A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings

Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on $f(n)$, the maximum number of stable matchings that a stable matching instance with $n$ men and $n$ women can have. It has been a long-standing open problem to understand the asymptotic behavior of $f(n)$ as $n\to\infty$, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately $2.28^n$, and the best upper bound was $2^{n\log n- O(n)}$. In this paper, we show that for all $n$, $f(n) \leq c^n$ for some universal constant $c$. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call "mixing". The latter might be of independent interest

安定マッチングの最大数と安定結婚グラフについて

福井大学教育地域科学部紀要（自然科学　数学編） , 4, 201

Gale-Shapleyev algoritam, varijacije i primjene

U ovom diplomskom radu opisuje se problem stabilnih brakova te se definira njegovo rješenje – stabilno sparivanje, kao i ostala njegova svojstva. Obrađuje se i algoritam koji rješava problem stabilnih brakova – Gale Shapleyev algoritam. Govori se o njegovoj složenosti i svojstvima. Promatramo i poopćenja osnovnog problema stabilnih brakova. Dajemo i definiciju stroge stabilnosti te navodimo algoritam koji pronalazi strogo stabilno rješenje, ukoliko ono postoji. Govorimo o problemu bolnice–specijalizanti, kao i o daljnjim poopćenjima problema stabilnih brakova.In this master thesis, we describe the stable marriage problem and we define its solution – the stable matching. We also define its properties. We elaborate an algorithm that solves the stable marriage problem – the Gale–Shapley algorithm. We talk about its complexity and its properties. We consider generalizations of the stable marriage problem, and we define the strong stability. We also give an algorithm which finds a strongly stable matching, if one exists. We present an insight into the hospital–residents problem, as well as into some further generalizations of the problem

Gale-Shapleyev algoritam, varijacije i primjene

U ovom diplomskom radu opisuje se problem stabilnih brakova te se definira njegovo rješenje – stabilno sparivanje, kao i ostala njegova svojstva. Obrađuje se i algoritam koji rješava problem stabilnih brakova – Gale Shapleyev algoritam. Govori se o njegovoj složenosti i svojstvima. Promatramo i poopćenja osnovnog problema stabilnih brakova. Dajemo i definiciju stroge stabilnosti te navodimo algoritam koji pronalazi strogo stabilno rješenje, ukoliko ono postoji. Govorimo o problemu bolnice–specijalizanti, kao i o daljnjim poopćenjima problema stabilnih brakova.In this master thesis, we describe the stable marriage problem and we define its solution – the stable matching. We also define its properties. We elaborate an algorithm that solves the stable marriage problem – the Gale–Shapley algorithm. We talk about its complexity and its properties. We consider generalizations of the stable marriage problem, and we define the strong stability. We also give an algorithm which finds a strongly stable matching, if one exists. We present an insight into the hospital–residents problem, as well as into some further generalizations of the problem

Gale-Shapleyev algoritam, varijacije i primjene

U ovom diplomskom radu opisuje se problem stabilnih brakova te se definira njegovo rješenje – stabilno sparivanje, kao i ostala njegova svojstva. Obrađuje se i algoritam koji rješava problem stabilnih brakova – Gale Shapleyev algoritam. Govori se o njegovoj složenosti i svojstvima. Promatramo i poopćenja osnovnog problema stabilnih brakova. Dajemo i definiciju stroge stabilnosti te navodimo algoritam koji pronalazi strogo stabilno rješenje, ukoliko ono postoji. Govorimo o problemu bolnice–specijalizanti, kao i o daljnjim poopćenjima problema stabilnih brakova.In this master thesis, we describe the stable marriage problem and we define its solution – the stable matching. We also define its properties. We elaborate an algorithm that solves the stable marriage problem – the Gale–Shapley algorithm. We talk about its complexity and its properties. We consider generalizations of the stable marriage problem, and we define the strong stability. We also give an algorithm which finds a strongly stable matching, if one exists. We present an insight into the hospital–residents problem, as well as into some further generalizations of the problem

Concerning the maximum number of stable matchings in the stable marriage problem

The function, f(n), represents the maximum number of stable matchings possible in an instance of size n of the stable marriage problem. It is shown that f(n) is a strictly increasing function of n, and a result of Knuth\u27s concerning the exponential growth of this function is generalized to apply to all positive integers, n. A method for constructing ranking matrices is used to produce instances with many stable matchings. A subproblem of the stable marriage problem developed by Eilers (Irvine Compiler Corporation Technical Report, ICC TR1999-2, 1999), called the pseudo-Latin marriage problem, plays a significant role as a tool and as motivation in the paper

Concerning the maximum number of stable matchings in the stable marriage problem

The function, f(n), represents the maximum number of stable matchings possible in an instance of size n of the stable marriage problem. It is shown that f(n) is a strictly increasing function of n, and a result of Knuth\u27s concerning the exponential growth of this function is generalized to apply to all positive integers, n. A method for constructing ranking matrices is used to produce instances with many stable matchings. A subproblem of the stable marriage problem developed by Eilers (Irvine Compiler Corporation Technical Report, ICC TR1999-2, 1999), called the pseudo-Latin marriage problem, plays a significant role as a tool and as motivation in the paper