Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies - Bressan, Marco; Roth, Marc (Proceedings / annual Symposium on Foundations of Computer Science, 2022)
We study the problems of counting the homomorphisms, the copies, and the induced copies of a k -vertex graph H in a d -degenerate n -vertex graph G . By leveraging a new family of graph-minor obstructions called F-gadgets, we establish explicit and exhaustive complexity classifications for counting copies and induced copies. For instance., we show that the copies of H in G can be counted in time f(k, d)n^{\max(1,\mathsf{imn}(H))} \log n , where f is some computable function and \mathsf{imn} (H) is the size of the largest induced matching of H ; and that whenever the class of allowed patterns has arbitrarily large induced matchings, no algorithm runs in time f(k, d)n^{o(\mathsf{imn}(H)/\log \mathsf{imn}(H))} for any function f , unless the Exponential Time Hypothesis fails. A similar result holds for counting induced copies, with the independence number \alpha(H) in place of \mathsf{imn}(H) . These results imply complexity dichotomies, into fixed-parameter tractable versus #W[1]-hard cases, which parallel the well-known dichotomies when d is not a parameter. Our results also imply the #W[1]-hardness of counting several patterns, such as k -matchings and k -trees, in d - degenerate graphs. We also give new hardness results and approximation algorithms for generalized pattern counting (i.e., counting patterns with a given property) in degenerate graphs.