Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six

We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted SUB-CNT_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for SUB-CNT_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can SUB_CNT_k be solved in linear time? We discover a chasm at k=6. Specifically, we prove that for k < 6, SUB_CNT_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ? 6, SUB-CNT_k cannot be solved even in near-linear time

Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams

We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles. For the important special case of counting triangles, we give a 4-pass, (1 +/- epsilon)-approximate, randomized algorithm using O-tilde(epsilon^(-2) m^(3/2) / T) space, where m is the number of edges and T is a promised lower bound on the number of triangles. This matches the space bound of a recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of Omega(min{m^(3/2)/

Graph Coloring via Degeneracy in Streaming and Other Space-Conscious Models

We study the problem of coloring a given graph using a small number of colors in several well-established models of computation for big data. These include the data streaming model, the general graph query model, the massively parallel computation (MPC) model, and the CONGESTED-CLIQUE and the LOCAL models of distributed computation. On the one hand, we give algorithms with sublinear complexity, for the appropriate notion of complexity in each of these models. Our algorithms color a graph $G$ using about $\kappa(G)$ colors, where $\kappa(G)$ is the degeneracy of $G$: this parameter is closely related to the arboricity $\alpha(G)$. As a function of $\kappa(G)$ alone, our results are close to best possible, since the optimal number of colors is $\kappa(G)+1$. On the other hand, we establish certain lower bounds indicating that sublinear algorithms probably cannot go much further. In particular, we prove that any randomized coloring algorithm that uses $\kappa(G)+1$ many colors, would require $\Omega(n^2)$ storage in the one pass streaming model, and $\Omega(n^2)$ many queries in the general graph query model, where $n$ is the number of vertices in the graph. These lower bounds hold even when the value of $\kappa(G)$ is known in advance; at the same time, our upper bounds do not require $\kappa(G)$ to be given in advance.Comment: 26 page