Lattice Linear Predicate Algorithms for the Constrained Stable Marriage Problem with Ties

TL;DR

This paper introduces lattice-linear predicate detection techniques to develop parallel and distributed algorithms for various constrained and tied variants of the stable marriage problem, demonstrating their generality and efficiency.

Contribution

It presents a unified lattice-linear predicate approach to solve constrained, super stable, and strongly stable marriage problems with ties, including asynchronous distributed algorithms.

Findings

Distributed algorithm for constrained stable marriage with O(n^2) messages

Parallel algorithms for stable marriage with ties without synchronization

Demonstrates the generality of lattice-linear predicate detection in stable matching problems

Abstract

We apply Lattice-Linear Predicate Detection Technique to derive parallel and distributed algorithms for various variants of the stable matching problem. These problems are: (a) the constrained stable marriage problem (b) the super stable marriage problem in presence of ties, and (c) the strongly stable marriage in presence of ties. All these problems are solved using the Lattice-Linear Predicate (LLP) algorithm showing its generality. The constrained stable marriage problem is a version of finding the stable marriage in presence of lattice-linear constraints such as ``Peter's regret is less than that of Paul.'' For the constrained stable marriage problem, we present a distributed algorithm that takes $O(n^2)$ messages each of size $O(\log n)$ where $n$ is the number of men in the problem. Our algorithm is completely asynchronous. Our algorithms for the stable marriage problem with ties are also parallel with no synchronization.