Complexity landscape for local certification

TL;DR

This work initiates a systematic study of the space complexity (certificate size) landscape for local certification in distributed graphs, focusing on how graph topology and information assumptions shape which certificate sizes are possible. It develops an automata- and arithmetic-based toolkit to classify gap and no-gap regimes across paths, cycles, and trees, proving a surprising $O(1)$ vs $oldsymbol{ ext{Theta}( ext{log log } n)}$ gap on anonymous paths and a separate $O(1)$ vs $oldsymbol{ ext{Theta}( ext{log } n)}$ gap on cycles, with a tree-generalization to diameter $d$ under radius-1 verification. The paper also identifies settings where gaps disappear (e.g., general graphs with radius $>1$, caterpillars), and analyzes how identifiers and knowledge of $n$ influence the landscape, showing that some gaps can be broken under precise information while others persist. The contributions combine novel automata-theoretic constructions, prime-product and Bézout-based arguments, and Chrobak normal form techniques to map the sublinear certificate-size regime, offering foundational insights for designing efficient proof-labeling schemes and understanding the limits of sublinear-space local certification.

Abstract

An impressive recent line of work has charted the complexity landscape of distributed graph algorithms. For many settings, it has been determined which time complexities exist, and which do not (in the sense that no local problem could have an optimal algorithm with that complexity). In this paper, we initiate the study of the landscape for space complexity of distributed graph algorithms. More precisely, we focus on the local certification setting, where a prover assigns certificates to nodes to certify a property, and where the space complexity is measured by the size of the certificates. Already for anonymous paths and cycles, we unveil a surprising landscape: - There is a gap between complexity $O(1)$ and $Θ(\log \log n)$ in paths. This is the first gap established in local certification. - There exists a property that has complexity $Θ(\log \log n)$ in paths, a regime that was not known to exist for a natural property. - There is a gap between complexity $O(1)$ and $Θ(\log n)$ in cycles, hence a gap that is exponentially larger than for paths. We then generalize our result for paths to the class of trees. Namely, we show that there is a gap between complexity $O(1)$ and $Θ(\log \log d)$ in trees, where $d$ is the diameter. We finally describe some settings where there are no gaps at all. To prove our results we develop a new toolkit, based on various results of automata theory and arithmetic, which is of independent interest.

Figures (5)

• Figure 1: The automaton corresponding to the certificates used to certify that the length of a path is divisible by $3$. The states corresponding to the tuples $(0,0)$, $(1,1)$ and $(2,2)$ are not represented because the have no incoming nor outgoing transitions. The final state is the state $f$.
• Figure 2: In this directed graph, $ADEDEFA$ is a closed walk of length $6$, which is not an elementary cycle. The closed walks $FAGF$ and $CDEFABC$ are elementary cycles and have length $3$ and $6$ respectively. Moreover, $FAGF$ is a subwalk of $DEFAGFAD$.
• Figure 4: A tree $T$ and a parsing of $T$. The bold vertices are the roots of the trees.
• Figure 5: An finite automata $\mathcal{A}$ with transitions labeled by rooted trees, and a tree $T$ such that $\sigma(T,u,v)$ is accepted by $\mathcal{A}$. Here the state with the incoming arrow is the only initial state, and the black state is the only final state.
• Figure 7: An automaton in Chrobak normal form. Black states are final states.

Theorems & Definitions (47)

• Theorem 1
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10
• Theorem 15
• Lemma 16
• Lemma 17