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