[Paper Review] A Generalization of Birkhoff's Theorem for Distributive Lattices, with Applications to Robust Stable Matchings.

Tung Mai, Vijay V. Vazirani|arXiv (Cornell University)|Apr 16, 2018

Game Theory and Voting Systems5 citations

TL;DR

This paper generalizes Birkhoff's theorem for distributive lattices by introducing the concept of compression on partial orders, enabling efficient characterization of sublattices in stable matching problems. It presents algorithms to compute robust stable matchings under preference list perturbations, including fully robust matchings that remain stable across multiple perturbed instances.

ABSTRACT

Birkhoff's theorem, which has also been called {\em the fundamental theorem for finite distributive lattices}, states that the elements of any such lattice $\cal L$ are isomorphic to the closed sets of a partial order, say $\Pi$. We generalize this theorem to showing that each sublattice of $\cal L$ is isomorphic to a distinct partial order that can be obtained from $\Pi$ via the operation of {\em compression}, defined in this paper. Let $A$ be an instance of stable matching, with $\cal L$ 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 $\mathcal{M}_A$ and $\mathcal{M}_B$ be their sets of stable matchings. Our results are the following: - We show that $\mathcal{M}_A \cap \mathcal{M}_B$ is a sublattice of $\mathcal{L}$ and $\mathcal{M}_{A} \setminus \mathcal{M}_{B}$ is a semi-sublattice of $\mathcal{L}$. - Using our generalization of Birkhoff's Theorem, we give an efficient algorithm for finding the compression of $\Pi$ that is isomorphic to the lattice of $\mathcal{M}_A \cap \mathcal{M}_B$. - 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 {\em fully robust stable matching}. - If yes, the set of all such matchings forms a sublattice of $\cal L$ 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.

Motivation & Objective

• To extend Birkhoff's fundamental theorem for finite distributive lattices to sublattices via a new compression operation on partial orders.
• To analyze the structure of stable matchings when one agent's preference list is permuted, showing that the intersection of stable matchings forms a sublattice.
• To develop an efficient algorithm for computing the partial order corresponding to the sublattice of stable matchings common to original and perturbed instances.
• To determine whether a fully robust stable matching exists across a polynomial-sized set of perturbed preference lists, and to compute such matchings efficiently.

Proposed method

• Introduce the operation of compression on a partial order Π to derive a new partial order isomorphic to any sublattice of the original distributive lattice.
• Prove that the set of stable matchings common to an original instance A and a perturbed instance B (where one agent's preference list is permuted) forms a sublattice of the original lattice L.
• Use the generalized Birkhoff theorem to compute the compressed partial order corresponding to the sublattice M_A ∩ M_B efficiently.
• Design a polynomial-time algorithm to check for the existence of a fully robust stable matching across a given polynomial-sized domain of perturbed instances.
• Show that if such a matching exists, the set of all fully robust matchings forms a sublattice, and compute its generating partial order.
• Leverage the computed partial order to find a fully robust matching that optimizes a given weight function (maximizes or minimizes it).

Experimental results

Research questions

• RQ1 How can Birkhoff's theorem be generalized to characterize sublattices of a distributive lattice using a novel compression operation on partial orders?
• RQ2 What is the lattice-theoretic structure of the intersection of stable matchings between an original stable matching instance and one with a single permuted preference list?
• RQ3 Can an efficient algorithm be constructed to compute the partial order corresponding to the sublattice of common stable matchings after a preference list perturbation?
• RQ4 Given a polynomial-sized set of preference list perturbations, is there an efficient way to determine whether a fully robust stable matching exists across all resulting instances?
• RQ5 If such a matching exists, can we efficiently compute the full sublattice of all fully robust stable matchings and optimize over it?

Key findings

• The intersection of stable matchings M_A ∩ M_B between an original instance A and a perturbed instance B (with one agent's preference list permuted) forms a sublattice of the original lattice L.
• The set M_A ∖ M_B forms a semi-sublattice of L, indicating a structured decomposition of the lattice under perturbation.
• An efficient algorithm exists to compute the compressed partial order that is isomorphic to the sublattice M_A ∩ M_B, using the generalized Birkhoff theorem.
• For any polynomial-sized domain D of preference list perturbations, a fully robust stable matching (stable across all perturbed instances) can be determined efficiently.
• If a fully robust stable matching exists, the set of all such matchings forms a sublattice of L, and its generating partial order can be computed efficiently.
• The computed partial order enables optimization over the set of fully robust stable matchings to find one that maximizes or minimizes a given weight function.