Download presentation

Presentation is loading. Please wait.

Published byMaya Garcia Modified over 4 years ago

1
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 Manlove Department of Computing Science University of Glasgow, UK

2
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

3
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

4
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.

5
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

6
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

7
7 Application: kidney exchange d1d1 p1p1

8
8 d1d1 p1p1 d2d2 p2p2

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

10
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

11
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

12
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

13
13 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 6 2: 3 6 5 1 4 3: 4 5 2 6 1 4: 2 6 5 1 3 5: 6 2 3 4 1 6: 3 1 4 5 2 Example: insoluble SR instance {1,3,6}, {2,4,5} is a partition of the agents

14
14 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 i C 1: 2 5 4 3 6 2: 3 6 5 1 4 3: 4 5 2 6 1 4: 2 6 5 1 3 5: 6 2 3 4 1 6: 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

15
15 A partition P is B-stable if it admits no blocking coalition 1: 2 5 4 3 6 2: 3 6 5 1 4 3: 4 5 2 6 1 4: 2 6 5 1 3 5: 6 2 3 4 1 6: 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

16
16 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 6 2: 3 6 5 1 4 3: 4 5 2 6 1 4: 2 6 5 1 3 5: 6 2 3 4 1 6: 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

17
17 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

18
18 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 6 2: 3 6 5 1 4 3: 4 5 2 6 1 4: 2 6 5 1 3 5: 6 2 3 4 1 6: 3 1 4 5 2 Example = 1,4,6, 2,3,5

19
19 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 6 2: 3 6 5 1 4 3: 4 5 2 6 1 4: 2 6 5 1 3 5: 6 2 3 4 1 6: 3 1 4 5 2 Example = 1,4,6, 2,5

20
20 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 6 2: 3 6 5 1 4 3: 4 5 2 6 1 4: 2 6 5 1 3 5: 6 2 3 4 1 6: 3 1 4 5 2 Example = 1,4, 2,5

21
21 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 6 2: 5 1 4 4: 2 5 1 3 5: 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

22
22 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 set of blocking pairs of matching M |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 Stable partition 1,2,3, 4,5,6 Coping with insoluble SR instances: 3. Almost stable matchings

23
23 1: 2 3 4 2: 3 1 4 3: 1 2 4 4: 1 2 3 Coping with insoluble SR instances: Method 2 vs method 3 Recall the insoluble instance I 2 of SR: Tans algorithm for method 2 constructs a matching of size 1, e.g. M 1 ={{1,2}} Any such matching M has |bp(M)| 2 1: 2 3 4 2: 3 1 4 3: 1 2 4 4: 1 2 3 Method 3: consider e.g. M 2 ={{1,2},{3,4}} |bp(M 2 )|=1 1: 2 3 4 2: 3 1 4 3: 1 2 4 4: 1 2 3 So |M 2 |=2|M 1 | and |bp(M 2 )|=½|bp(M 1 )|

24
24 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 n ½-, 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

25
25 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 n ½-

26
26 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 n 1-, 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

27
27 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(m K+1 ) algorithm finds a matching M where |bp(M)|=K or reports that none exists Idea: 1. For each subset B of agent pairs {a i, a j } where |B|=K 2. Try to construct a matching M in I such that bp(M)=B Step 1: O(m K ) subsets to consider Step 2: O(m) time

28
28 Outline of the algorithm Let {a i, a j } B where |B|=K Preference list of a i : … … a k … … a j … … If {a i, a j } bp(M) then {a i, a k } M Delete {a i, a k } but must not introduce new blocking pairs Preference list of a k : … … a i … … a j … … If {a i, a k } B then {a j, a k } M Delete {a k, a j } and mark a k Check whether there is a stable matching M in reduced SR instance such that all marked agents are matched in M

29
29 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 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

30
30 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

31
31 Open problems 1.Is there an approximation algorithm for MIN-BP-SR that has performance guarantee o(n 2 ) ? An upper bound is 2C(n-1) 2.Are the upper/lower bounds for bp(I) tight? 3.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}

32
32 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 information D.J. Abraham, P. Biró, D.F.M., Almost stable matchings in the Roommates problem, to appear in Proceedings of WAOA 05: the 3 rd Workshop on Approximation and Online Algorithms, Lecture Notes in Computer Science, Springer-Verlag, 2005 Further details

Similar presentations

OK

1 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt Synthetic.

1 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt Synthetic.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on wind energy Ppt on non biodegradable waste wikipedia Free download ppt on railway reservation system Ppt on network theory definition Ppt on google docs Ppt on earth movements and major landforms in china Ppt on index numbers in mathematics Ppt on minimum wages act mumbai Ppt on carburetor shop Ppt on supreme court of india