Recognizing Hereditary Properties in the Presence of Byzantine Nodes

TL;DR

This work extends the Byzantine congested clique model to the recognition of hereditary graph classes, proposing a randomized algorithm that, with high probability, accepts inputs in a hereditary class $\\mathcal{G}$ and rejects those that are $|B|$-far from $\\mathcal{G}$. The core methodology combines committee-based communication with reconstruction of agreement and disagreement graphs; a Measuring Gap procedure then determines membership by testing feasible edge-relabelings consistent with at most $|B|$ Byzantine nodes. The main result shows a general upper bound of $O\left(\left(\frac{\log|\\mathcal{G}_n|}{n} + |B|\right)\cdot \mathrm{polylog}(n)\right)$ rounds for any hereditary class, with many classes achieving $O(|B|\,\mathrm{polylog}(n))$ rounds due to subquadratic growth. The paper also proves a tight impossibility for forests, establishing that when the gap $f$ is smaller than $|B|$, no randomized algorithm can distinguish forests from graphs with $|B|$ disjoint cycles. Overall, the results demonstrate the resilience of hereditary graph properties in adversarial distributed settings and lay groundwork for broader algorithmic exploration under Byzantine faults.

Abstract

Augustine et al. [DISC 2022] initiated the study of distributed graph algorithms in the presence of Byzantine nodes in the congested clique model. In this model, there is a set $B$ of Byzantine nodes, where $|B|$ is less than a third of the total number of nodes. These nodes have complete knowledge of the network and the state of other nodes, and they conspire to alter the output of the system. The authors addressed the connectivity problem, showing that it is solvable under the promise that either the subgraph induced by the honest nodes is connected, or the graph has $2|B|+1$ connected components. In the current work, we continue the study of the Byzantine congested clique model by considering the recognition of other graph properties, specifically hereditary properties. A graph property is hereditary if it is closed under taking induced subgraphs. Examples of hereditary properties include acyclicity, bipartiteness, planarity, and bounded (chromatic, independence) number, etc. For each class of graphs ${\bf G}$ satisfying a hereditary property (a hereditary graph-class), we propose a randomized algorithm which, with high probability, (1) accepts if the input graph $G$ belongs to ${\bf G}$, and (2) rejects if $G$ contains at least $|B| + 1$ disjoint subgraphs not belonging to ${\bf G}$. The round complexity of our algorithm is $$O\left(\left(\dfrac{\log \left(\left|{\bf G}_n\right|\right)}{n} +|B|\right)\cdot\textrm{polylog}(n)\right),$$ where ${\bf G}_n$ is the set of $n$-node graphs in ${\bf G}$. Finally, we obtain an impossibility result that proves that our result is tight. Indeed, we consider the hereditary class of acyclic graphs, and we prove that there is no algorithm that can distinguish between a graph being acyclic and a graph having $|B|$ disjoint cycles.

Figures (2)

• Figure 1: Example of the graph $G$ and the possible graph $G^F$ obtained that belongs to the class of forests.
• Figure 2: Examples of the input graphs $G_{\text{Yes}}$ and $G_{\text{No}}$, and representation of the indistinguishable instances obtained from them

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Lemma 7
• Lemma 8
• Lemma 9
• Theorem 9