Stable Matching with Deviators and Conformists
Frederik Glitzner, David Manlove
TL;DR
The paper studies stability in classic matching problems under a designated set of deviator agents that may block, distinguishing them from conformists. It proves strong intractability results, showing NP-hardness even for zero-blocking Deviator problems in both bipartite and general settings, and complements these with fixed-parameter tractable (FPT) and polynomial-time results for restricted inputs, notably small deviator sets and short preference lists. The authors introduce and analyze k-Deviator variants, provide configuration-enumeration-based FPT algorithms leveraging maximum-weight matchings, and derive efficient algorithms for instances with very short lists ($d_{ ext{max}} o2$). They also extend the framework to minimise the number of blocking deviators or blocking agents, resolving several open questions and mapping a nuanced tractability frontier. Overall, the work delineates when deviator-stable matchings are computationally feasible and when they are not, offering algorithmic tools for practical multi-agent systems under stability constraints.
Abstract
In the fundamental Stable Marriage and Stable Roommates problems, there are inherent trade-offs between the size and stability of solutions. While in the former problem, a stable matching always exists and can be found efficiently using the celebrated Gale-Shapley algorithm, the existence of a stable matching is not guaranteed in the latter problem, but can be determined efficiently using Irving's algorithm. However, the computation of matchings that minimise the instability, either due to the presence of additional constraints on the size of the matching or due to restrictive preference cycles, gives rise to a collection of infamously intractable almost-stable matching problems. In practice, however, not every agent is able or likely to initiate deviations caused by blocking pairs. Suppose we knew, for example, due to a set of requirements or estimates based on historical data, which agents are likely to initiate deviations - the deviators - and which are likely to comply with whatever matching they are presented with - the conformists. Can we decide efficiently whether a matching exists in which no deviator is blocking, i.e., in which no deviator has an incentive to initiate a deviation? Furthermore, can we find matchings in which only a few deviators are blocking? We characterise the computational complexity of this question in bipartite and non-bipartite preference settings. Surprisingly, these problems prove computationally intractable in strong ways: for example, unlike in the classical setting, where every agent is considered a deviator, in this extension, we prove that it is NP-complete to decide whether a matching exists where no deviator is blocking. On the positive side, we identify polynomial-time and fixed-parameter tractable cases, providing novel algorithmics for multi-agent systems where stability cannot be fully guaranteed.