A Generalization of Birkhoff's Theorem for Distributive Lattices, with Applications to Robust Stable Matchings
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Birkhoff's theorem, which has also been called the fundamental theorem for finite distributive lattices, states that the elements of any such lattice L are isomorphic to the closed sets of a partial order, say Π. We generalize this theorem to showing that each sublattice of L is isomorphic to a distinct partial order that can be obtained from Π via the operation of compression, defined in this paper. Let A be an instance of stable matching, with L being its lattice of stable matchings, and let B be the instance obtained by permuting the preference list of any one boy or any one girl. Let M_A and M_B be their sets of stable matchings. Our results are the following: - We show that M_A ∩M_B is a sublattice of L and M_A∖M_B is a semi-sublattice of L. - Using our generalization of Birkhoff's Theorem, we give an efficient algorithm for finding the compression of Π that is isomorphic to the lattice of M_A ∩M_B. - Given a polynomial sized domain D of such errors (of permuting one of the preference lists), we give an efficient algorithm that checks if there is a stable matching for A that is stable for each such resulting instance B. We call this a fully robust stable matching. - If yes, the set of all such matchings forms a sublattice of L and our algorithm finds its partial order as well. Using the latter, we can obtain a matching that optimizes (maximizes or minimizes) the weight among all fully robust stable matchings.
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