1 Overview of Graph Theory Addendum “The Stable Marriage Problem” Instructor: Carlos Pomalaza-Ráez Fall 2003 University of Oulu, Finland

2 The Stable Marriage Problem Problem: Given N men and N women, find a "suitable" matching between men and women The problem and the solution can be represented as a bipartite graph Raimo Saku Tarmo Urho Vesa Eeva Anne Miina Inka Katri MenWomen Edges represent all possible matching. Matching (M) means to select some of the edges according to some criteria. Perfect Matching: each man gets exactly one woman; each woman gets exactly one man A matching is unstable if there is a pair, e.g. (Raimo, Anne), who like each other more than their spouses; they can improve their situation by dumping spouses and eloping Gale-Shapley Theorem: A stable marriage always possible, and found in O(n 2 ) time.

3 Matchmaker, Matchmaker, Make Me A Match! Man1 st 2 nd 3 rd 4 th 5 th RaimoAnneEevaInkaKatriMiina SakuInkaAnneEevaMiinaKatri TarmoAnneKatriMiinaInkaEeva UrhoEevaInkaMiinaAnneKatri VesaAnneInkaEevaKatriMiina Woman 1 st 2 nd 3 rd 4 th 5 th EevaVesaRaimoSakuUrhoTarmo AnneTarmoSakuUrhoRaimoVesa MiinaSakuTarmoUrhoVesaRaimo InkaRaimoVesaUrhoTarmoSaku KatriUrhoSakuVesaTarmoRaimo Women’s Preference ListMen’s Preference List best worst The Gale-Shapley Algorithm  Each man lists women in order of preference from best to worst  Each woman lists men in order of preference  All people begin unengaged  While there are unengaged men, each proposes until a woman accept  Unengaged women accept 1 st proposal they get  If an engaged woman receives a proposal she likes better, she breaks old engagement and accepts new proposal; dumped man begins proposing where he left off

4 Gale-Shapley Algorithm Results of G-S algorithm are always stable  It requires two people of opposite sex in different couples to break up a marriage  If a man wants to leave for some woman, then he already proposed to her and she rejected him, so she won’t leave her husband for him Man1 st 2 nd 3 rd TarmoEevaAnneMiina UrhoAnneEevaMiina VesaEevaAnneMiina Woman 1 st 2 nd 3 rd EevaUrhoTarmoVesa AnneTarmoUrhoVesa MiinaTarmoUrhoVesa is a perfect matching but unstable because (Anne, Tarmo) prefer each other to current partners Man1 st 2 nd 3 rd TarmoEevaAnneMiina UrhoAnneEevaMiina VesaEevaAnneMiina Woman 1 st 2 nd 3 rd EevaUrhoTarmoVesa AnneTarmoUrhoVesa MiinaTarmoUrhoVesa is a perfect stable matching  When men propose we call it M 0  When woman propose we call it M 1

5 Gale-Shapley Algorithm Man1 st 2 nd 3 rd TarmoEevaAnneMiina UrhoAnneEevaMiina VesaEevaAnneMiina Woman 1 st 2 nd 3 rd EevaUrhoTarmoVesa AnneTarmoUrhoVesa MiinaTarmoUrhoVesa is a also perfect stable matching but now Eeva and Anne are better of than on the previous matching  Each man has the best partner he can have in any stable marriage  Each woman has the worst partner she can have in any stable marriage  G-S always produces same stable marriage - order of proposals is irrelevant What happens when men do the proposing?  Historically, men propose to women. Why it has to be that way?  Men: propose early and often  Women: ask out the guys Final observations Reference: The Stable Marriage Problem: Structure and Algorithms (Foundations of Computing) by Dan Gusfield and Robert Irving, MIT Press, 1989.