Testing perfection is hard

A graph property P is strongly testable if for every fixed \epsilon>0 there is a one-sided \epsilon-tester for P whose query complexity is bounded by a function of \epsilon. In classifying the strongly testable graph properties, the first author and Shapira showed that any hereditary graph property (such as P the family of perfect graphs) is strongly testable. A property is easily testable if it is strongly testable with query complexity bounded by a polynomial function of \epsilon^{-1}, and otherwise it is hard. One of our main results shows that testing perfectness is hard. The proof shows that testing perfectness is at least as hard as testing triangle-freeness, which is hard. On the other hand, we show that induced P_3-freeness is easily testable. This settles one of the two exceptional graphs, the other being C_4 (and its complement), left open in the characterization by the first author and Shapira of graphs H for which induced H-freeness is easily testable.

💡 Research Summary

The paper investigates the query complexity of testing two fundamental graph properties: perfectness and induced‑P₃‑freeness. A property P is called strongly testable if for every fixed ε > 0 there exists a one‑sided ε‑tester whose query complexity depends only on ε, not on the size of the input graph. If the query complexity can be bounded by a polynomial in ε⁻¹, the property is said to be easily testable; otherwise it is classified as hard.

The authors first recall that every hereditary graph property (i.e., closed under taking induced subgraphs) is strongly testable, a result due to Alon and Shapira. However, the distinction between easy and hard within this class remained largely open. In particular, the status of perfect graphs and comparability graphs was unknown, as was the status of induced‑H‑freeness for the two exceptional graphs H = P₃ (the three‑vertex path) and H = C₄ (the four‑cycle) together with their complements.

Hardness of testing perfect graphs.
The paper shows that testing whether a graph is perfect is at least as hard as testing triangle‑freeness, a property known to be hard. The argument proceeds via a reduction: starting from a graph G that is ε‑far from triangle‑free, the authors construct a larger graph G′ (of size 5|V(G)|) that is ε/25‑far from being induced‑C₅‑free. They then prove that a random sample of vertices from G′ is, with probability at least 1/2, a comparability graph. Since every comparability graph is perfect and every perfect graph is induced‑C₅‑free, any tester for perfectness would also distinguish triangle‑free graphs from those far from triangle‑free. Consequently, any one‑sided tester for perfectness must incur the same super‑polynomial query lower bound known for triangle‑freeness (the bound is essentially ε⁻ᶜ·log ε⁻¹). The same reduction also yields hardness for testing comparability graphs.

Easily testable induced‑P₃‑freeness.
The second main contribution is an explicit polynomial‑time tester for induced‑P₃‑freeness (also known as cographs). The authors introduce the notion of a β‑cut: a bipartition (V₁,V₂) of the vertex set such that either the edge density between the parts is at most β or at least 1 − β. They prove a structural lemma (Theorem 2.1) stating that any n‑vertex graph without a β‑cut must contain at least (β/100)¹²·n⁴ induced copies of P₃. Using this lemma, they design a tester that samples O(β⁻¹²) vertices uniformly at random. If the sampled subgraph contains a P₃, the algorithm rejects; otherwise it accepts. By setting β proportional to ε, they guarantee that a graph ε‑far from being induced‑P₃‑free will be rejected with probability at least 2/3, while any induced‑P₃‑free graph is always accepted. The query complexity of this tester is O(ε⁻¹²), a polynomial in ε⁻¹, establishing that induced‑P₃‑freeness is easily testable.

Implications and context.
These results resolve one of the two remaining open cases in the classification of induced‑H‑freeness for hereditary properties: P₃ is now known to be easy, while C₄ (and its complement) remain hard. Moreover, the hardness result for perfect graphs shows that even though perfectness is a hereditary property and therefore strongly testable, it does not admit a polynomial‑query tester. This demonstrates a clear separation between strong testability and easy testability within hereditary properties. The β‑cut technique may be of independent interest for analyzing other induced‑subgraph‑free properties.

In summary, the paper establishes that (i) testing perfect graphs (and comparability graphs) is hard, matching the lower bounds for triangle‑freeness, and (ii) testing induced‑P₃‑freeness is easy, with a concrete O(ε⁻¹²) query algorithm. These contributions sharpen our understanding of the landscape of graph property testing, delineating precisely where efficient testers are possible and where inherent combinatorial complexity prevents them.