Title:
1Almost 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
2Stable 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
3Example SR Instance (1)
Example SR instance I1 1 3 2 4 2 4 3
1 3 2 1 4 4 1 3 2
4Example SR Instance (1)
Example SR instance I1 1 3 2 4 2 4 3
1 3 2 1 4 4 1 3 2
5Example SR Instance (2)
Example SR instance I2 1 2 3 4 2 3 1
4 3 1 2 4 4 1 2 3
6Example 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.
7Application kidney exchange
d1
p1
8Application kidney exchange
d1
d2
p2
p1
9Application kidney exchange
d1
d2
A. Roth, T. Sönmez, U. Ünver, Pairwise Kidney
Exchange, Journal of Economic Theory, to appear
p2
p1
10Application 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
11Efficient 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
12Empirical 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
13Coping 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
14Almost 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
15Hardness 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
16Outline 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-?
17Preference 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
18Polynomial-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
19Outline 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
20Interpolation 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
21Open 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