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Strong Stability in the Hospitals/Residents Problem

Strong Stability in the Hospitals/Residents Problem. Robert W. Irving, David F. Manlove and Sandy Scott. University of Glasgow Department of Computing Science Supported by EPSRC grant GR/R84597/01 and Nuffield Foundation Award NUF-NAL-02. Hospitals/Residents problem (HR): Motivation.

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Strong Stability in the Hospitals/Residents Problem

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  1. Strong Stability in the Hospitals/Residents Problem Robert W. Irving,David F. Manloveand Sandy Scott University of Glasgow Department of Computing Science Supported by EPSRC grant GR/R84597/01 and Nuffield Foundation Award NUF-NAL-02

  2. Hospitals/Residents problem(HR): Motivation • Graduating medical students or residents seek hospital appointments • Centralised matching schemes are in operation • Schemes produce stable matchingsof residents to hospitals • National Resident Matching Program (US) • other large-scale matching schemes, both educational and vocational BCTCS 19

  3. Hospitals/Residents problem(HR): Definition • a set H of hospitals,a setRofresidents • each resident r ranks a subset of H in strict order of preference • each hospital h has ph posts, and ranks in strict order those residents who have ranked it • a matching M is a subset of the acceptable pairs of RH such that |{h: (r,h) M}|  1 for all r and |{r: (r,h) M}| ph for all h BCTCS 19

  4. r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 h1 h3 r6: h3 h1:3: r2 r1 r3 r5 h2:2: r3 r2 r1 r4 r5 h3:1: r4 r5 r1 r3 r6 An instance of HR BCTCS 19

  5. r1: h2 h3 h1 r2: h2h1 r3: h3 h2h1 r4: h2 h3 r5: h2 h1 h3 r6: h3 h1:3: r2 r1r3 r5 h2:2: r3 r2r1 r4r5 h3:1: r4 r5 r1 r3r6 A matching in HR BCTCS 19

  6. Indifference in the ranking • ties: h1 : r7 (r1 r3) r5 • version of HR with ties is HRT • more general form of indifference involves partial orders • version of HR with partial orders is HRP BCTCS 19

  7. r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6 An instance of HRT BCTCS 19

  8. r1: (h2 h3) h1 r2: h2h1 r3: h3 h2h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1r3) r5 h2:2: r3 r2 (r1 r4r5) h3:1: (r4 r5) (r1 r3) r6 A matching in HRT BCTCS 19

  9. r1: (h2 h3) h1 r2: h2h1 r3: h3 h2h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1r3) r5 h2:2: r3 r2 (r1r4r5) h3:1: (r4 r5) (r1 r3) r6 A blocking pair r4 and h2 form a blocking pair BCTCS 19

  10. Stability • a matching M is stable unless there is an acceptable pair (r,h) M such that, if they joined together • both would be better off (weak stability) • neither would be worse off (super-stability) • one would be better off and the other no worse off (strong stability) • such a pair constitutes a blocking pair • hereafter consider only strong stability BCTCS 19

  11. r1: (h2h3) h1 r2: h2h1 r3: h3 h2h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1r3) r5 h2:2: r3 r2 (r1 r4r5) h3:1: (r4 r5) (r1 r3) r6 Another blocking pair r1 and h3 form a blocking pair BCTCS 19

  12. r1: (h2 h3) h1 r2: h2 h1 r3: h3h2 h1 r4: h2h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6 A strongly stable matching BCTCS 19

  13. State of the art for HRT / HRP • weak stability: • weakly stable matching always exists • efficient algorithm (Gale and Shapley (AMM, 1962), Gusfield and Irving (MIT Press, 1989)) • matchings may vary in size (Manlove et al. (TCS, 2002)) • many NP-hard problems, including finding largest weakly stable matching (Iwama et al. (ICALP, 1999), Manlove et al. (TCS, 2002)) BCTCS 19

  14. State of the art for HRT / HRP • super-stability • super-stable matching may or may not exist • efficient algorithm (Irving, Manlove and Scott (SWAT, 2000)) • strong stability • strongly stable matching may or may not exist • efficient algorithm for HRT • in contrast, problem is NP-complete in HRP (Irving, Manlove and Scott (STACS, 2003)) BCTCS 19

  15. The algorithm in brief repeat provisionally assign all free residents to hospitals at head of list form reduced provisional assignment graph find critical set of residents and make corresponding deletions until critical set is empty form a feasible matching check if feasible matching is strongly stable BCTCS 19

  16. Properties of the algorithm • algorithm has complexity O(a2), where a is the number of acceptable pairs • bounded below by complexity of finding a perfect matching in a bipartite graph • matching produced by the algorithm is resident-optimal • same set of residents matched and posts filled in every strongly stable matching BCTCS 19

  17. Strong stability in HRP • HRP under strong stability is NP-complete • even if all hospitals have just one post, and every pair is acceptable • reduction from RESTRICTED 3-SAT: • Boolean formula B in CNF where each variable v occurs in exactly two clauses as literal v, and exactly two clauses as literal ~v BCTCS 19

  18. Open problems • find a weakly stable matching with minimum number of strongly stable blocking pairs • size of strongly stable matchings relative to possible sizes of weakly stable matchings • hospital-oriented algorithm BCTCS 19

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