From One Stable Marriage to the Next: How Long Is the Way?

Abstract

The diameter of the stable matching (stable marriage) polytope is bounded from above by n 2 , where n is the number of men (or women) involved in the matching; this bound is attainable. Introduction.The stable matching (SM) problem (also known as the stable marriage problem [10]) asks for a matching from a set M of men to a set W of women that is stable under given preference lists, i.e., when men have preferences with regard to women, and vice versa. In 1962, Gale and Shapley introduced the problem in the setting of college admissions [10] and opened a new research chapter in the fields of combinatorics, mathematical economics, and social behavior [15,12,20]. Its significance has been recognized in the faces of Roth and Shapley, who received the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design" [17]. The current work adds another page to this chapter by studying the diameter of the polytope P S defined as the convex hull of incidence vectors of stable matchings.The polytope P S bears a wider interest from a combinatorial-theoretic point of view since, as shown in [4,13], every finite distributive lattice is a set of stable matchings. The polytope was initially investigated in [25], where a description in terms of linear inequalities is given. This description is simplified and extended in [21]. Further work in the same context includes [19,2,1,23,24,8]. In [18], the faces of P S are characterized in terms of the lattice of stable matchings with the use of a directed line graph, namely, the marriage graph Γ (see [18, Definition 6]). A minimal linear description of P S in terms of its minimal equation system as well as all the classes of facets is given in [7].The diameter of P S (diam(P S )) has been examined in neither of the abovementioned papers nor (to the best of our knowledge) in any other work appearing in the literature. We provide an upper bound on diam(P S ) equal to min{|M|,|W |}

On the diameter of the polytope of the stable marriage with ties

“…In this paper, we give an upper bound of ⌊ n 3 ⌋ on the diameter of the polytope of the stable marriage problem with ties (which we call P SMT ), and give a family of instances for which our bound holds tight. Our result generalizes what is known for P SM , meaning that if all preference orderings are strict, then it recovers the bound given in [8]. However, it relies on different and new ingredients, which we are going to describe next.…”

“…diameter results for matchings, TSP, or network flows and transportation in [2,7,11,21,24,4,6,3,5,26]). For the stable marriage polytope, which we call P SM here, Eirinakis et al [8] proved a diameter upper bound of ⌊n/4⌋, where n := |M ∪ W |. The authors also show the existence of instances for which this bound holds tight.…”

“…A key tool used in [8] to bound the diameter of P SM is the so-called stable marriage graph, introduced in [17]. The stable marriage graph is an auxiliary graph that one can construct (in polynomial time) for a given instance of the standard stable marriage problem.…”

The Diameter of the Fractional Matching Polytope and Its Hardness Implications

“…Furthermore, since K is a cycle, K also contains an even number of nodes that are witnesses in W. It follows that K contains at least two nodes in V (K) ∩ T ( ) that are not witnesses of any cycle, and therefore can pay for the move. This shows that invariant (12) is maintained. After the move is performed, the component K is no longer a component of the target graph, while all other components of the target graph remain the same.…”

“…Besides providing general bounds, many researchers in the past 50 years have given bounds and/or characterizations of the diameter of polytopes that correspond to the set of feasible solutions of classical combinatorial optimization problems. Just to mention a few, such problems include matching [3,8], TSP [19,20], edge cover [13], fractional stable set [18], network flow and transportation problems [2,5,6,7], stable marriage [12], and many more.…”