3 6 5 1 43: 4 5 2 6 14: 2 6 5 1 35: 6 2 3 4 16: 3 1 4 5 2

• Example:=1,4,6, 2,3,5

(i) shown in green -1(i) shown in red

E.g. 4 prefers 5 to -1(4)=1 5 prefers -1(5)=3 to 4

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• 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)

Properties of stable partitions

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• Given a stable partition • Delete a single agent from each odd cycle• Decompose each even cycles into pairs

Maximum matching where the matched pairs are stable within themselves

1: 2 5 4 3 62: 3 6 5 1 43: 4 5 2 6 14: 2 6 5 1 35: 6 2 3 4 16: 3 1 4 5 2

• Example

=1,4,6, 2,3,5

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• Given a stable partition • Delete a single agent from each odd cycle• Decompose each even cycles into pairs

Maximum matching where the matched pairs are stable within themselves

1: 2 5 4 3 62: 3 6 5 1 43: 4 5 2 6 14: 2 6 5 1 35: 6 2 3 4 16: 3 1 4 5 2

• Example

=1,4,6, 2,5

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• Given a stable partition • Delete a single agent from each odd cycle• Decompose each even cycles into pairs

Maximum matching where the matched pairs are stable within themselves

1: 2 5 4 3 62: 3 6 5 1 43: 4 5 2 6 14: 2 6 5 1 35: 6 2 3 4 16: 3 1 4 5 2

• Example

=1,4, 2,5

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• Given a stable partition • Delete a single agent from each odd cycle• Decompose each even cycles into pairs

Maximum matching where the matched pairs are stable within themselves

1: 2 5 4 3 62: 5 1 44: 2 5 1 35: 2 4 1

• Example

• 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

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• The following instance I3 of SR is insoluble1: 2 3 5 6 42: 3 1 6 4 53: 1 2 4 5 64: 5 6 2 3 15: 6 4 3 1 26: 4 5 1 2 3

• Let bp(M) denote the set of blocking pairs of matching M

|bp(M2)|=12

1: 2 3 5 6 42: 3 1 6 4 53: 1 2 4 5 64: 5 6 2 3 15: 6 4 3 1 26: 4 5 1 2 3

1: 2 3 5 6 42: 3 1 6 4 53: 1 2 4 5 64: 5 6 2 3 15: 6 4 3 1 26: 4 5 1 2 3

|bp(M1)|=2

• Stable partition 1,2,3, 4,5,6

Coping with insoluble SR instances:3. “Almost stable” matchings

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1: 2 3 42: 3 1 43: 1 2 44: 1 2 3

Coping with insoluble SR instances:Method 2 vs method 3

• 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 42: 3 1 43: 1 2 44: 1 2 3

• Method 3: consider e.g. M2={{1,2},{3,4}}

• |bp(M2)|=1

1: 2 3 42: 3 1 43: 1 2 44: 1 2 3

• So |M2|=2|M1| and |bp(M2)|=½|bp(M1)|

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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

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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-

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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

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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:1. For each subset B of agent pairs {ai, aj} where |B|=K2. 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

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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

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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 5 6 42: 3 1 6 4 53: 1 2 4 5 64: 5 6 2 3 15: 6 4 3 1 26: 4 5 1 2 3

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

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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

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Open problems

1. Is there an approximation algorithm for MIN-BP-SR that has performance guarantee o(n2)?

An upper bound is 2C(n-1)

2. Are the upper/lower bounds for bp(I) tight?

3. 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}

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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 037301)• EPSRC (grant GR/R84597/01)• RSE / Scottish Executive Personal Research Fellowship• RSE International Exchange Programme

More informationD.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

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