Every finite distributive lattice is a set of stable matchings for a small stable marriage instance

Adaptivity to changing environments and constraints is key to success in modern society. We address this by proposing “incrementalized versions” of Stable Marriage and Stable Roommates. That is, we try to answer the following question: for both problems, what is the computational cost of adapting an existing stable matching after some of the preferences of the agents have changed. While doing so, we also model the constraint that the new stable matching shall be not too different from the old one. After formalizing these incremental versions, we provide a fairly comprehensive picture of the computational complexity landscape of Incremental Stable Marriage and Incremental Stable Roommates. To this end, we exploit the parameters “degree of change” both in the input (difference between old and new preference profile) and in the output (difference between old and new stable matching). We obtain both hardness and tractability results, in particular showing a fixed-parameter tractability result with respect to the parameter “distance between old and new stable matching”.

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The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

The Electronic Journal of Combinatorics ◽

10.37236/5980 ◽

2016 ◽

Vol 23 (4) ◽

Cited By ~ 1

Author(s):

Samuel Braunfeld

Keyword(s):

Distributive Lattice ◽

Equivalence Relations ◽

Electronic Journal ◽

Finite Distributive Lattice ◽

Linear Orders ◽

Finite Dimensional ◽

Homogeneous Structures ◽

Special Case

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2.

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Linear and nonlinear inequalities concerning a certain combinatorial sequence which arises from counting the number of chains of a finite distributive lattice

Discrete Applied Mathematics ◽

10.1016/0166-218x(91)90084-a ◽

1991 ◽

Vol 34 (1-3) ◽

pp. 145-150 ◽

Cited By ~ 1

Author(s):

Takayuki Hibi

Keyword(s):

Distributive Lattice ◽

Finite Distributive Lattice ◽

Linear And Nonlinear

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Stable Marriage Problem Mapped to Lotka-volterra Model: Stable Equilibria Mapped from Stable Matchings

Procedia Computer Science ◽

10.1016/j.procs.2014.08.189 ◽

2014 ◽

Vol 35 ◽

pp. 1398-1406 ◽

Cited By ~ 1

Author(s):

Yoshikazu Hata ◽

Yoshiteru Ishida

Keyword(s):

Volterra Model ◽

Stable Marriage ◽

Stable Marriage Problem ◽

Stable Matchings ◽

Marriage Problem ◽

Stable Equilibria

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Congrunce lattices of rectangular lattices1

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-191220 ◽

2020 ◽

Vol 39 (3) ◽

pp. 2831-2843

Author(s):

Peng He ◽

Xue-Ping Wang

Keyword(s):

Distributive Lattice ◽

Upper Bound ◽

Congruence Lattice ◽

Semimodular Lattice ◽

Rectangular Lattice ◽

Finite Distributive Lattice ◽

Irreducible Elements

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp.

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