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“Almost stable” matchings in the Roommates problem

David AbrahamComputer Science Department

Carnegie-Mellon University, USA

Péter Biró Department of Computer Science and Information Theory

Budapest University of Technology and Economics, Hungary

David ManloveDepartment of Computing Science

University of Glasgow, UKSupported by EPSRC grant GR/R84597/01,RSE / Scottish Exec Personal Research Fellowship

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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 4

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

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

2: 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

d1

p1

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Application: kidney exchange

d1

p1

d2

p2

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Application: kidney exchange

d1

p1

d2

p2

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

appear

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Application: kidney exchange

d1

p1

d2

p2

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

olub

le

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

• Coalition formation games 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

• Stable partition Every SR instance admits such a structure Tan 1991 (Journal of Algorithms) Can be used to find a maximum matching such that the matched pairs are stable within themselves Tan 1991 (International Journal of Computer Mathematics)

• Matching with the fewest number of blocking pairs

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“Almost stable” matchings

• 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}

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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)• 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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Open problems

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

Trivial upper bound is 2n(n-1)

2. 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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