A Simple 1.5-Approximation Algorithm for a Wide Range of Max-SMTI Problems

TL;DR

The paper tackles Max-SMTI and a wide range of generalizations (ties, Delta-stability, critical vertices/edges, free edges, and matroid constraints) by introducing an edge-duplication technique that reduces generalized instances to standard stable matching problems. The authors present a simple $3/2$-approximation algorithm that runs via Gale–Shapley on an extended instance and projects back to the original problem, and extend the approach to matroid constraints using Fleiner’s algorithm. They prove that the method yields a $3/2$-approximation for Max-$c\gamma$-smti and its matroid extensions, with linear or near-linear runtimes depending on the structure (e.g., laminar matroids). This unifies multiple complex models under a single, elegant framework and resolves an open question on Max-SMTI with free edges, suggesting broad potential for future applications in matching markets and beyond.

Abstract

We give a simple approximation algorithm for a common generalization of many previously studied extensions of the maximum size stable matching problem with ties. These generalizations include the existence of critical vertices in the graph, amongst whom we must match as much as possible, free edges, that cannot be blocking edges and $Δ$-stabilities, which mean that for an edge to block, the improvement should be large enough on one or both sides. We also introduce other notions to generalize these even further, which allows our framework to capture many existing and future applications. We show that the edge duplicating technique allows us to treat these different types of generalizations simultaneously, while also making the algorithm, the proofs and the analysis much simpler and shorter than in previous approaches. In particular, we answer an open question by Askalidis et al. (2013) about the existence of a $\frac{3}{2}$-approximation algorithm for the MAX-SMTI problem with free edges. This demonstrates that this technique can grasp the underlying essence of these problems quite well and have the potential to be able to solve many future applications.

Theorems & Definitions (11)

• theorem 1
• proof
• theorem 2
• proof
• remark 1
• theorem 3
• lemma 1
• proof
• theorem 4
• proof