On efficient distance approximation for graph properties

A distance-approximation algorithm for a graph property P in the adjacency-matrix model is given an approximation parameter ε ∈ (0, 1) and query access to the adjacency matrix of a graph G = (V, E). It is required to output an estimate of the distance between G and the closest graph G⁰ = (V, E⁰) that satisfies P, where the distance between graphs is the size of the symmetric difference between their edge sets, normalized by |V |². In this work we introduce property covers, as a basis for a methodology that uses distance-approximation algorithms for “simple” properties to design distance-approximation algorithms for more “complex” properties. Applying this methodology we present distance-approximation algorithms with poly(1/ε) query complexity for induced P₃-freeness, induced P₄-freeness, and Chordality. For induced C₄-freeness our algorithm has query complexity exp(poly(1/ε)). These complexities essentially match the corresponding known results for testing these properties and provide an exponential improvement on previously known results.