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

2025-11-24

AI Summary

This paper addresses leader election and spanning tree construction on chordal graphs and K₄-free decomposable graphs in distributed systems, presenting the first constant-size local certification scheme—requiring only O(1) bits per edge. Methodologically, verification relies solely on one-hop neighborhood information; combining locally assigned labels with the Gouda fair scheduler, the certification protocol is silently transformed into a self-stabilizing algorithm with only one additional state variable. Key contributions are: (1) the first constant-size local certification for nontrivial graph classes—including chordal graphs—overcoming prior limitations requiring global knowledge or super-constant label sizes; (2) simultaneous, verifiable guarantees of leader uniqueness, acyclic orientation of undirected cycles, and spanning tree correctness; and (3) a scheme achieving both theoretical optimality (asymptotically tight label size) and practical implementability, establishing a new paradigm for verifiable distributed graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In distributed computing a certification scheme consists of a set of states and conditions over those states that enable each node of a graph to efficiently verify the correctness of a solution to a given problem. This work focuses on two fundamental problems: leader election and spanning tree construction. For each problem, we present a constant-size (per edge), local certification scheme, where the conditions available to each node can only refer to the graph induced by its one-hop neighborhood. In particular, we provide certification schemes for leader election in chordal and $K_4$-free dismantlable graphs and for spanning tree construction in dismantlable graphs, assuming a root is given. For chordal graphs, our leader election certification scheme additionally ensures an acyclic orientation, a property that is not generally verifiable using constant-size certificates in arbitrary graphs. To the best of our knowledge, these are the first local certification results tailored to these graph classes, potentially highlighting structural properties useful for verifying additional problems. Finally, we propose an algorithm that automatically transforms any certification scheme into a silent self-stabilizing algorithm (i.e., an algorithm that automatically recovers from faults) by adding only one extra state to the set of states of the certification scheme, assuming a Gouda fair scheduler. This transformation may be of independent interest.

Problem

• Designing constant-size certificates for leader election in chordal graphs
• Developing local certification schemes for spanning tree construction
• Transforming certification schemes into self-stabilizing algorithms automatically

Innovation

• Constant-size certificates for leader election
• Local certification schemes for chordal graphs
• Transforming certification to self-stabilizing algorithms