DeepAI AI Chat
Log In Sign Up

A Natural Generalization of Stable Matching Solved via New Insights into Ideal Cuts

by   Tung Mai, et al.

We study a natural generalization of stable matching to the maximum weight stable matching problem and we obtain a combinatorial polynomial time algorithm for it by reducing it to the problem of finding a maximum weight ideal cut in a DAG. We give the first polynomial time algorithm for the latter problem; this algorithm is also combinatorial. The combinatorial nature of our algorithms not only means that they are efficient but also that they enable us to obtain additional structural and algorithmic results: - We show that the set, M', of maximum weight stable matchings forms a sublattice L' of the lattice L of all stable matchings. - We give an efficient algorithm for finding boy-optimal and girl-optimal matchings in M'. - We generalize the notion of rotation, a central structural notion in the context of the stable matching problem, to macro-rotation. Just as rotations help traverse the lattice of all stable matchings, macro-rotations help traverse the sublattice over M'.


page 1

page 2

page 3

page 4


A Matroid Generalization of the Super-Stable Matching Problem

A super-stable matching, which was introduced by Irving, is a solution c...

Marriage and Roommate

This paper has two objectives. One is to give a linear time algorithm th...

Finding Stable Matchings that are Robust to Errors in the Input

Given an instance A of stable matching, let B be the instance that resul...

A Generalization of Birkhoff's Theorem for Distributive Lattices, with Applications to Robust Stable Matchings

Birkhoff's theorem, which has also been called the fundamental theorem ...

Stabilizing Weighted Graphs

An edge-weighted graph G=(V,E) is called stable if the value of a maximu...

A Structural and Algorithmic Study of Stable Matching Lattices of Multiple Instances

Recently MV18a identified and initiated work on the new problem of under...

Finding Robust Solutions to Stable Marriage

We study the notion of robustness in stable matching problems. We first ...