An Approximation Algorithm for Maximum Stable Matching with Ties and Constraints

TL;DR

This paper introduces a polynomial-time approximation algorithm for maximum stable matching with ties and laminar constraints, achieving a 1.5 approximation ratio and strategy-proofness in certain cases.

Contribution

It presents a novel 3/2-approximation algorithm for complex many-to-many stable matchings with constraints, using matroid-kernel and auxiliary instances.

Findings

Achieves a 1.5 approximation ratio for the problem.

Provides a strategy-proof mechanism in specific cases.

Uses matroid theory to analyze and guarantee approximation bounds.

Abstract

We present a polynomial-time $\frac{3}{2}$-approximation algorithm for the problem of finding a maximum-cardinality stable matching in a many-to-many matching model with ties and laminar constraints on both sides. We formulate our problem using a bipartite multigraph whose vertices are called workers and firms, and edges are called contracts. Our algorithm is described as the computation of a stable matching in an auxiliary instance, in which each contract is replaced with three of its copies and all agents have strict preferences on the copied contracts. The construction of this auxiliary instance is symmetric for the two sides, which facilitates a simple symmetric analysis. We use the notion of matroid-kernel for computation in the auxiliary instance and exploit the base-orderability of laminar matroids to show the approximation ratio. In a special case in which each worker is assigned at most one contract and each firm has a strict preference, our algorithm defines a $\frac{3}{2}$-approximation mechanism that is strategy-proof for workers.