The GaleShapley Algorithm Gale Shapley 1962 Menpropose 1

The Gale-Shapley Algorithm [Gale & Shapley 1962] 　　　　(Men-propose） 1: a c b d e a: 2 1 × 3 2: c a e b d b: 2 1 4 ×b ×a ×c b ×c d e d c c: 1 2 3 d e a d: 3 1 4 2 5 b e a e: 4 3 1 2 5 3: 4: 5: 4 5 × 3 × 5 × 4 5 Theorem [Gale & Shapley 1962, Gusfield & Irving 1989] 　　The Gale-Shapley algorithm finds a stable matching.

証明 Suppose not stable. There is a blocking pair. × 2: ・ ・ ・　e　 ・ ・ ・ e: ・ ・ ・　2　・ ・ ・　 During the algorithm execution, “ 2” made a proposal to “e”, but he was rejected. At this moment, “e” had a partner better than “ 2”. × e: ・ ・ ・　2　・ ・ ・　 After that, she may change a partner, but never becomes worse. a contradiction (Q. E. D. )

安定結婚問題の例題の制限緩和 希望リストの制限 希望リストは全順序かつ完全。 2: c a e b d Original Stable Marriage problem (SM) 希望リストの制限緩和 (1) 同順位リスト 2: (c a) (e b) d Stable Marriage with Ties (SMT) (2) 不完全リスト 2: c a e Stable Marriage with Incomplete lists (SMI)

定理 [Gusfield & Irving 1989] 任意のSMT例題は、少なくとも1つ安定マッチングを持つ。 （証明） 1: a (c b d) e a: 2: c a e b d b: ( 2 3: b a (e 4: c b 5: c (d b) e d) c d (e 2 c: 1 3 4 5 1) 4 5 3 2 3 5 4 1 a) d: ( 3 1 4) (2 a e: 3 1 4 SMT　例題 5) 2 5 同順位を適当にばらす 1: a b c d e a: 2 1 3 4 5 2: c a e b d b: 1 2 4 5 3 3: b a e d c c: 1 2 3 5 4 4: c b d a e d: 3 1 4 2 5 5: c d b e a e: 4 3 1 2 5 SM　例題

Overview of the algorithm (Local Search) Mi M’i Procedure Increase M i+1 Procedure Stabilize Mi ：stable M’i ：not necessarily stable (but with a good property) ：stable M i+1 |M i | < |M’i | ≦ |M i+1 | m w blocking pair (m, w) m w m is single or w is single m w

Approximation ratio Mi M i+1 M’i Procedure Increase Never fails Procedure Stabilize OPT 1 Mi+1 + , we can obtain 　　　. |M | While |M i | < i 16 2 previously c log |M i |, and c√ |M i | The size of stable matching M finally obtained |M| ≧ OPT 2 1 + 16 |M| ≧ 8 15 OPT 15 Approximation ratio ≦ = 1. 875 8

Procedure Increase についてのみ説明する Stabilize は説明しない（前と一緒） - Input: Stable matching M i. - Output: Matching M’i. such that ・ |M’i | > |Mi | ・ There is no prohibited blocking pair (m, w) m w

Strategy in [Iwama et. al. 2004] Current matching Goal 　・Increase the matching size. 　・Prohibited blocking pairs may not arise. Mi × × Remove ｔ edges and add 2ｔ edges 　　→　Increase the size by t Remove prohibited blocking pairs, by removing edges. If # removed edges < t, then it is OK. Problems 　・There is no guarantee that every 2 t persons find a partner. 　・Many prohibited blocking pairs may arise. If we remove some pairs to resolve prohibited blocking pairs, the size may decrease. We construct a new matching so that the number of removed edges is at most t-1.

Strategy in [Iwama et. al. 2004] Good edges M opt Mi a good edge

Strategy in [Iwama et. al. 2004] Property of good edges M opt current partner OPT-partner Mi 1: … a … １ a a: … ( 1… 2 … ) 2: … b … a … ２ b b: … 2 … OPT-partner current partner 嬉しいこと 　　(1) Both 2 and a write a person who is currently single (b and 1, respectively). 　　(2) Both 2 and a are currently matched with a person at least as good as the partner in OPT (OPT-partner). 　　(3) For either 2 or a, OPT-partner and current partner are tied.