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

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.