Beating Treewidth for Average-Case Subgraph Isomorphism

Author Gregory Rosenthal

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Gregory Rosenthal
  • University of Toronto, Canada


Thanks to Benjamin Rossman for introducing me to this topic, and for having many helpful discussions about the research and about drafts of this paper. Thanks to Henry Yuen for providing feedback on a draft of this paper as well. Part of this work was done while the author was visiting the Simons Institute for the Theory of Computing.

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Gregory Rosenthal. Beating Treewidth for Average-Case Subgraph Isomorphism. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


For any fixed graph G, the subgraph isomorphism problem asks whether an n-vertex input graph has a subgraph isomorphic to G. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of the problem, denoted G-SUB, and then solves G-SUB in time O(n^{tw(G)+1}) where tw(G) is the treewidth of G. Marx (2010) conjectured that G-SUB requires time Omega(n^{const * tw(G)}) and, assuming the Exponential Time Hypothesis, proved a lower bound of Omega(n^{const * emb(G)}) for a certain graph parameter emb(G) = Omega(tw(G)/log tw(G)). With respect to the size of AC^0 circuits solving G-SUB, Li, Razborov and Rossman (2017) proved an unconditional average-case lower bound of Omega(n^{kappa(G)}) for a different graph parameter kappa(G) = Omega(tw(G)/log tw(G)). Our contributions are as follows. First, we show that emb(G) is at most O(kappa(G)) for all graphs G. Next, we show that kappa(G) can be asymptotically less than tw(G); for example, if G is a hypercube then kappa(G) is Theta(tw(G)/sqrt{log tw(G)}). Finally, we construct AC^0 circuits of size O(n^{kappa(G)+const}) that solve G-SUB in the average case, on a variety of product distributions. This improves an O(n^{2 kappa(G)+const}) upper bound of Li et al., and shows that the average-case complexity of G-SUB is n^{o(tw(G))} for certain families of graphs G such as hypercubes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
  • subgraph isomorphism
  • average-case complexity
  • AC^0
  • circuit complexity


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