Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies

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...

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Vydáno v:Proceedings / annual Symposium on Foundations of Computer Science s. 276 - 285
Hlavní autoři: Bressan, Marco, Roth, Marc
Médium: Konferenční příspěvek
Jazyk:angličtina
Vydáno: IEEE 01.02.2022
Témata:
Approximation algorithms
Complexity theory
Computer science
counting problems
degenerate graphs
fine-grained complexity theory
parameterized algorithms
Pattern matching
ISSN:2575-8454
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Shrnutí: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.
ISSN:2575-8454
DOI:10.1109/FOCS52979.2021.00036