[Paper] Constant-Size Certificates for Leader Election in Chordal Graphs and Related Classes

Source: arXiv - 2511.19208v1

Overview

This paper tackles a classic challenge in distributed systems: how can each node in a network locally verify that a global solution—such as a leader election or a spanning‑tree construction—is correct, using only a tiny amount of extra information (certificates)? The authors present constant‑size, per‑edge certificates that work for important graph families like chordal and $K_4$‑free dismantlable graphs, and they show how to turn any such certification into a silent self‑stabilizing algorithm.

Key Contributions

• Constant‑size local certificates for leader election on chordal graphs and on $K_4$‑free dismantlable graphs.
• Constant‑size certificates for spanning‑tree construction on dismantlable graphs (assuming the root is known).
• For chordal graphs, the leader‑election certificates also guarantee an acyclic orientation, a property that cannot be certified with constant size in general graphs.
• A generic transformation that converts any certification scheme into a silent self‑stabilizing algorithm by adding just one extra state, under a Gouda‑fair scheduler.
• The first certification results that exploit the structural properties of chordal and dismantlable graph classes, opening the door to similar schemes for other problems.

Methodology

1. Graph‑class exploitation – The authors study chordal and dismantlable graphs, which have well‑understood decomposition properties (e.g., perfect elimination orderings for chordal graphs). These structural insights let them design compact certificates that can be checked locally.
2. Certificate design – Each edge carries a constant‑size label (the “certificate”). A node’s verification rule looks only at its immediate neighbors and the labels on incident edges, ensuring locality (one‑hop view).
3. Verification predicates – For leader election, the predicates ensure that exactly one node declares itself leader, that every node can trace a path to the leader, and that the orientation induced by the certificates is acyclic. For spanning trees, the predicates verify parent‑child relationships and that the tree spans all nodes.
4. Self‑stabilizing conversion – Building on the certification, they add a single “reset” state. When a node detects a violation of the predicates, it switches to this state, which propagates until the system converges back to a legitimate certified configuration, all without further communication (hence “silent”).

Results & Findings

• Certificate size: The authors prove that for the targeted graph classes, a constant number of bits per edge suffices—independent of the graph’s size.
• Correctness: Formal proofs show that if all local checks pass, the global configuration indeed corresponds to a valid leader election (or spanning tree) and, in the chordal case, an acyclic orientation.
• Self‑stabilization: The transformation adds only one extra state, yet guarantees that any transient fault (e.g., corrupted certificates) will be automatically healed, and the system will converge to a silent, correct configuration.
• Impossibility contrast: The paper highlights that the same constant‑size certification is impossible for arbitrary graphs, underscoring the importance of the structural restrictions.

Practical Implications

• Lightweight verification in sensor/IoT networks – Devices often have severe memory constraints; constant‑size certificates mean they can still locally verify that a leader (e.g., a coordinator) has been correctly elected without storing large tables.
• Fault‑tolerant protocols – The silent self‑stabilizing wrapper can be layered on top of existing distributed algorithms, giving them automatic recovery from transient errors with negligible overhead.
• Network topology‑aware design – In many real‑world deployments (e.g., hierarchical data‑center topologies, road‑network overlays) the underlying graph is chordal or dismantlable. Engineers can exploit these results to build certifiable protocols tailored to those topologies.
• Reduced communication – Since verification only needs one‑hop information, there’s no need for costly multi‑hop consensus rounds just to confirm correctness, saving bandwidth and energy.

Limitations & Future Work

• Graph‑class restriction – The schemes rely on chordal or dismantlable structure; they do not extend to arbitrary networks, limiting applicability to topologies that can be forced or recognized as such.
• Root assumption for spanning trees – The spanning‑tree certification presumes a known root; handling root election jointly with tree construction remains open.
• Scheduler assumption – The self‑stabilizing transformation assumes a Gouda‑fair scheduler, which may not hold in all asynchronous environments.
• Extending to other problems – The authors suggest that the same structural insights could be used for certifying additional distributed tasks (e.g., coloring, matching), a promising direction for follow‑up research.

Bottom line: By marrying graph‑theoretic structure with distributed certification, this work delivers ultra‑compact, locally checkable proofs for leader election and spanning trees—plus a neat recipe for making any such proof self‑healing. For developers building fault‑resilient, resource‑constrained distributed systems, especially on chordal‑like topologies, these results offer a practical new toolbox.

Authors

• Jérémie Chalopin
• Maria Kokkou

Paper Information

• arXiv ID: 2511.19208v1
• Categories: cs.DC
• Published: November 24, 2025
• PDF: Download PDF