A distance-approximation algorithm for a graph property P in the adjacency-matrix model is given an approximation parameter ∊ ∊ (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 G′ = (V, E′) that satisfies , 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/∊) query complexity for induced P3-freeness, induced P4-freeness, and Chordality. For induced C4-freeness our algorithm has query complexity exp(poly(1/∊)). These complexities essentially match the corresponding known results for testing these properties and provide an exponential improvement on previously known results.

In this paper we give sublinear-time distributed algorithms in the CONGEST model for subgraph detection for two classes of graphs: cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time. For even-length cycles, C_{2k}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from O~(n^{5/6}) to O~(n^{3/4}) rounds. We also show two obstacles on proving lower bounds for C_{2k}-freeness: First, we use the new connection to extremal combinatorics to show that the current lower bound of Omega~(sqrt{n}) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant delta in (0,1/2) such that for any k, a Omega(n^{1/2+delta}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity. For general subgraphs, it was shown in [Orr Fischer et al., 2018] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Omega~(n^{2-Theta(1/k)}) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n^{2 - Theta(1/k)}) rounds, nearly matching the lower bound of [Orr Fischer et al., 2018].