Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems

TL;DR

This paper investigates the parameterized complexity of incremental stable matching problems, revealing hardness results and identifying tractable cases when preferences change over time.

Contribution

It answers open questions about the complexity of incremental stable matchings, providing new hardness results and identifying conditions for polynomial-time solvability.

Findings

Incremental Stable Roommates is W[1]-hard parameterized by preference changes.

Incremental Stable Marriage with Ties is W[1]-hard parameterized by the number of ties.

Certain cases based on preference list similarity are polynomial-time solvable or fixed-parameter tractable.

Abstract

When computing stable matchings, it is usually assumed that the preferences of the agents in the matching market are fixed. However, in many realistic scenarios, preferences change over time. Consequently, an initially stable matching may become unstable. Then, a natural goal is to find a matching which is stable with respect to the modified preferences and as close as possible to the initial one. For Stable Marriage/Roommates, this problem was formally defined as Incremental Stable Marriage/Roommates by Bredereck et al. [AAAI '20]. As they showed that Incremental Stable Roommates and Incremental Stable Marriage with Ties are NP-hard, we focus on the parameterized complexity of these problems. We answer two open questions of Bredereck et al. [AAAI '20]: We show that Incremental Stable Roommates is W[1]-hard parameterized by the number of changes in the preferences, yet admits an intricate XP-algorithm, and we show that Incremental Stable Marriage with Ties is W[1]-hard parameterized by the number of ties. Furthermore, we analyze the influence of the degree of "similarity" between the agents' preference lists, identifying several polynomial-time solvable and fixed-parameter tractable cases, but also proving that Incremental Stable Roommates and Incremental Stable Marriage with Ties parameterized by the number of different preference lists are W[1]-hard.