On the testability and repair of hereditary hypergraph properties

TL;DR

This paper refines and extends the understanding of testability and local repairability of hereditary hypergraph properties, showing positive results for certain classes and limitations for others, with implications for graph theory and property testing.

Contribution

It strengthens existing results to cover directed and multi-colored hypergraphs, and demonstrates local repairability in continuous graphs, while identifying cases where it fails.

Findings

Hereditary properties of undirected graphs are testable with one-sided error.

Local repair algorithms are efficient, depending on limited data, for certain hypergraph classes.

Local repairability fails for directed graphs and 3-uniform hypergraphs.

Abstract

Recent works of Alon-Shapira and R\"odl-Schacht have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally". In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs. In the case of undirected graphs, we extend the result to continuous graphs on probability spaces, and show that the repair algorithm is "local" in the sense that it only depends on a bounded amount of data; in particular, the graph can be repaired in a time linear in the number of edges. We also show that local repairability also holds for monotone or partite hypergraph properties (this latter result is also implicitly in work of Ishigami). In the negative direction, we show that local repairability breaks down for directed graphs, or for undirected 3-uniform hypergraphs. The reason for this contrast in behavior stems from (the limitations of) Ramsey theory.