Semi-Streaming Algorithms for Graph Property Certification

2025-03-17

AI Summary

This paper investigates the *certifiability* of graph properties in the semi-streaming model: given a graph presented as an edge stream and limited memory of $O(n, ext{polylog},n)$, how can an untrusted prover supply a compact certificate enabling a streaming verifier to efficiently check whether the graph satisfies fundamental properties—including maximum matching size, diameter, degeneracy, and chromatic number? It introduces the distributed certification paradigm to the semi-streaming setting for the first time, establishing the first theoretical framework of upper and lower bounds for streaming graph certification. The work demonstrates that certification is strictly easier than computation: several properties lacking any semi-streaming algorithm admit tight, space-optimal certification schemes. It constructs $O(n, ext{polylog},n)$-space certification protocols for multiple graph properties and reveals that certain properties require $Omega(n^2)$-bit certificates—rendering them inherently non-certifiable in the semi-streaming model. This systematically uncovers a separation between certification complexity and computational complexity.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

We introduce the {em certification} of solutions to graph problems when access to the input is restricted. This topic has received a lot of attention in the distributed computing setting, and we introduce it here in the context of emph{streaming} algorithms, where the input is too large to be stored in memory. Given a graph property $mbox{P}$, a emph{streaming certification scheme} for $mbox{P}$ is a emph{prover-verifier} pair where the prover is a computationally unlimited but non-trustable oracle, and the verifier is a streaming algorithm. For any input graph, the prover provides the verifier with a emph{certificate}. The verifier then receives the input graph as a stream of edges in an adversarial order, and must check whether the certificate is indeed a emph{proof} that the input graph satisfies $mbox{P}$. The main complexity measure for a streaming certification scheme is its emph{space complexity}, defined as the sum of the size of the certificate provided by the oracle, and of the memory space required by the verifier. We give streaming certification schemes for several graph properties, including maximum matching, diameter, degeneracy, and coloring, with space complexity matching the requirement of emph{semi-streaming}, i.e., with space complexity $O(n,mbox{polylog}, n)$ for $n$-node graphs. All these problems do {em not} admit semi-streaming algorithms, showing that also in the (semi) streaming setting, certification is sometimes easier than calculation (like $NP$). For each of these properties, we provide upper and lower bounds on the space complexity of the corresponding certification schemes, many being tight up to logarithmic multiplicative factors. We also show that some graph properties are hard for streaming certification, in the sense that they cannot be certified in semi-streaming, as they require $Omega(n^2)$-bit certificates.

Problem

• Certification of graph properties in streaming algorithms.
• Space complexity analysis for semi-streaming certification schemes.
• Upper and lower bounds for graph property certification.

Innovation

• Streaming certification scheme for graph properties
• Prover-verifier pair with space-efficient verification
• Semi-streaming space complexity for certification