Finding Stable Matchings that are Robust to Errors in the Input

TL;DR

This paper develops a polynomial time algorithm to find stable matchings that are robust to input errors, introduces structural relationships between solution lattices of similar instances, and characterizes the set of robust solutions as a sublattice.

Contribution

It presents a novel polynomial time algorithm for robust stable matchings under certain error classes and explores the structural lattice properties of these solutions.

Findings

Polynomial time algorithm for robust stable matchings.

Robust matchings form a sublattice of all stable matchings.

Efficient representation and retrieval of robust matchings.

Abstract

We study the problem of finding solutions to the stable matching problem that are robust to errors in the input and we obtain a polynomial time algorithm for a special class of errors. In the process, we also initiate work on a new structural question concerning the stable matching problem, namely finding relationships between the lattices of solutions of two "nearby" instances. Our main algorithmic result is the following: We identify a polynomially large class of errors, $D$, that can be introduced in a stable matching instance. Given an instance $A$ of stable matching, let $B$ be the random variable that represents the instance that results after introducing {\em one} error from $D$, chosen via a given discrete probability distribution. The problem is to find a stable matching for $A$ that maximizes the probability of being stable for $B$ as well. Via new structural properties of the type described in the question stated above, we give a combinatorial polynomial time algorithm for this problem. We also show that the set of robust stable matchings for instance $A$, under probability distribution $p$, forms a sublattice of the lattice of stable matchings for $A$. We give an efficient algorithm for finding a succinct representation for this set; this representation has the property that any member of the set can be efficiently retrieved from it.