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 ManloveDepartment of Computing
  ScienceUniversity of Glasgow, UK

Supported 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 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

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Example SR Instance (1)
Example SR instance I1 1 3 2 4 2 4 3
1 3 2 1 4 4 1 3 2
4
Example SR Instance (1)
Example SR instance I1 1 3 2 4 2 4 3
1 3 2 1 4 4 1 3 2
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Example SR Instance (2)
Example SR instance I2 1 2 3 4 2 3 1
4 3 1 2 4 4 1 2 3
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Example SR Instance (2)
Example SR instance I2 1 2 3 4 2 3 1
4 3 1 2 4 4 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.
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Application kidney exchange
d1
p1
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Application kidney exchange
d1
d2
p2
p1
9
Application kidney exchange
d1
d2
A. Roth, T. Sönmez, U. Ünver, Pairwise Kidney
Exchange, Journal of Economic Theory, to appear
p2
p1
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Application kidney exchange
d1
d2
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

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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,
  6577-595
• 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 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
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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 insoluble

1 2 3 5 6 4 2 3 1 6 4 5 3 1 2 4
5 6 4 5 6 2 3 1 5 6 4 3 1 2 6 4 5
1 2 3

• Stable partition
• ?1,2,3?, ?4,5,6?

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

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Hardness results for SR

• Let I be an SR instance
• Define bp(I)minbp(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 ? gt 0, unless PNP
• 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
  pV
• If G admits no maximal matching of size K, then
  bp(I) gt 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 ? gt 0, unless PNP, 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(mK1) 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
  BK
• 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 BK
• 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)k2, is
  there a matching M3 in I such that bp(M3)k1 ?
• Not in general!

1 2 3 5 6 4 2 3 1 6 4 5 3 1 2 4
5 6 4 5 6 2 3 1 5 6 4 3 1 2 6 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

• Is there an approximation algorithm for MIN-BP-SR
  that has performance guarantee o(n2)?
• Trivial upper bound is 2n(n-1)
• 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 maxbp(I) I is an SR instance with 2n
  agents