Markov chains simplify matching models using graph symmetries
Strong aggregation of the Markov chains associated with matching models based on the automorphism group of their compatibility graphs
Performance
Summary
Matching problems involve pairing items based on which ones can be matched together, often modeled by graphs. This paper studies how to group similar states in these complicated matching models using symmetries (automorphisms) in the graph, making analysis easier. The authors prove that under certain conditions, the Markov chains describing these systems can be simplified while preserving key behavior. They test their ideas on specific graph types called odd rings and extend results to various matching rules.
What this means in practice
- •For network schedulers: Improve analysis of resource matching in networks with symmetrical connection patterns by simplifying state models using graph automorphisms.
- •For inventory managers: Model and analyze complex inventory matching and pairing systems more efficiently when compatibility has symmetrical patterns.
A theory result. No direct application yet.
Authors
Moyi Yang, Jean-Michel Fourneau
Abstract
We extend the analysis of strong aggregation to general compatibility graphs, focusing on item counts rather than positions, and exploring generalized greedy matching disciplines. We prove that under a condition of automorphism-based transition consistency, the associated Markov chain is strongly aggregable for an arbitrary graph with a non-trivial automorphism group. Furthermore, we extend our analysis to non-greedy matching disciplines, distinguishing scenarios where compatible items can or cannot coexist within the same state. This result is illustrated with a simple compatibility graph with a rich automorphism structure: the odd rings. For all scenarios, we investigate the strong aggregation properties of the resulting Markov chains. This work enhances the theoretical understanding of lumpability in stochastic matching models and provides a foundation for analyzing complex graph structures.