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1 Almost stable matchings in the Roommates problem David Abraham Computer Science Department Carnegie-Mellon University Péter Biró Department of Computer Science and Information Theory Budapest University of Technology and Economics David Manlove University of Glasgow Department of Computing Science Supported by EPSRC grant GR/R84597/01, RSE / Scottish Exec Personal Research Fellowship

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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

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

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

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5 Example SR Instance (2) Example SR instance I 2 : 1: 2 3 4 2: 3 1 4 3: 1 2 4 4: 1 2 3

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6 Example SR Instance (2) Example SR instance I 2 : 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 I 2 has no stable matching. 1: 2 3 4 2: 3 1 4 3: 1 2 4 4: 1 2 3 1: 2 3 4 2: 3 1 4 3: 1 2 4 4: 1 2 3

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7 Application: kidney exchange d1d1 p1p1

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8 d1d1 p1p1 d2d2 p2p2

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9 d1d1 p1p1 d2d2 p2p2 A. Roth, T. Sönmez, U. Ünver, Pairwise Kidney Exchange, Journal of Economic Theory, to appear

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10 Application: kidney exchange d1d1 p1p1 d2d2 p2p2 A. Roth, T. Sönmez, U. Ünver, Pairwise Kidney Exchange, Journal of Economic Theory, to appear (d 1, p 1 ) (d 2, p 2 ) Create a vertex for each donor- patient pair Edges represent compatibility Preference lists can take into account degrees of compatibility

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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, 6:577-595 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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12 Empirical results Instance size % soluble Experiments based on taking average of s 1, s 2, s 3 where s j is number of soluble instances among 10,000 randomly generated instances, each of given size 2n Results due to Colin Sng Instance size 4205010020050010002000400060008000 % soluble 96.382.973.164.355.145.138.832.227.825.023.6

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13 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 Abraham, Biró, DFM, 2005

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14 Almost stable matchings The following instance I 3 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 Let bp(M) denote the number of blocking pairs of matching M Stable partition { 1,2,3, 4,5,6 } bp(M 2 )=12 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 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 bp(M 1 )=2

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15 Algorithmic results Let I be an SR instance Define bp(I)=min{bp(M) : M is a matching in I} Problem of determining bp(I) is NP-hard Given an integer n, problem of deciding whether bp(I)=n is NP-complete Given a fixed integer K 0, problem of deciding whether bp(I)=K is solvable in O(n 2(K+1) ) time Problem of finding bp(I) is not approximable within n ½-, for any > 0, unless P=NP Abraham, Biró, DFM, 2005

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16 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 DFM, Journal of Algorithms, 2002 Problem of finding bp(I) is NP-hard Follows immediately from no-ties case Given a fixed integer K 0, problem of deciding whether bp(I)=K is NP-complete Problem of finding bp(I) is not approximable within n 1-, for any > 0, unless P=NP Abraham, Biró, DFM, 2005

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17 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 M 1, M 2 such that bp(M 1 )=k and bp(M 2 )=k+2, is there a matching M 3 in I such that bp(M 3 )=k+1 ? 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 I 3 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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18 Open problems 1.Is there an approximation algorithm for the problem of determining bp(I) that has performance guarantee o(n 2 ) ? Trivial upper bound is 2n(n-1) 2.Determine values of k n and obtain a characterisation of I n such that I n is an SR instance with 2n agents in which bp(I n )=k n and k n = max{bp(I) : I is an SR instance with 2n agents}

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